One stick has the juice boxes at the ends of the stick (high moment of inertia) and one stick has them taped to the middle of the stick (low moment of inertia).
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From that, you could calculate the change in the moment of inertia.
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The moment of inertia tells you how difficult it is to change its rotational motion.
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It's the product of the angular velocity (how fast it spins—represented with the symbol ω) and the moment of inertia (using the symbol I). I think most people are OK with the idea of the angular velocity—but the moment of inertia thing is a bit more complicated.
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Oh, to make things more fun I gave the higher moment of inertia stick to the stronger girl.
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Angular velocity and moment of inertia are the two concepts at work here, and they have an inverse relationship.
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Basically, the moment of inertia is a property of an object that depends on the distribution of the mass about the rotation axis.
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It is a product of the moment of inertia and the angular velocity—which is a measure of how fast something is rotating.
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You will have to estimate the mass and the moment of inertia for the Navoo (or you could assume it's a hollow cylinder).
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"It's all about the change in moment of inertia," says John Di Bartolo, an applied physicist at New York University Tandon School of Engineering.
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Moment of inertia is a term physicists use to describe a body's tendency to resist rotation; the smaller it is, the easier an object spins.
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If you have more mass further away from the axis of rotation, the moment of inertia is larger than if that was was close to the axis.
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To increase the speed of a flip as they fall toward the water, divers crunch their bodies into compact balls to reduce the "moment of inertia," or the tendency to resist accelerated, angular momentum.
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Guthrie reduces his moment of inertia (and increases his spin rate) by sweeping his arms and legs far from his body throughout the touchdown raiz, then drawing them sharply inward as he leaves the ground and initiates the quad cork.
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Scientists from the University of València in Spain have figured out to use neutrino particle data—the smallest, most abundant particle in the universe—from Antarctica's "IceCube" observatory at the South Pole in to order to calculate the mass of the Earth, the mass of the earth's core, prove that the core is denser than the mantle, and determine earth's moment of inertia (oppositional force during rotation).
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J is denoted as 2nd moment of inertia/polar moment of inertia.
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In planetary sciences, the moment of inertia factor or normalized polar moment of inertia is a dimensionless quantity that characterizes the radial distribution of mass inside a planet or satellite. Since a moment of inertia must have dimensions of mass times length squared, the moment of inertia factor is the coefficient that multiplies these.
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The moment of inertia of an object, symbolized by I, is a measure of the object's resistance to changes to its rotation. The moment of inertia is measured in kilogram metre² (kg m2). It depends on the object's mass: increasing the mass of an object increases the moment of inertia. It also depends on the distribution of the mass: distributing the mass further from the center of rotation increases the moment of inertia by a greater degree.
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The SI unit for polar moment of inertia, like the area moment of inertia, is meters to the fourth power (m4), and inches to the fourth power (in4) in U.S. Customary units and imperial units.
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In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.
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The moment of inertia factor provides an important constraint for models representing the interior structure of a planet or satellite. At a minimum, acceptable models of the density profile must match the volumetric mass density and moment of inertia factor of the body.
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The greater the magnitude of the polar moment of inertia, the greater the torsional resistance of the object.
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There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor. The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw.
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In a paper published in 2007 it was claimed that the axial dimensionless moment of inertia coefficient was 0.4. Such a value indicated that Rhea had an almost homogeneous interior (with some compression of ice in the center) while the existence of a rocky core would imply a moment of inertia of about 0.34. In the same year another paper claimed the moment of inertia was about 0.37. Rhea being either partially or fully differentiated would be consistent with the observations of the Cassini probe.
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These wheels are sometimes referred to as "Z-balances". A temperature increase makes the arms bend inward toward the center of the wheel, and the shift of mass inward reduces the moment of inertia of the balance, similar to the way a spinning ice skater can reduce her moment of inertia by pulling in her arms. This reduction in the moment of inertia compensated for the reduced torque produced by the weaker balance spring. The amount of compensation is adjusted by moveable weights on the arms.
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True polar wander may have been observed in other planetary bodies. Data suggests that Mars's polar wander resembles Earth's true polar wander; that is, when Mars had an active lithosphere its structure allowed slow polar drift to stabilize the moment of inertia. Unlike the Earth and Mars, Venus’s structure does not seem to allow the same slow polar wander; when observed the maximum moment of inertia of Venus is largely offset from the geographic pole. Therefore, the deviation of the maximum moment of inertia will remain for longer periods of time.
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As the angular velocity increases (up to five revolutions per second), the moment of inertia is decreased as the arms and free leg move towards the center of the spin. At this point, the center of gravity reaches its maximum as the skater stretches vertically, the moment of inertia is at its minimum, and the angular velocity is at its maximum. The skater ends the spin by opening his or her arms, which increases the moment of inertia, and he or she exits the spin on a curve.
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Another parameter which is less commonly a factor is the trailer moment of inertia. Even if the center of mass is forward of the wheels, a trailer with a long load, and thus large moment of inertia, may be unstable. Some vehicles are equipped with a Trailer Stability Program that may be able to compensate for improper loading.
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The extension of the booms increased the moment of inertia of the spacecraft, permitting the acceleration level to remain below 10−3g.
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The parallel axes rule also applies to the second moment of area (area moment of inertia) for a plane region D: :I_z = I_x + Ar^2, where is the area moment of inertia of D relative to the parallel axis, is the area moment of inertia of D relative to its centroid, is the area of the plane region D, and is the distance from the new axis to the centroid of the plane region D. The centroid of D coincides with the centre of gravity of a physical plate with the same shape that has uniform density.
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Tightrope walkers use the moment of inertia of a long rod for balance as they walk the rope. Samuel Dixon crossing the Niagara River in 1890. The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
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It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis. For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters.
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When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared (kg·m2) in SI units and pound-foot-second squared (lbf·ft·s2) in imperial or US units. Moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion.
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By his study of the oscillation period of compound pendulums Huygens made pivotal contributions to the development of the concept of moment of inertia.
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Rotating solids are affected considerably by the mass distribution, either if they are homogeneous or inhomogeneous - see Torque, moment of inertia, wobble, imbalance and stability.
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For a single particle of mass m a distance r from the axis of rotation, the moment of inertia is given by :I = mr^2.
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Besides Earth, the Moon is the only planetary body with a seismic observation network in place. Analysis of lunar seismic data have helped constrain the thickness of the crust (~45 km) and mantle, as well as the core radius (~330 km). # Moment of inertia parameters. True (physical) libration of the Moon measured via Lunar laser ranging constrains the normalized polar moment of inertia to 0.394 ± 0.002.
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The parallel axis theorem, also known as Huygens-Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes.
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However, a particular benefit of the tubular-clincher design is that the risk of pinch flats is very low (like the tubular tire), yet it allows the use of the more popular clincher wheel. Wheel moment of inertia is a controversial subject. In this article: wheel theory, the author does some calculations on wheel effects. Moment of inertia changes result in a decrease in watts of between .
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The space coordinates are treated as unmoving, while the body coordinates are considered embedded in the moving body. Calculations involving acceleration, angular acceleration, angular velocity, angular momentum, and kinetic energy are often easiest in body coordinates, because then the moment of inertia tensor does not change in time. If one also diagonalizes the rigid body's moment of inertia tensor (with nine components, six of which are independent), then one has a set of coordinates (called the principal axes) in which the moment of inertia tensor has only three components. The angular velocity of a rigid body takes a simple form using Euler angles in the moving frame.
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It is a measure of rotational inertia. Moment of inertia (shown here), and therefore angular momentum, is different for every possible configuration of mass and axis of rotation. Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation. Therefore, the total moment of inertia, and the angular momentum, is a complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits.
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A schematic showing how the polar moment of inertia is calculated for an arbitrary shape about an axis O. Where \rho is the radial distance to the element dA. : Note: While it has become common to find the term moments of inertia used to describe the polar and planar second moments of area, this is primarily a construct of engineering fields. The term moment of inertia, within physics and mathematics fields, is strictly the mass moment of inertia, or second moment of mass, used to describe a massive object's resistance to rotational motion, not its resistance to torsional deformation. While the polar and planar second moments of inertia are integrated over all the infinitesimal elements of a given area in some two-dimensional cross-section, the mass moment of inertia is integrated over all the infinitesimal elements of mass in a three-dimensional space occupied by an object.
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The most important of details include: mass, center of mass, moment of inertia, thruster positions, thrust vectors, thrust curves, specific impulse, thrust centroid offsets, and fuel consumption.
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For bodies in hydrostatic equilibrium, the Darwin–Radau relation can provide estimates of the moment of inertia factor on the basis of shape, spin, and gravity quantities.
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The polar moment of inertia is insufficient for use to analyze beams and shafts with non-circular cross-sections, due their tendency to warp when twisted, causing out-of-plane deformations. In such cases, a torsion constant should be substituted, where an appropriate deformation constant is included to compensate for the warping effect. Within this, there are articles that differentiate between the polar moment of inertia, I_z, and the torsional constant, J_t, no longer using J to describe the polar moment of inertia. In objects with significant cross-sectional variation (along the axis of the applied torque), which cannot be analyzed in segments, a more complex approach may have to be used.
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However, according to quantum mechanics, the energy difference between the allowed (quantized) rotation states is inversely proportional to the moment of inertia about the corresponding axis of rotation. Because the moment of inertia of a single atom is exceedingly small, the activation temperature for its rotational modes is extremely high. The same applies to the moment of inertia of a diatomic molecule (or a linear polyatomic one) about the internuclear axis, which is why that mode of rotation is not active in general. On the other hand, electrons and nuclei can exist in excited states and, in a few exceptional cases, they may be active even at room temperature, or even at cryogenic temperatures.
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The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by mr^2, where r is the distance of the point from the axis, and m is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.
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Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and moment of inertia and polar moment of inertia for stiffness. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below.
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Moment of inertia, denoted by , measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. Mass moments of inertia have units of dimension ML2([mass] × [length]2). It should not be confused with the second moment of area, which is used in beam calculations. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass.
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The internal structure of an icy astronomical body is generally deduced from measurements of its bulk density, gravity moments, and shape. Determining the moment of inertia of a body can help assess whether it has undergone differentiation (separation into rock-ice layers) or not. Shape or gravity measurements can in some cases be used to infer the moment of inertia – if the body is in hydrostatic equilibrium (i.e. behaving like a fluid on long timescales).
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In this case the water+ice layer can be as thick as 250–300 km. Failing an ocean, the icy lithosphere may be somewhat thicker, up to about 300 km. Beneath the lithosphere and putative ocean, Callisto's interior appears to be neither entirely uniform nor particularly variable. Galileo orbiter data (especially the dimensionless moment of inertiaThe dimensionless moment of inertia referred to is I / (mr^2), where is the moment of inertia, the mass, and the maximal radius.
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Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum p is proportional to mass m and linear speed :p = mv, angular momentum L is proportional to moment of inertia I and angular speed \omega measured in radians per second. :L = I\omega. Unlike mass, which depends only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation and the shape of the matter.
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The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis.H. Williams, Measuring the inertia tensor, presented at the IMA Mathematics 2007 Conference. The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.
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The moment of inertia for a differentiated planet is less than 0.4, because the density of the planet is concentrated in the center. Mercury has a moment of inertia of 0.346, which is evidence for a core. Conservation of energy calculations as well as magnetic field measurements can also constrain composition, and surface geology of the planets can characterize differentiation of the body since its accretion. Mercury, Venus, and Mars’ cores are about 75%, 50%, and 40% of their radius respectively.
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For raster shapes, i.e. shapes composed of pixels or cells, some tests involve distinguishing between exterior and interior edges (or faces). More sophisticated measures of compactness include calculating the shape's moment of inertia or boundary curvature.
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This is because, although the rotation axis is fixed in space (by the conservation of angular momentum), it is not necessarily fixed in the body of the object itself. As a result of this, the moment of inertia of the object around the rotation axis can vary, and hence the rate of rotation can vary (because the product of the moment of inertia and the rate of rotation is equal to the angular momentum, which is fixed). For example, Hyperion, a satellite of Saturn, exhibits this behaviour, and its rotation period is described as chaotic.
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For simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression. Typically this occurs when the mass density is constant, but in some cases the density can vary throughout the object as well. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems.
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As a consequence, the polar diameter of Earth increases, and the equatorial diameter decreases (Earth's volume must remain the same). This means that mass moves closer to the rotation axis of Earth, and that Earth's moment of inertia decreases. This process alone leads to an increase of the rotation rate (phenomenon of a spinning figure skater who spins ever faster as they retract their arms). From the observed change in the moment of inertia the acceleration of rotation can be computed: the average value over the historical period must have been about −0.6 ms/century.
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It is especially prominent at lower speeds, with the symptom of jerkiness. Cogging torque results in torque as well as speed ripple; however, at high speed the motor moment of inertia filters out the effect of cogging torque.
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This changes moment of inertia of the swing and hence the resonance frequency, and children can quickly reach large amplitudes provided that they have some amplitude to start with (e.g., get a push). Standing and squatting at rest, however, leads nowhere.
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The curling process is used to form an edge on a ring. This process is used to remove sharp edges. It also increases the moment of inertia near the curled end. The flare/burr should be turned away from the die.
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The advantages claimed for this design are that, without projecting weight screws, the diameter of the balance can be increased, giving it a larger moment of inertia, and that it has less air resistance. The Gyromax balance has six to eight small turnable weights that fit on pins located in recesses around the top of the balance wheel rim. Each of the weights, called collets, has a cutout making it heavier on one side. When the collet's cutout points to the outside of the balance wheel, the heavier side is toward the center which decreases the wheel's moment of inertia, increasing its speed.
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If the body shown in the illustration is a homogeneous disc, this moment of inertia is \scriptstyle I=m r^2 /2 . If the disc has the mass 0,5 kg and the radius 0,8 m, the moment of inertia is 0,16 kgm2. If the amount of force is 2 N, and the lever arm 0,6 m, the amount of torque is 1,2 Nm. At the instant shown, the force gives to the disc the angular acceleration α = /I = 7,5 rad/s2, and to its center of mass it gives the linear acceleration a = F/m = 4 m/s2.
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Simply put, the polar and planar second moments of inertia are an indication of rigidity, and the mass moment of inertia is the rotational motion resistance of a massive object. The equation describing the polar moment of inertia is a multiple integral over the cross-sectional area, A, of the object. : J = \iint\limits_A r^2 dA where r is the distance to the element dA. Substituting the x and y components, using the Pythagorean theorem: : J = \iint\limits_A (x^2+y^2) dxdy : J = \iint\limits_A x^2 dxdy + \iint\limits_A y^2 dxdy Given the planar second moments of area equations, where: : I_x = \iint\limits_A y^2 dxdy : I_y = \iint\limits_A x^2 dxdy It is shown that the polar moment of inertia can be described as the summation of the x and y planar moments of inertia, I_x and I_y :\therefore J = I_z = I_x + I_y This is also shown in the perpendicular axis theorem.
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This diagram of true polar wander shows the present-day Earth rotating with respect to its rotational axis True polar wander is a solid-body rotation of a planet or moon with respect to its spin axis, causing the geographic locations of the north and south poles to change, or "wander". If a body is not totally rigid (as is the case of the earth), then in a stable state, the largest moment of inertia axis will be aligned with the spin axis, with the smaller two moments of inertia axes lying in the plane of the equator. If the body is not in this steady state, true polar wander will occur: the planet or moon will rotate as a rigid body to realign the largest moment of inertia axis with the spin axis. (See .) If the body is near the steady state but with the angular momentum not exactly lined up with the largest moment of inertia axis, the pole position will oscillate.
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The polar moment of inertia is traditionally determined by combining measurements of spin quantities (spin precession rate and/or obliquity) with gravity quantities (coefficients of a spherical harmonic representation of the gravity field). These geodetic data usually require an orbiting spacecraft to collect.
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"Cord front-drive car is here", The New York Times. April 12, 1936. p. XX7. (The heaviest component is near the centre of the car, making the main component of its moment of inertia relatively low). Another result of this design is a lengthened chassis.
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There were also two rear facing openings to feed air to the carburetors on each side, and a horizontal slot in the rear deck for cooling air. The concentration of masses in the centre of the car resulted in a low polar moment of inertia.
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A molecule in the gas phase is free to rotate relative to a set of mutually orthogonal axes of fixed orientation in space, centered on the center of mass of the molecule. Free rotation is not possible for molecules in liquid or solid phases due to the presence of intermolecular forces. Rotation about each unique axis is associated with a set of quantized energy levels dependent on the moment of inertia about that axis and a quantum number. Thus, for linear molecules the energy levels are described by a single moment of inertia and a single quantum number, J, which defines the magnitude of the rotational angular momentum.
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Schematic cross- section of the internal structure of the Moon Cornell University's Ask an Astronomer, run by volunteers in the Astronomy Department, answered the question "Can we prove that the Moon isn't hollow?". There, physicist Suniti Karunatillake suggests that there are at least two ways to determine the distribution of mass within a body. One involves moment of inertia parameters, the other involves seismic observations. In the case of the former, Karunatillake points out that the moment of inertia parameters indicate that the core of the moon is both dense and small, with the rest of the moon consisting of material with nearly-constant density.
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Due to mobility of automata the following new parameters of cellular automata have to be included into consideration: Ri - radius-vector of automaton; Vi - velocity of automaton; ωi - rotation velocity of automaton; θi - rotation vector of automaton; mi - mass of automaton; Ji - moment of inertia of automaton.
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In computational anatomy, Av was first called the Eulerian or diffeomorphic shape momentum since when integrated against Eulerian velocity v gives energy density, and since there is a conservation of diffeomorphic shape momentum which holds. The operator A is the generalized moment of inertia or inertial operator.
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Alexander Balankin and coauthors have studied the behavior of toilet paper under tensile stress and during wetting and burning. Toilet paper has been used in physics education to demonstrate the concepts of torque, moment of inertia, and angular momentum; and the conservation of momentum and energy.
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For a rigid Earth which is an oblate spheroid to a good approximation, the figure axis F would be its geometric axis defined by the geographic north and south pole, and identical with the axis of its polar moment of inertia. The Euler period of free nutation is (1) τE = 1/νE = A/(C − A) sidereal days ≈ 307 sidereal days ≈ 0.84 sidereal years νE = 1.19 is the normalized Euler frequency (in units of reciprocal years), C = 8.04 × 1037 kg m2 is the polar moment of inertia of the Earth, A is its mean equatorial moment of inertia, and C - A = 2.61 × 1035 kg m2. The observed angle between the figure axis of the Earth F and its angular momentum M is a few hundred milliarcseconds (mas). This rotation can be interpreted as a linear displacement of either geographical pole amounting to several meters on the surface of the Earth: 100 mas subtends an arc length of 3.082 m, when converted to radians and multiplied by the Earth's polar radius (6,356,752.3 m).
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It distributes mass away from the pivot point, thereby increasing the moment of inertia. This reduces angular acceleration, so a greater torque is required to rotate the performer over the wire. The result is less tipping. In addition, the performer can also correct sway by rotating the pole.
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This is very close to the value for a solid object with radially constant density, which would be 2/5 = 0.4 (for comparison, Earth's value is 0.33). The normalized polar moment of inertia for a hollow Moon would be close to the value 2/3 for a thin sphere.
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Therefore, friction coefficients of all parts involved in force transmission also play a major role in system design. Also, close attention has to be paid to the ratio of the moment of inertia of the robot body and its ball in order to prevent undesired ball spin, especially while yawing.
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Orbiting Frog Otolith (OFO) with booms. Booms out increased the moment of inertia. Frog Otolith Experiment Package The Frog Otolith Experiment Package (FOEP) contains all apparatus necessary to assure survival of two frogs. Specimens are housed in a water-filled, self-contained centrifuge which supplies the test acceleration during orbit.
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This may affect many systems that otherwise require a lot of energy.Bertels, 2006. The amount of time that an object such as the ISS can remain safely in free-drift varies depending on moment of inertia, perturbation torques, tidal gradients, etc. The ISS itself generally can last about 45 minutes in this mode.
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The bullet density in the equation above is implicit in m through the moment of inertia approximation. Finally, note that the denominator of Miller's formula is based upon the relative shape of a modern bullet. The term l(1+l^2) roughly indicates a shape similar to that of an American football.
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The loss of mass in the region around the ice sheet would decrease the gravitational potential there, reducing the amount of local sea level rise or even causing local sea level fall. The loss of the localized mass would also change the moment of inertia of the Earth, as flow in the Earth's mantle will require 10–15 thousand years to make up the mass deficit. This change in the moment of inertia results in true polar wander, in which the Earth's rotational axis remains fixed with respect to the sun, but the rigid sphere of the Earth rotates with respect to it. This changes the location of the equatorial bulge of the Earth and further affects the geoid, or global potential field.
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RMR layout; the engine is located in front of the rear axle. Rear Mid-engine transversely-mounted / Rear-wheel drive In automotive design, a RMR or Rear Mid-engine, Rear-wheel-drive layout (now simply known as MR or Mid-engine, Rear-wheel-drive layout) is one in which the rear wheels are driven by an engine placed just in front of them, behind the passenger compartment. In contrast to the rear-engined RR layout, the center of mass of the engine is in front of the rear axle. This layout is typically chosen for its low moment of inertia and relatively favorable weight distribution (the heaviest component is within the wheelbase, making the main component of its moment of inertia relatively low).
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In the general case of rotation of an unsymmetric body, which has different values of the moment of inertia about the three principal axes, the rotational motion can be quite complex unless the body is rotating around a principal axis. As described in the tennis racket theorem, rotation of an object around its first or third principal axis is stable, while rotation around its second principal axis (or intermediate axis) is not. The motion is simplified in the case of an axisymmetric body, in which the moment of inertia is the same about two of the principal axes. These cases include rotation of a prolate spheroid (the shape of an American football), or rotation of an oblate spheroid (the shape of a pancake).
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Ctirad Uher during a physics lecture on moment of inertia in winter 2012 Professor Ctirad Uher is the C. Wilbur Peters Collegiate Professor at the University of Michigan in Ann Arbor. Born in Prague, Czech Republic, he graduated from the University of New South Wales, Australia in 1972 and earned his Ph.D. from there in 1979.
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A ballistic galvanometer is a type of sensitive galvanometer; commonly a mirror galvanometer. Unlike a current-measuring galvanometer, the moving part has a large moment of inertia, thus giving it a long oscillation period. It is really an integrator measuring the quantity of charge discharged through it. It can be either of the moving coil or moving magnet type.
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In classical mechanics, analytical dynamics, or more briefly dynamics, is concerned with the relationship between motion of bodies and its causes, namely the forces acting on the bodies and the properties of the bodies, particularly mass and moment of inertia. The foundation of modern-day dynamics is Newtonian mechanics and its reformulation as Lagrangian mechanics and Hamiltonian mechanics.
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Precession and nutation are caused principally by the gravitational forces of the Moon and Sun acting upon the non-spherical figure of the Earth. Precession is the effect of these forces averaged over a very long period of time, and a time-varying moment of inertia (If an object is asymmetric about its principal axis of rotation, the moment of inertia with respect to each coordinate direction will change with time, while preserving angular momentum), and has a timescale of about 26,000 years. Nutation occurs because the forces are not constant, and vary as the Earth revolves around the Sun, and the Moon revolves around the Earth. Basically, there are also torques from other planets that cause planetary precession which contributes to about 2% of the total precession.
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For a collection of objects revolving about a center, for instance all of the bodies of the Solar System, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random. In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by, :I=k^2m, :where k is the radius of gyration, the distance from the axis at which the entire mass m may be considered as concentrated.
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However, if the moment of inertia around one of the two axes close to the equator becomes nearly equal to that around the polar axis, the constraint on the orientation of the object (the Earth) is relaxed. This situation is like a rugby football or an American football spinning around an axis running through its "equator". (Note that the "equator" of the ball does not correspond to the equator of the Earth.) Small perturbations can move the football, which then spins around another axis through the same "equator". In the same way, conditions can make the Earth (both the crust and the mantle) slowly reorient until a new geographic point moves to the North Pole, with the axis of low moment of inertia being kept very near the equator.
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94mm chassis are also popular for racing due to lower polar moment of inertia and weight distribution. Classic bodies include Lancia Stratos, Lotus Europa, Porsche 934 and 935, Lamborghini Countach, Shelby Cobra and many others. Mini-Z comes with parts to adjust wheelbase and motor location to fit the body. Different bodies will handle differently due to wheelbase and distribution of the masses.
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Science: 305. 827–829. Using Doppler tracking and two-way ranging, scientists added earlier measurements from the Viking landers to determine that the non-hydrostatic component of the polar moment of inertia is due to the Tharsis bulge and that the interior is not melted. The central metallic core is between 1300 km and 2000 km in radius.Golombek, M. et al. 1997.
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The population of vibrationally excited states follows a Boltzmann distribution, so low- frequency vibrational states are appreciably populated even at room temperatures. As the moment of inertia is higher when a vibration is excited, the rotational constants (B) decrease. Consequently, the rotation frequencies in each vibration state are different from each other. This can give rise to "satellite" lines in the rotational spectrum.
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The magnesium monohydride molecule is a simple diatomic molecule with a magnesium atom bonded to a hydrogen atom. The distance between hydrogen and magnesium atoms is 1.7297Å. The ground state of magnesium monohydride is X2Σ+. Due to the simple structure the symmetry point group of the molecule is C∞v. The moment of inertia of one molecule is 4.805263×10−40 g cm2.
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A schematic of the curling process The four steps to create a full curl Curling is a sheet metal forming process used to form the edges into a hollow ring. Curling can be performed to eliminate sharp edges and increase the moment of inertia near the curled end. Other parts are curled to perform their primary function, such as door hinges.
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Cabell and Bateman, p. 22 Factors such as angular momentum, the moment of inertia, angular acceleration, and the skater's center of mass determines if a jump is successfully completed.Cabell and Bateman, p. 27 Unlike jumping from dry land, which is fundamentally a linear movement, jumping on the ice is more complicated because of angular momentum. For example, most jumps involve rotation.Petkevich, p.
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A ballistic galvanometer is a type of sensitive galvanometer for measuring the quantity of charge discharged through it. In reality it is an integrator, unlike a current-measuring galvanometer, the moving part has a large moment of inertia that gives it a long oscillation period. It can be either of the moving coil or moving magnet type; commonly it is a mirror galvanometer.
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The magnitude of angular momentum remains constant throughout the dive, but since ::angular momentum = rotational velocity × moment of inertia, and the moment of inertia is larger when the body has an increased radius, the speed of rotation may be increased by moving the body into a compact shape, and reduced by opening out into a straight position. Since the tucked shape is the most compact, it gives the most control over rotational speed, and dives in this position are easier to perform. Dives in the straight position are hardest, since there is almost no scope for altering the speed, so the angular momentum must be created at take-off with a very high degree of accuracy. (A small amount of control is available by moving the position of the arms and by a slight hollowing of the back).
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Thouless made many important contributions to the theory of many-body problems. For atomic nuclei, he cleared up the concept of 'rearrangement energy' and derived an expression for the moment of inertia of deformed nuclei. In statistical mechanics, he contributed many ideas to the understanding of ordering, including the concept of 'topological ordering'. Other important results relate to localised electron states in disordered lattices.
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The calculated maximum moment of inertia of a uniformly dense object the same shape as Ida coincides with the spin axis of the asteroid. This suggests that there are no major variations of density within the asteroid. Ida's axis of rotation precesses with a period of 77 thousand years, due to the gravity of the Sun acting upon the nonspherical shape of the asteroid.
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Stabilizers can reduce noise and vibration. These energies are absorbed by viscoelastic polymers, gels, powders, and other materials used to build stabilizers. Stabilizers improve the forgiveness and accuracy by increasing the moment of inertia of the bow to resist movement during the shooting process. Lightweight carbon stabilizers with weighted ends are desirable because they improve the moment of interia while minimizing the weight added.
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In the case of pure racing cars, this is typically between "40/60" and "35/65". This gives the front tires an advantage in overcoming the car's moment of inertia (yaw angular inertia), thus reducing corner-entry understeer. Using wheels and tires of different sizes (proportional to the weight carried by each end) is a lever automakers can use to fine tune the resulting over/understeer characteristics.
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Yuko Kawaguti, 2010. Illustration of angular momentum: As a skater pulls in her arms, she reduces her moment of inertia and rotates faster. As The New York Times says, "While jumps look like sport, spins look more like art. While jumps provide the suspense, spins provide the scenery, but there is so much more to the scenery than most viewers have time or means to grasp".
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Using this and the known moment of inertia of the flywheel, the computer is able to calculate speed, power, distance and energy usage. Some ergometers can be connected to a personal computer using software, and data on individual exercise sessions can be collected and analysed. In addition, some software packages allows users to connect multiple ergometers either directly or over the internet for virtual races and workouts.
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This type of chaotic pattern of rapidly changing land, ice, saltwater and freshwater has been proposed as the likely model for the Baltic and Scandinavian regions, as well as much of central North America at the end of the last glacial maximum, with the present-day coastlines only being achieved in the last few millennia of prehistory. Also, the effect of elevation on Scandinavia submerged a vast continental plain that had existed under much of what is now the North Sea, connecting the British Isles to Continental Europe. The redistribution of ice-water on the surface of the Earth and the flow of mantle rocks causes changes in the gravitational field as well as changes to the distribution of the moment of inertia of the Earth. These changes to the moment of inertia result in a change in the angular velocity, axis, and wobble of the Earth's rotation.
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The RX-8 was designed as a front mid-engine, rear- wheel-drive, four-door, four-seater quad coupé. The car has a near 50:50 front-rear weight distribution and a low polar moment of inertia, achieved by mounting the engine behind the front axle and by placing the fuel tank ahead of the rear axle. The front suspension uses double wishbones and the rear suspension is multi-link.
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A simple example of a parametric oscillator is a child pumping a playground swing by periodically standing and squatting to increase the size of the swing's oscillations. Note: In real-life playgrounds, swings are predominantly driven, not parametric, oscillators. The child's motions vary the moment of inertia of the swing as a pendulum. The "pump" motions of the child must be at twice the frequency of the swing's oscillations.
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There exist two main passive control types for satellites. The first one uses gravity gradient, and it leads to four stable states with the long axis (axis with smallest moment of inertia) pointing towards Earth. As this system has four stable states, if the satellite has a preferred orientation, e.g. a camera pointed at the planet, some way to flip the satellite and its tether end-for-end is needed.
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Therefore B in this article corresponds to \bar B = B/hc in the Rigid rotor article. and the moment of inertia of the molecule, and, knowing the atomic masses, can be used to determine the bond length directly. For diatomic molecules this process is straightforward. For linear molecules with more than two atoms it is necessary to measure the spectra of two or more isotopologues, such as 16O12C32S and 16O12C34S.
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Drop tank storage aboard The primary disadvantage with drop tanks is that they impose a drag penalty on the aircraft carrying them. External fuel tanks will also increase the moment of inertia, thereby reducing roll rates for air maneuvres. Some of the drop tank's fuel is used to overcome the added drag and weight of the tank itself. Drag in this sense varies with the square of the aircraft's speed.
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Astrojax is a toy consisting of three balls on a string. One ball is fixed at each end of the string, and the center ball is free to slide along the string between the two end balls. Inside each ball is a metal weight. The metal weight lowers the moment of inertia of the center ball so it can rotate rapidly in response to torques applied by the string.
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This is further generalised to quadratic forms in linear spaces via the inner product. The inertia tensor in mechanics is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to the size (length). There are infinitely many Pythagorean triples, sets of three positive integers such that the sum of the squares of the first two equals the square of the third.
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This robotic configuration is a parallel manipulator. It is a parallel configuration robot as it is composed of two controlled Serial manipulators connected to the endpoint. Unlike a Serial manipulator, this configuration has the advantage of having both motors grounded at the base link. As the motor can be quite massive, this significantly decreases the total moment of inertia of the linkage and improves backdrivability for haptic feedback applications.
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The mid-engine, rear-wheel drive layout (abbreviated as MR layout) is one where the rear wheels are driven by an engine placed just in front of them, behind the passenger compartment. In contrast to the rear-engined RR layout, the center of mass of the engine is in front of the rear axle. This layout is typically chosen for its low moment of inertia and relatively favorable weight distribution.
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As mentioned above in the section on balance, one effect of turning the front wheel is a roll moment caused by gyroscopic precession. The magnitude of this moment is proportional to the moment of inertia of the front wheel, its spin rate (forward motion), the rate that the rider turns the front wheel by applying a torque to the handlebars, and the cosine of the angle between the steering axis and the vertical. For a sample motorcycle moving at 22 m/s (50 mph) that has a front wheel with a moment of inertia of 0.6 kg·m2, turning the front wheel one degree in half a second generates a roll moment of 3.5 N·m. In comparison, the lateral force on the front tire as it tracks out from under the motorcycle reaches a maximum of 50 N. This, acting on the 0.6 m (2 ft) height of the center of mass, generates a roll moment of 30 N·m.
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Similarly, for a point mass m the moment of inertia is defined as, :I=r^2m :where r is the radius of the point mass from the center of rotation, and for any collection of particles m_i as the sum, :\sum_i I_i = \sum_i r_i^2m_i Angular momentum's dependence on position and shape is reflected in its units versus linear momentum: kg⋅m2/s, N⋅m⋅s, or J⋅s for angular momentum versus kg⋅m/s or N⋅s for linear momentum. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity. (When performing dimensional analysis, it may be productive to use orientational analysis which treats radians as a base unit, but this is outside the scope of the International system of units). Angular momentum's units can be interpreted as torque⋅time or as energy⋅time per angle.
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Schematic animation of the motion involved Cats falling at normal gravity and with no gravity After determining down from up visually or with their vestibular apparatus (in the inner ear), cats manage to twist themselves to face downward without changing their net angular momentum. They are able to accomplish this with these key steps: #Bend in the middle so that the front half of their body rotates about a different axis from the rear half. #Tuck their front legs in to reduce the moment of inertia of the front half of their body and extend their rear legs to increase the moment of inertia of the rear half of their body so that they can rotate their front by as much as 90° while the rear half rotates in the opposite direction as little as 10°. #Extend their front legs and tuck their rear legs so that they can rotate their rear half further while their front half rotates in the opposite direction less.
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The engine of the 550 is mounted in front of the rear axle making it mid-engined. This gives it a more balanced weight distribution, and allows for largely neutral handling. On the other hand, the low mass moment of inertia about the vehicle's vertical axis can lead to a sudden, difficult to control rotation of the car. Ferdinand Porsche had pioneered this design layout with the Auto Union Grand Prix car of the 1930s.
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This relates to the handicap theory of evolution of ornamental male traits. However, studies cite a lack of evidence to support the idea that the extended eye-stalks are a handicap to male Diopsid flies. While the eye-stalk increases the moment of inertia of male flies, they were not found to suffer a flight performance decrement. Specifically T. dalmanni male flies were found to perform better than females in free-flying turning behavior.
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Another variation of the upright spin, is the sideways leaning spin, in which the skater's head and shoulders are leaning sideways and his or her upper body is arched. The free leg, as it is for the layback spin, is optional.S&P;/ID 2018, p. 102–103 The angular velocity of an upright spin is low, about one revolution per second, but its moment of inertia is large during its balancing stage.
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A quantity related to inertia is rotational inertia (→ moment of inertia), the property that a rotating rigid body maintains its state of uniform rotational motion. Its angular momentum remains unchanged, unless an external torque is applied; this is also called conservation of angular momentum. Rotational inertia is often considered in relation to a rigid body. For example, a gyroscope uses the property that it resists any change in the axis of rotation.
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At the beginning of his scientific career, Wittenbauer worked on kinematic geometry. His main contribution lay in applying graphic methods of kinematic geometry to dynamics. In 1904, he started publishing treatises which were preliminary works for his almost 800 page book on Graphische Dynamik (Graphical Dynamics), which he completed only shortly before his death. In 1905, Wittenbauer first published his internationally acclaimed and still valid method for a graphic determination of the flywheel moment of inertia.
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However, some panels require stricter movement restrictions, or certainly those that prohibit a torque-like motion. Deflection in mullions is controlled by different shapes and depths of curtain wall members. The depth of a given curtain wall system is usually controlled by the area moment of inertia required to keep deflection limits under the specification. Another way to limit deflections in a given section is to add steel reinforcement to the inside tube of the mullion.
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Stabilisers in use at an archery competition In archery, a stabiliser is a general term for various types of weights, usually on rods, mounted on the bow to increase stability i.e. lessen movement on release, thereby increasing precision. Stabilisers help reduce inconsistencies of the archer's release by increasing the moment of inertia of the bow.Charles E. Phelps Archery Stabilizers – Theory and Practice 2006 If the shooting technique of the archer were perfect, no stabilisers would be required.
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The mass distribution of the Earth is not spherically symmetric, and the Earth has three different moments of inertia. The axis around which the moment of inertia is greatest is closely aligned with the rotation axis (the axis going through the geographic North and South Poles). The other two axes are near the equator. That is similar to a brick rotating around an axis going through its shortest dimension (a vertical axis when the brick is lying flat).
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Pendulums used in Mendenhall gravimeter apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth. A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of.
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For the same object, different axes of rotation will have different moments of inertia about those axes. In general, the moments of inertia are not equal unless the object is symmetric about all axes. The moment of inertia tensor is a convenient way to summarize all moments of inertia of an object with one quantity. It may be calculated with respect to any point in space, although for practical purposes the center of mass is most commonly used.
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When power is disconnected from the motor, the rotor spins freely until friction slows it to a stop. Large rotors and loads with a high moment of inertia may take a significant amount of time to stop through inherent friction alone. To reduce downtime, or possibly as an emergency safety feature, DC injection braking can be used to quickly stop the rotor. A DC injection brake system can be used as an alternative to a friction brake system.
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The closer in the balls are, the smaller the moment of inertia of the torsion pendulum and the faster it will turn, like a spinning ice skater who pulls in her arms. This causes the clock to speed up. One oscillation of the torsion pendulum usually takes 12, 15, or 20 seconds. The escapement mechanism, that changes the rotational motion of the clock's gears to pulses to drive the torsion pendulum, works rather like an anchor escapement.
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The speed of rotation is given by the angular frequency (rad/s) or frequency (turns per time), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s²), caused by torque. The ratio of the two (how heavy is it to start, stop, or otherwise change rotation) is given by the moment of inertia. The angular velocity vector (an axial vector) also describes the direction of the axis of rotation.
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Soil liquefaction in Kōtō, Tokyo The Earth's axis shifted by estimates of between and . This deviation led to a number of small planetary changes, including the length of a day, the tilt of the Earth, and the Chandler wobble. The speed of the Earth's rotation increased, shortening the day by 1.8 microseconds due to the redistribution of Earth's mass. The axial shift was caused by the redistribution of mass on the Earth's surface, which changed the planet's moment of inertia.
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Heavier wheels have a greater moment of inertia and their spinning takes away energy that would otherwise contribute to the speed of the car. A standard wheel has a mass of 2.6 g, but this can be reduced to as little as 1 g by removing material from the inside of the wheel. A raised wheel can reduce the rotational energy up to one-quarter, but this advantage is less with a bumpy track. Another consideration is the track itself.
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Consider a beam whose cross-sectional area increases in two dimensions, e.g. a solid round beam or a solid square beam. By combining the area and density formulas, we can see that the radius of this beam will vary with approximately the inverse of the square of the density for a given mass. By examining the formulas for area moment of inertia, we can see that the stiffness of this beam will vary approximately as the fourth power of the radius.
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As the flow is already supersonic, increasing the speed even more would not be beneficial for the wing structure. Reducing the thickness of the wing brings the top and bottom stringers closer together, reducing the total moment of inertia of the structure. This increases is axial load in the stringers, and thus the area, and weight, of the stringers must be increased. Some designs for hypersonic missiles have used liquid cooling of the leading edges (usually the fuel en route to the engine).
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The equation L=rmv combines a moment (a mass m turning moment arm r) with a linear (straight-line equivalent) speed v. Linear speed referred to the central point is simply the product of the distance r and the angular speed \omega versus the point: v=r\omega, another moment. Hence, angular momentum contains a double moment: L=rmr\omega. Simplifying slightly, L=r^2m\omega, the quantity r^2m is the particle's moment of inertia, sometimes called the second moment of mass.
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In mathematics (in particular geometry and trigonometry) and all natural sciences (e.g. astronomy and geophysics), the angular distance (a. k. a. angular separation, apparent distance, or apparent separation) between two point objects, as viewed from a location different from either of these objects, is the angle of length between the two directions originating from the observer and pointing toward these two objects. Angular distance shows up in the classical mechanics of rotating objects alongside angular velocity, angular acceleration, angular momentum, moment of inertia and torque.
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While several experiments yielded negative results, in the 1980s, John Goodkind discovered the first anomaly in a solid by using ultrasound. Inspired by his observation, in 2004 Eun-Seong Kim and Moses Chan at Pennsylvania State University saw phenomena which were interpreted as supersolid behavior. Specifically, they observed a non- classical rotational moment of inertia of a torsional oscillator. This observation could not be explained by classical models but was consistent with superfluid-like behavior of a small percentage of the helium atoms contained within the oscillator.
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The increased moment of inertia makes the long gun slower and more difficult to traverse and elevate, and it is thus slower and more difficult to adjust the aim. However, this also results in greater stability in aiming. The greater amount of material in a long gun tends to make it more expensive to manufacture, other factors being equal. The greater size makes it more difficult to conceal, and more inconvenient to use in confined quarters, as well as requiring a larger storage space.
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The partial differentiation of Callisto (inferred e.g. from moment of inertia measurements) means that it has never been heated enough to melt its ice component. Therefore, the most favorable model of its formation is a slow accretion in the low-density Jovian subnebula—a disk of the gas and dust that existed around Jupiter after its formation. Such a prolonged accretion stage would allow cooling to largely keep up with the heat accumulation caused by impacts, radioactive decay and contraction, thereby preventing melting and fast differentiation.
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The most fundamental design parameters of a ballbot are its height, mass, its center of gravity and the maximum torque its actuators can provide. The choice of those parameters determine the robot's moment of inertia, the maximum pitch angle and thus its dynamic and acceleration performance and agility. The maximum velocity is a function of actuator power and its characteristics. Besides the maximum torque, the pitch angle is additionally upper bounded by the maximum force which can be transmitted from the actuators to the ground.
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He was responsible for curriculum and administration of educational programs provided to employees of Bell Laboratories. In this new role, he invented the Shive wave machine (also known as the Shive wave generator). The wave generator illustrates wave motion using a series of steel rods joined by a thin torsion wire which transmits energy from one rod to the next. The high moment of inertia of each rod ensures the wave takes several seconds to traverse the entire series of rods, making the dynamics easily visible.
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However, a desmodromic system must deal with the inertia of the two rocker arms per valve, so this advantage depends greatly on the skill of the designer. Another disadvantage is the contact point between the cams and rocker arms. It is relatively easy to use roller tappets in conventional valvetrains, although it does add considerable moving mass. In a desmodromic system the roller would be needed at one end of the rocker arm, which would greatly increase its moment-of-inertia and negate its "effective mass" advantage.
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FMR layout, the engine is located behind the front axle. In automotive design, a front mid- engine, rear-wheel-drive layout (FMR) is one that places the engine in the front, with the rear wheels of vehicle being driven. In contrast to the front- engine, rear-wheel-drive layout (FR), the engine is pushed back far enough that its center of mass is to the rear of the front axle. This aids in weight distribution and reduces the moment of inertia, improving the vehicle's handling.
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The added weight of the butter has no effect on the falling process, since the butter spreads throughout the slice. A piece of toast has inertia as it flips towards the floor, preventing its spin from stopping easily; it is usually only stopped by hitting the floor. This moment of inertia is determined by the speed at which the toast is flipping, combined with the size and mass of the toast. Because most toast is relatively uniform, they often land in a similar manner.
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This is difficult to measure directly because of the smallness of the force. Cavendish accomplished this by a method widely used since: measuring the resonant vibration period of the balance. If the free balance is twisted and released, it will oscillate slowly clockwise and counterclockwise as a harmonic oscillator, at a frequency that depends on the moment of inertia of the beam and the elasticity of the fiber. Since the inertia of the beam can be found from its mass, the spring constant can be calculated.
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Ganymede , a satellite of Jupiter (Jupiter III), is the largest and most massive of the Solar System's moons. The ninth-largest object in the Solar System, it is the largest without a substantial atmosphere. It has a diameter of , making it 26% larger than the planet Mercury by volume, although it is only 45% as massive. Possessing a metallic core, it has the lowest moment of inertia factor of any solid body in the Solar System and is the only moon known to have a magnetic field.
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The variety of shapes it can be built into has further increased stiffness and also allowed aerodynamic tube sections. CFRP forks including suspension fork crowns and steerers, handlebars, seatposts, and crank arms are becoming more common on medium as well as higher-priced bicycles. CFRP rims remain expensive but their stability compared to aluminium reduces the need to re-true a wheel and the reduced mass reduces the moment of inertia of the wheel. CFRP spokes are rare and most carbon wheelsets retain traditional stainless steel spokes.
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Then radius of gyration can be used to characterize the typical distance travelled by this point. Suppose a body consists of n particles each of mass m. Let r_1, r_2, r_3, \dots , r_n be their perpendicular distances from the axis of rotation. Then, the moment of inertia I of the body about the axis of rotation is :I = m_1 r_1^2 + m_2 r_2^2 + \cdots + m_n r_n^2 If all the masses are the same (m), then the moment of inertia is I=m(r_1^2+r_2^2+\cdots+r_n^2). Since m = M/n (M being the total mass of the body), :I=M(r_1^2+r_2^2+\cdots+r_n^2)/n From the above equations, we have :MR_g^2=M(r_1^2+r_2^2+\cdots+r_n^2)/n Radius of gyration is the root mean square distance of particles from axis formula :R_g^2=(r_1^2+r_2^2+\cdots+r_n^2)/n Therefore, the radius of gyration of a body about a given axis may also be defined as the root mean square distance of the various particles of the body from the axis of rotation.
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One effect of turning the front wheel is a roll moment caused by gyroscopic precession. The magnitude of this moment is proportional to the moment of inertia of the front wheel, its spin rate (forward motion), the rate that the rider turns the front wheel by applying a torque to the handlebars, and the cosine of the angle between the steering axis and the vertical. For a sample motorcycle moving at 22 m/s (50 mph) that has a front wheel with a moment of inertia of 0.6 kgm2, turning the front wheel one degree in half a second generates a roll moment of 3.5 Nm. In comparison, the lateral force on the front tire as it tracks out from under the motorcycle reaches a maximum of 50 N. This, acting on the 0.6 m (2 ft) height of the center of mass, generates a roll moment of 30 Nm. While the moment from gyroscopic forces is only 12% of this, it can play a significant part because it begins to act as soon as the rider applies the torque, instead of building up more slowly as the wheel out- tracks. This can be especially helpful in motorcycle racing.
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In 2004, Jim Woodhouse and Paul Galluzzo of Cambridge University described the motion of a bowed string as being "the only stick-slip oscillation which is reasonably well understood". The length, weight, and balance point of modern bows are standardized. Players may notice variations in sound and handling from bow to bow, based on these parameters as well as stiffness and moment of inertia. A violinist or violist would naturally tend to play louder when pushing the bow across the string (an 'up-bow'), as the leverage is greater.
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The symmetric and asymmetric stretching vibrations are close to each other, so the rotational fine structures of these bands overlap. The bands at shorter wavelength are overtones and combination bands, all of which show rotational fine structure. Medium resolution spectra of the bands around 1600 cm−1 and 3700 cm−1 are shown in Banwell and McCash, p91. Ro-vibrational bands of asymmetric top molecules are classed as A-, B- or C- type for transitions in which the dipole moment change is along the axis of smallest moment of inertia to the highest.
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These breaks provide a significant decrease in the thermal conductivity of the curtain wall. However, since the thermal break interrupts the aluminum mullion, the overall moment of inertia of the mullion is reduced and must be accounted for in the structural analysis and deflection analysis of the system. Thermal conductivity of the curtain wall system is important because of heat loss through the wall, which affects the heating and cooling costs of the building. On a poorly performing curtain wall, condensation may form on the interior of the mullions.
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Some, known as Rocket On Rotor systems, involve placing rockets on the tips of the rotor blades that are fueled from an onboard fuel tank. If the helicopter's engine fails, the tip jets on the rotor increase the moment of inertia, hence permitting it to store energy, which makes performing a successful autorotation landing somewhat easier. However, the tip jet also typically generates significant extra air drag, which demands a higher sink rate and means that a very sudden transition to the landing flare must occur for survival, with little room for error.
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In debut season of 2010 Super GT, HSV-010 GT got both titles of Team's Championship and Drivers' Championship: by Weider Honda Racing with Takashi Kogure and Loïc Duval. In 2011 season, radiator of HSV-010 GT was divided in to two and mounted on both sides, aiming quick cornering with reduced yaw moment of inertia. However, the center of gravity became higher, and configuration and adjusting for rounds of races became difficult and time- consuming task. In 2013 season, the last season under 2009 regulations, radiator was moved back to front with lightened equipment.
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Radio Doppler measurements were taken with Viking and twenty years later with Mars Pathfinder, and in each case the axis of rotation of Mars was estimated. By combining this data the core size was constrained, because the change in axis of rotation over 20 years allowed a precession rate and from that the planet's moment of inertia to be estimated. InSight measurements of crust thickness, mantle viscosity, core radius and density, and seismic activity should result in a three- to tenfold increase in accuracy compared to current data.
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Consider a beam whose cross-sectional area increases in one dimension, e.g. a thin-walled round beam or a rectangular beam whose height but not width is varied. By combining the area and density formulas, we can see that the radius or height of this beam will vary with approximately the inverse of the density for a given mass. By examining the formulas for area moment of inertia, we can see that the stiffness of this beam will vary approximately as the third power of the radius or height.
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The car has options for several types of engine. The smallest recommended powerplants included the 1.6 liter, 2 liter, or 2.3 liter 4-cylinder Ford Pinto / Bobcat engines used by the smaller Blakely Bantam. The Bernardi chassis is also amenable to larger engines like a 2.8 liter Ford or Chevrolet / Pontiac V6, and even the small-block Ford V8. All engine types mount behind the front axle, making the chassis a front-mid-engine design with the attendant benefits in front-rear weight balance and low polar moment of inertia.
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Ganymede appears to be fully differentiated, with an internal structure consisting of an iron- sulfide–iron core, a silicate mantle and outer layers of water ice and liquid water. The precise thicknesses of the different layers in the interior of Ganymede depend on the assumed composition of silicates (fraction of olivine and pyroxene) and amount of sulfur in the core. Ganymede has the lowest moment of inertia factor, 0.31, among the solid Solar System bodies. This is a consequence of its substantial water content and fully differentiated interior.
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In classical mechanics, the stretch rule (sometimes referred to as Routh's rule) states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to an axis of rotation that is a principal axis, provided that the distribution of mass remains unchanged except in the direction parallel to the axis. This operation leaves cylinders oriented parallel to the axis unchanged in radius. This rule can be applied with the parallel axis theorem and the perpendicular axes rule to find moments of inertia for a variety of shapes.
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Deviation of day length from SI- based day Some recent large-scale events, such as the 2004 Indian Ocean earthquake, have caused the length of a day to shorten by 3 microseconds by reducing Earth's moment of inertia.Sumatran earthquake sped up Earth's rotation, Nature, 30 December 2004. Post-glacial rebound, ongoing since the last Ice age, is also changing the distribution of Earth's mass, thus affecting the moment of inertia of Earth and, by the conservation of angular momentum, Earth's rotation period. The length of the day can also be influenced by manmade structures.
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124 As clocks were made smaller, first as bracket clocks and lantern clocks and then as the first large watches after 1500, balance wheels began to be used in place of foliots., p. 92 Since more of its weight is located on the rim away from the axis, a balance wheel could have a larger moment of inertia than a foliot of the same size, and keep better time. The wheel shape also had less air resistance, and its geometry partly compensated for thermal expansion error due to temperature changes.
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Air resistance models use vanes on the flywheel to provide the flywheel braking needed to generate resistance. As the flywheel is spun faster, the air resistance increases. An adjustable vent can be used to control the volume of air moved by the vanes of the rotating flywheel, therefore a larger vent opening results in a higher resistance, and a smaller vent opening results in a lower resistance. The energy dissipated can be accurately calculated given the known moment of inertia of the flywheel and a tachometer to measure the deceleration of the flywheel.
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These processes change the Earth's moment of inertia, affecting the rate of rotation due to the conservation of angular momentum. Some of these redistributions increase Earth's rotational speed, shorten the solar day and oppose tidal friction. For example, glacial rebound shortens the solar day by 0.6 ms/century and the 2004 Indian Ocean earthquake is thought to have shortened it by 2.68 microseconds. It is evident from the figure that the Earth's rotation has slowed at a decreasing rate since the initiation of the current system in 1971, and the rate of leap second insertions has therefore been decreasing.
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Although a pendulum can theoretically be any shape, any rigid object swinging on a pivot, clock pendulums are usually made of a weight or bob attached to the bottom end of a rod, with the top attached to a pivot so it can swing. The advantage of this construction is that it positions the centre of mass close to the physical end of the pendulum, farthest from the pivot. This maximizes the moment of inertia, and minimises the length of pendulum required for a given period. Shorter pendulums allow the clock case to be made smaller, and also minimize the pendulum's air resistance.
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Radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated. Mathematically the radius of gyration is the root mean square distance of the object's parts from either its center of mass or a given axis, depending on the relevant application. It is actually the perpendicular distance from point mass to the axis of rotation. One can represent a trajectory of a moving point as a body.
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The resonant amplification of the low-frequency forced nutations depends sensibly on the size, moment of inertia, and flattening of the core. This amplification is expected to correspond to a displacement of between a few to forty centimeters on Mars surface. Observing the amplification allows to confirm the liquid state of the core and to determine some core properties. LaRa will also measure variations in the rotation angular momentum due to the redistribution of masses, such as the migration of ice from the polar caps to the atmosphere and the sublimation/condensation cycle of atmospheric CO2.
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They calculated that the neck musculature was strong enough to absorb the force of two individuals colliding with their heads frontally at a speed of 5.7 m/s each. Fernando Novas (2009) interpreted several skeletal features as adaptations for delivering blows with the head. He suggested that the shortness of the skull might have made head movements quicker by reducing the moment of inertia, while the muscular neck would have allowed strong head blows. He also noted an enhanced rigidity and strength of the spinal column that may have evolved to withstand shocks conducted by the head and neck.
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The swing weight of a baseball bat deals with how heavy the bat "feels" when swinging. The swing weight is measured around a certain pivot point along the bat. Once a pivot point is determined (usually 6 inches for baseball bats) the bats balance point, total weight and the amount of time it takes for the bat to swing from side to side like a pendulum are used to determine its 'swing weight', or as some refer to it, its mass moment of inertia. Bat manufacturers can adjust a composite bat's swing weight by changing how the weight is distributed along the bat.
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At one end of each spoke is a specialized nut, called a nipple, which is used to connect the spoke to the rim and adjust the tension in the spoke. The nipple is usually located at the rim end of the spoke but on some wheels is at the hub end to move its weight closer to the axis of the wheel, reducing the moment of inertia. A variant of this is integrating nipples into the hub, its flange containing the threads for usually bladed spokes. Until recently there were only two types of nipples: brass and aluminum (often referred to as "alloy").
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However, with engines built by Al Bartz, Falconer & Dunn and Traco Engineering, the pinnacle of the 302's use in professional racing was its being the primary engine that powered the outstanding but overshadowed 1968-1976 Formula 5000 Championship Series, a SCCA Formula A open-wheel class designed for lower cost. The engine was also popular in Formula 5000 racing around the world, especially in Australia and New Zealand where it proved more powerful than the Repco-Holden V8. Weighing , with a iron block and head engine positioned near the car's polar moment of inertia for responsive turn pivoting, a Hewland 5-spd.
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In US mechanical engineering, torque is defined mathematically as the rate of change of angular momentum of an object (in physics it is called "net torque"). The definition of torque states that one or both of the angular velocity or the moment of inertia of an object are changing. Moment is the general term used for the tendency of one or more applied forces to rotate an object about an axis, but not necessarily to change the angular momentum of the object (the concept which is called torque in physics). Kane, T.R. Kane and D.A. Levinson (1985).
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The Curtiss YA-10 Shrike was the first YA-8 fitted with a Pratt & Whitney R-1690-9 (R-1690D) Hornet radial engine. The conversion was carried out in September 1932, and it was found that the aircraft's performance was not degraded by the change of engine, and low-level maneuverability was improved due to the lower mass moment of inertia with the short radial engine. The USAAC preferred radials to inline engines for the ground attack role, due to the vulnerability of the latter's cooling system to anti-aircraft fire. The US Navy also preferred radials for carrier-borne operations.
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The cores of the rocky planets were initially characterized by analyzing data from spacecraft, such as NASA's Mariner 10 that flew by Mercury and Venus to observe their surface characteristics. The cores of other planets cannot be measured using seismometers on their surface, so instead they have to be inferred based on calculations from these fly-by observation. Mass and size can provide a first-order calculation of the components that make up the interior of a planetary body. The structure of rocky planets is constrained by the average density of a planet and its moment of inertia.
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It also allowed contra-rotating engines, where gas generator core and power turbine revolved in opposite directions, reducing the overall moment of inertia. For the helicopter engine replacement market, this ability allowed previous engines of either direction to be replaced simply. Some turboshaft engines' omni-angle freedom of their installation angle also allowed installation into existing helicopter designs, no matter how the previous engines had been arranged. In time though, the move towards axial LP compressors and so smaller diameter engines encouraged a move to the now standard layout of one or two engines set side-by-side, horizontally above the cabin.
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Inverse dynamics is an inverse problem. It commonly refers to either inverse rigid body dynamics or inverse structural dynamics. Inverse rigid-body dynamics is a method for computing forces and/or moments of force (torques) based on the kinematics (motion) of a body and the body's inertial properties (mass and moment of inertia). Typically it uses link-segment models to represent the mechanical behaviour of interconnected segments, such as the limbs of humans, animals or robots, where given the kinematics of the various parts, inverse dynamics derives the minimum forces and moments responsible for the individual movements.
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At the time, there was no way of synchronising such a weapon with the propeller, or of mounting it elsewhere than the fuselage, so a pusher configuration was necessary, the pilot sitting in a nacelle with the gun in its nose. In order to make the aircraft more manoeuvrable and in particular to increase its roll rate, a triplane configuration was chosen. This provided about the same total wing area as that of the biplane Scout with a lower moment of inertia about the roll axis. The Triplane had single-bay wings with heavy stagger, carrying six ailerons.
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A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force. A familiar example of parametric oscillation is "pumping" on a playground swing. A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing and squatting, in rhythm with the swing's oscillations. The varying of the parameters drives the system.
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While this scheme worked well enough to allow Harrison to meet the standards set by the Longitude Act, it was not widely adopted. Around 1765, Pierre Le Roy (son of Julien Le Roy) invented the compensation balance, which became the standard approach for temperature compensation in watches and chronometers. In this approach, the shape of the balance is altered, or adjusting weights are moved on the spokes or rim of the balance, by a temperature-sensitive mechanism. This changes the moment of inertia of the balance wheel, and the change is adjusted such that it compensates for the change in modulus of elasticity of the balance spring.
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Similar to planar second moment of area calculations (I_x,I_y, and I_{xy}), the polar second moment of area is often denoted as I_z. While several engineering textbooks and academic publications also denote it as J or J_z, this designation should be given careful attention so that it does not become confused with the torsion constant, J_t, used for non-cylindrical objects. Simply put, the polar moment of inertia is a shaft or beam's resistance to being distorted by torsion, as a function of its shape. The rigidity comes from the object's cross-sectional area only, and does not depend on its material composition or shear modulus.
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High (or highpoint) kicks target the opponent's head or neck; they are often responsible for knockouts in competition. Some Thai camps emphasize targeting the side of the neck with the high angle kick cutting down from its highest point to compress the carotid artery and so shock the opponent, weakening his or her fighting ability or knocking him out. There are several traits which give the muay Thai roundhouse a very different feel and look. The main methodological difference is that the hips are rotated into the kick in order to convey more moment of inertia in the kick, and the abdominal muscles are strongly recruited in the act of rotation.
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Not only do larger gears occupy more space, but the mass and rotational inertia (moment of inertia) of a gear is quadratic in proportion to its radius. Instead of idler gears, of course, a toothed belt or a roller chain can be used to transmit torque over distance. For short distances, a train of idlers may be used; whether an odd or even number is used determines whether the final output gear rotates the same direction as the input gear or not. For longer distances, a roller chain or belt is quieter and creates less friction, although gears are typically stronger, depending on the strength of the roller chain.
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To a first approximation, which neglects deformations due to its elasticity, the wheel is a rigid rotor that is constrained to rotate about its axle. If a principal axis of the wheel's moment of inertia is not aligned with the axle, due to an asymmetric mass distribution, then an external torque, perpendicular to the axle, is necessary to force the wheel to rotate about the axle. This additional torque must be provided by the axle and its orientation rotates continuously with the wheel. The reaction to this torque, by Newton's Third Law is applied to the axle, which transfers it to the suspension and can cause it to vibrate.
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Tharsis is so large and massive that it has likely affected the planet's moment of inertia, possibly causing a change in the orientation of the planet's crust with respect to its rotational axis over time. According to one recent study, Tharsis originally formed at about 50°N latitude and migrated toward the equator between 4.2 and 3.9 billion years ago. Such shifts, known as true polar wander, would have caused dramatic climate changes over vast areas of the planet. A more recent study reported in Nature agreed with the polar wander, but the authors thought the eruptions at Tharsis happened at a slightly different time.
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Marine chronometer balance wheels from the mid-1800s, with various 'auxiliary compensation' systems to reduce middle temperature error The standard Earnshaw compensation balance dramatically reduced error due to temperature variations, but it didn't eliminate it. As first described by J. G. Ulrich, a compensated balance adjusted to keep correct time at a given low and high temperature will be a few seconds per day fast at intermediate temperatures. pp. 176–177 The reason is that the moment of inertia of the balance varies as the square of the radius of the compensation arms, and thus of the temperature. But the elasticity of the spring varies linearly with temperature.
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When correctly adjusted and set in motion, it exhibits a curious motion in which periods of purely rotational oscillation gradually alternate with periods of purely up and down oscillation. The energy stored in the device shifts slowly back and forth between the translational 'up and down' oscillation mode and the torsional 'clockwise and counterclockwise' oscillation mode, until the motion eventually dies away. Despite the name, in normal operation it does not swing back and forth as ordinary pendulums do. The mass usually has opposing pairs of radial 'arms' sticking out horizontally, threaded with small weights that can be screwed in or out to adjust the moment of inertia to 'tune' the torsional vibration period.
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One proposed solution to account for this imbalance is that if the difference between the maximum moment of inertia and rotation axis exceeds a certain limit, the planet will undergo a larger degree of oscillation to realign its maximum of inertia with its rotation axis. If this is indeed the case, then the timescale at which this correction happens must be fairly short. Europa, a moon of Jupiter, has been modelled to have a crust that is decoupled from its mantle; that is, the outer icy crust may be floating on a covered ocean. If this is true, then models predict that the shell could display the polar wander trace on its surface as its crust realigns.
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Deviation of day length from SI based day Any change of the axial component of the atmospheric angular momentum (AAM) must be accompanied by a corresponding change of the angular momentum of Earth's crust and mantle (due to the law of conservation of angular momentum). Because the moment of inertia of the system mantle-crust is only slightly influenced by atmospheric pressure loading, this mainly requires a change in the angular velocity of the solid Earth; i.e., a change of LOD. The LOD can presently be measured to a high accuracy over integration times of only a few hours, and general circulation models of the atmosphere allow high precision determination of changes in AAM in the model.
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Unified framework is a general formulation which yields nth \- order expressions giving mode shapes and natural frequencies for damaged elastic structures such as rods, beams, plates, and shells. The formulation is applicable to structures with any shape of damage or those having more than one area of damage. The formulation uses the geometric definition of the discontinuity at the damage location and perturbation to modes and natural frequencies of the undamaged structure to determine the mode shapes and natural frequencies of the damaged structure. The geometric discontinuity at the damage location manifests itself in terms of discontinuities in the cross- sectional properties, such as the depth of the structure, the cross-sectional area or the area moment of inertia.
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Static representation of a parallel robot is often akin to that of a pin-jointed truss: the links and their actuators feel only tension or compression, without any bending or torque, which again reduces the effects of any flexibility to off-axis forces. A further advantage of the parallel manipulator is that the heavy actuators may often be centrally mounted on a single base platform, the movement of the arm taking place through struts and joints alone. This reduction in mass along the arm permits a lighter arm construction, thus lighter actuators and faster movements. This centralisation of mass also reduces the robot's overall moment of inertia, which may be an advantage for a mobile or walking robot.
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In the absence of external torques, the vector of the angular momentum M of a rotating system remains constant and is directed toward a fixed point in space. If the earth were perfectly symmetrical and rigid, M would remain aligned with its axis of symmetry, which would also be its axis of rotation. In the case of the Earth, it is almost identical with its axis of rotation, with the discrepancy due to shifts of mass on the planet's surface. The vector of the figure axis F of the system (or maximum principal axis, the axis which yields the largest value of moment of inertia) wobbles around M. This motion is called Euler's free nutation.
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There exist, in addition, polar motions with smaller periods of the order of decades. Finally, a secular polar drift of about 0.10 m per year in the direction of 80° west has been observed which is due to mass redistribution within the Earth's interior by continental drift, and/or slow motions within mantle and core which gives rise to changes of the moment of inertia. The annual variation was discovered by Karl Friedrich Küstner in 1885 by exact measurements of the variation of the latitude of stars, while S.C. Chandler found the free nutation in 1891. Both periods superpose, giving rise to a beat frequency with a period of about 5 to 8 years (see Figure 1).
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By contrast, if a beam's weight is fixed, its cross-sectional dimensions are unconstrained, and increased stiffness is the primary goal, the performance of the beam will depend on Young's modulus divided by either density squared or cubed. This is because a beam's overall stiffness, and thus its resistance to Euler buckling when subjected to an axial load and to deflection when subjected to a bending moment, is directly proportional to both the Young's modulus of the beam's material and the second moment of area (area moment of inertia) of the beam. Comparing the list of area moments of inertia with formulas for area gives the appropriate relationship for beams of various configurations.
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Alternatively, animals and aircraft which depend on maneuverability (fighters, predators and the preyed upon, and those who live amongst trees and bushes, insect catchers, etc.) need to be able to roll fast to turn, and the high moment of inertia of long narrow wings, as well as the high angular drag and quick balancing of aileron lift with wing lift at a low rotation rate, produces lower roll rates. For them, short-span, broad wings are preferred. Additionally, ground handling in aircraft is a significant problem for very high aspect ratios and flying animals may encounter similar issues. The highest aspect ratio man-made wings are aircraft propellers, in their most extreme form as helicopter rotors.
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As the core of a massive star is compressed during a Type II supernova or a Type Ib or Type Ic supernova, and collapses into a neutron star, it retains most of its angular momentum. But, because it has only a tiny fraction of its parent's radius (and therefore its moment of inertia is sharply reduced), a neutron star is formed with very high rotation speed, and then over a very long period it slows. Neutron stars are known that have rotation periods from about 1.4 ms to 30 s. The neutron star's density also gives it very high surface gravity, with typical values ranging from 1012 to 1013 m/s2 (more than 1011 times that of Earth).
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MF layout In automotive design, a Front Mid-engine, Front-wheel-drive layout (sometimes called FMF or just MF) is one in which the front road wheels are driven by an internal-combustion engine placed just behind them, in front of the passenger compartment. In contrast to the Front-engine, front-wheel-drive layout (FF), the center of mass of the engine is behind the front axle. This layout is typically chosen for its better weight distribution (the heaviest component is near the center of the car, lowering its moment of inertia). Since the differences between the FF and MF layouts are minor, most people consider the MF layout to be the same as the FF layout.
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The car was constructed with aluminium panels over a tubular steel spaceframe, as was typical in F1 at the time,Ferrari official website retrieved 21 May 2009 but featured a large number of new design features, the most interesting of which was the transverse-mounted gearbox – the T in the car's name stood for Trasversale. The gearbox design allowed it to be positioned ahead of the rear axle, in order to give a low polar moment of inertia. The suspension was also significantly different from that of the 312B3, and the front of the chassis was much narrower. The handling of the car was found to be inherently neutral, not suffering from the persistent understeer which blighted the 312B3.
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Angular position servo and signal flow graph. θC = desired angle command, θL = actual load angle, KP = position loop gain, VωC = velocity command, VωM = motor velocity sense voltage, KV = velocity loop gain, VIC = current command, VIM = current sense voltage, KC = current loop gain, VA = power amplifier output voltage, LM = motor inductance, IM = motor current, RM = motor resistance, RS = current sense resistance, KM = motor torque constant (Nm/amp), T = torque, M = moment of inertia of all rotating components α = angular acceleration, ω = angular velocity, β = mechanical damping, GM = motor back EMF constant, GT = tachometer conversion gain constant,. There is one forward path (shown in a different color) and six feedback loops. The drive shaft assumed to be stiff enough to not treat as a spring.
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A year later yet another paper claimed that the moon may not be in hydrostatic equilibrium meaning that the moment of inertia can not be determined from the gravity data alone. In 2008 an author of the first paper tried to reconcile these three disparate results. He concluded that there is a systematic error in the Cassini radio Doppler data used in the analysis, but after restricting the analysis to a subset of data obtained closest to the moon, he arrived at his old result that Rhea was in hydrostatic equilibrium and had the moment inertia of about 0.4, again implying a homogeneous interior. The triaxial shape of Rhea is consistent with a homogeneous body in hydrostatic equilibrium rotating at Rhea's angular velocity.
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The fourth part of the book is concerned with the study of the center of oscillation. The derivations of propositions in this part is based on a single assumption: that the center of gravity of heavy objects cannot lift itself, which Huygens used as a virtual work principle. In the process, Huygens obtained solutions to dynamical problems such as the period of an oscillating pendulum as well as a compound pendulum, center of oscillation and its interchangeability with the pivot point, and the concept of moment of inertia. gives a detailed description of Huygen's methods The last part of the book gives propositions regarding bodies in uniform circular motion, without proof, and states the laws of centrifugal force for uniform circular motion.
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More formally, this follows from the properties of the vector product, and shows that rotational effect of the force depends only on the position of its line of application, and not on the particular choice of the point of application along that line. The torque vector is perpendicular to the plane defined by the force and the vector \scriptstyle \vec r , and in this example it is directed towards the observer; the angular acceleration vector has the same direction. The right hand rule relates this direction to the clockwise or counter-clockwise rotation in the plane of the drawing. The moment of inertia \scriptstyle I is calculated with respect to the axis through the center of mass that is parallel with the torque.
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After spinning upright (in the so-called "sleep" position) for an extended period, the angular momentum will gradually lessen (mainly due to friction), leading to ever increasing precession, finally causing the top to topple and roll some distance on its side. In the "sleep" period, and only in it, provided it is ever reached, less friction means longer "sleep" time (whence the common error that less friction implies longer global spinning time). The total spinning time of a top is generally increased by increasing its moment of inertia and lowering its center of gravity.. These variables however are constrained by the need to prevent the body from touching the ground. Asymmetric tops of virtually any shape can also be created and designed to balance.
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If a ship floods, the loss of stability is caused by the increase in KB, the centre of buoyancy, and the loss of waterplane area - thus a loss of the waterplane moment of inertia - which decreases the metacentric height. This additional mass will also reduce freeboard (distance from water to the deck) and the ship's angle of down flooding (minimum angle of heel at which water will be able to flow into the hull). The range of positive stability will be reduced to the angle of down flooding resulting in a reduced righting lever. When the vessel is inclined, the fluid in the flooded volume will move to the lower side, shifting its centre of gravity toward the list, further extending the heeling force.
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The appeal of the 404 stems from its aerodynamics and a low rim weight (resulting in a low moment of inertia) which make it a versatile wheel for flat and hilly terrain. In 2011 Zipp released a new line of its long-awaited carbon clincher wheelsets. The 303, 404, and 808 Firecrest carbon clinchers were designed to provide decreased aerodynamic drag and increased stability in crosswinds as well as improved braking performance on long descents. The revolutionary Firecrest rim shape, with its wide profile and flat, U-shaped spoke bed, was quickly copied throughout the industry. The same year Zipp also discontinued its 404 and 808 aluminum/carbon hybrid clinchers and retooled its Indianapolis factory to produce only carbon wheels.
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Jacques Philippe Marie Binet (; 2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical foundations of matrix algebra which would later lead to important contributions by Cayley and others. In his memoir on the theory of the conjugate axis and of the moment of inertia of bodies he enumerated the principle now known as Binet's theorem. He is also recognized as the first to describe the rule for multiplying matrices in 1812, and Binet's Formula expressing Fibonacci numbers in closed form is named in his honour, although the same result was known to Abraham de Moivre a century earlier.
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Chassis 353, Graham Hill's 1958 Monaco Grand Prix car The Lotus Twelve was the first to use the infamous Lotus 'Queerbox' transaxle. This was developed to be, in typical Colin Chapman fashion, the smallest and lightest five-speed transmission possible, also to have a low driveshaft line allowing a low driving position, thus lower centre of mass and air resistance. Chapman also chose a transaxle over the usual gearbox and rear axle layout, as had been used in the first Twelve, as this gave a lower polar moment of inertia. The initial design, the work of Chapman and Harry Mundy, began with the principle of the most compact layout, with the gear cluster arranged in a closely spaced stack, akin to a motorcycle transmission.
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Similarly, inverse dynamics in biomechanics computes the net turning effect of all the anatomical structures across a joint, in particular the muscles and ligaments, necessary to produce the observed motions of the joint. These moments of force may then be used to compute the amount of mechanical work performed by that moment of force. Each moment of force can perform positive work to increase the speed and/or height of the body or perform negative work to decrease the speed and/or height of the body. The equations of motion necessary for these computations are based on Newtonian mechanics, specifically the Newton–Euler equations of: : Force equal mass times linear acceleration, and : Moment equals mass moment of inertia times angular acceleration.
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A 2009 study estimated that ankylosaurids could swing their tails at 100 degrees laterally, and the mainly cancellous clubs would have had a lowered moment of inertia and been effective weapons. The study also found that while adult ankylosaurid tail clubs were capable of breaking bones, those of juveniles were not. Despite the feasibility of tail-swinging, the researchers could not determine whether ankylosaurids used their clubs for defense against potential predators, in intraspecific combat, or both. In 1993 Tony Thulborn proposed that the tail club of ankylosaurids primarily acted as a decoy for the head, as he thought the tail too short and inflexible to have an effective reach; the "dummy head" would lure a predator close to the tail, where it could be struck.
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The key to completing higher-rotation jumps is how a skater controls the moment of inertia. Since the tendency of an edge is toward the center of the circle created by that edge, a skater's upper body, arms, and free leg also have a tendency to be pulled along by the force of the edge. If the upper body, arms, and free leg are allowed to follow passively, they will eventually overtake the edge's rotational edge and will rotate faster, a principle that is also used to create faster spins. The inherent force of the edge and the force generated by a skater's upper body, arms, and free leg tend to increase rotation, so successful jumping requires precise control of these forces.
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If the balance could be made to shrink in diameter as it got warmer, the smaller moment of inertia would compensate for the weakening of the balance spring, keeping the period of oscillation the same. To accomplish this, the outer rim of the balance was made of a ‘sandwich’ of two metals; a layer of steel on the inside fused to a layer of brass on the outside. Strips of this bimetallic construction bend toward the steel side when they are warmed, because the thermal expansion of brass is greater than steel. The rim was cut open at two points next to the spokes of the wheel, so it resembled an S-shape (see figure) with two circular bimetallic ‘arms’.
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Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be simple pendulums or compound pendulums (also called complex pendulums) and the motion may be in three dimensions or restricted to the vertical plane. In the following analysis, the limbs are taken to be identical compound pendulums of length and mass , and the motion is restricted to two dimensions. Double compound pendulum Motion of the double compound pendulum (from numerical integration of the equations of motion) Trajectories of a double pendulum In a compound pendulum, the mass is distributed along its length. If the mass is evenly distributed, then the center of mass of each limb is at its midpoint, and the limb has a moment of inertia of about that point.
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The mass of an object of known density, the moment of inertia of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus. An example of the use of calculus in mechanics is Newton's second law of motion: historically stated it expressly uses the term "change of motion" which implies the derivative saying The change of momentum of a body is equal to the resultant force acting on the body and is in the same direction. Commonly expressed today as Force = Mass × acceleration, it implies differential calculus because acceleration is the time derivative of velocity or second time derivative of trajectory or spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.
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The radius of any such core cannot exceed 600 km, and the density may lie between 3.1 and 3.6 g/cm3. In this case, Callisto's interior would be in stark contrast to that of Ganymede, which appears to be fully differentiated. However, a 2011 reanalysis of Galileo data suggests that Callisto is not in hydrostatic equilibrium; its S22 coefficient from gravity data is an anomalous 10% of its C22 value, which is not consistent with a body in hydrostatic equilibrium and thus significantly increases the error bars on Callisto's moment of inertia. Further, an undifferentiated Callisto is inconsistent with the presence of a substantial internal ocean as inferred by magnetic data, and it would be difficult for an object as large as Callisto to fail to differentiate at any point.
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The period depends on the length of the pendulum, and also to a slight degree on its weight distribution (the moment of inertia about its own center of mass) and the amplitude (width) of the pendulum's swing. For a point mass on a weightless string of length L swinging with an infinitesimally small amplitude, without resistance, the length of the string of a seconds pendulum is equal to L = g/π2 where g is the acceleration due to gravity, with units of length per second squared, and L is the length of the string in the same units. Using the SI recommended acceleration due to gravity of g0 = 9.80665 m/s2, the length of the string will be approximately 993.6 millimeters, i.e. less than a centimeter short of one meter everywhere on Earth.
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A muscle back is the more traditional design and consists of a solid metal head, typically made of forged iron. The design of the club typically distributes the metal more evenly around the clubhead (though most designs still place more weight along the sole of the club), which makes the center of mass of the club higher and the moment of inertia (the clubhead's resistance to rotation) lower as compared to newer cavity-backed designs. As such, these clubs are said to have a smaller "sweet spot", requiring greater skill and a more consistent swing to make accurate, straight shots. Novice golfers with less consistent swing fundamentals can easily mis-hit these clubs, causing shots to launch or curve off of the intended line of play (such as "pushing", "pulling", "slicing" or "hooking").
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Activating the decompression lever locks the outlet valves in a slight down position, resulting in the engine not having any compression and thus allowing for turning the crankshaft over without resistance. When the crankshaft reaches a higher speed, flipping the decompression lever back into its normal position will abruptly re-activate the outlet valves, resulting in compression − the flywheel's mass moment of inertia then starts the engine. Other diesel engines, such as the precombustion chamber engine XII Jv 170/240 made by Ganz & Co., have a valve timing changing system that is operated by adjusting the inlet valve camshaft, moving it into a slight "late" position. This will make the inlet valves open with a delay, forcing the inlet air to heat up when entering the combustion chamber.
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Therefore, the front tires have a more difficult time overcoming the car's moment of inertia during corner entry at low speed, and much less difficulty as the cornering speed increases. So the natural tendency of any car is to understeer on entry to low-speed corners and oversteer on entry to high-speed corners. To compensate for this unavoidable effect, car designers often bias the car's handling toward less corner-entry understeer (such as by lowering the front roll center), and add rearward bias to the aerodynamic downforce to compensate in higher-speed corners. The rearward aerodynamic bias may be achieved by an airfoil or "spoiler" mounted near the rear of the car, but a useful effect can also be achieved by careful shaping of the body as a whole, particularly the aft areas.
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The gravity of the Earth decreases according to the inverse-square law, and by extending the long axis perpendicular to the orbit, the "lower" part of the orbiting structure will be more attracted to the Earth. The effect is that the satellite will tend to align its axis of minimum moment of inertia vertically. The first experimental attempt to use the technique on a human spaceflight was performed on September 13, 1966, on the US Gemini 11 mission, by attaching the Gemini spacecraft to its Agena target vehicle by a tether. The attempt was a failure, as insufficient gradient was produced to keep the tether taut. The technique was first successfully used in a near-geosynchronous orbit on the Department of Defense Gravity Experiment (DODGE) satellite in July 1967.
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Similarly, though manufacturer literature often describes specific core shapes, differently-shaped cores can make exactly the same contribution to ball motion if they have the same overall RG characteristics. "Weak" layouts ("pin down": pin between finger and thumb holes) hook sooner but have milder backend reaction, while "strong" layouts ("pin up": pin further from thumb hole than finger holes) enable greater skid lengths and more angular backend reaction. Manufacturers commonly cite specifications relating to a bowling ball's core, include radius of gyration (RG), differential of RG (commonly abbreviated differential), and intermediate differential (also called mass bias). Analytically, the United States Bowling Congress defines RG as "the distance from the axis of rotation at which the total mass of a body might be concentrated without changing its moment of inertia".
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An increase in temperature would actually make a spring stronger if it affected only its physical dimensions. However, a much larger effect in a balance spring made of plain steel is that the elasticity of the spring's metal decreases significantly as the temperature increases, the net effect being that a plain steel spring becomes weaker with increasing temperature. An increase in temperature also increases diameter of a steel or brass balance wheel, increasing its rotational inertia, its moment of inertia, making it harder for the balance spring to accelerate. The two effects of increasing temperature on physical dimensions of the spring and the balance, the strengthening of the balance spring and the increase in rotational inertia of the balance, have opposing effects and to an extent cancel each other.
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Cavity back style iron Cavity back, or perimeter weighted, irons are usually made by investment casting, which creates a harder metal allowing thinner surfaces while retaining durability, and also allows for more precise placement of metal than forging techniques. Cavity backs are so called because of the cavity created in the rear of the clubhead due to the removal of metal from the center of the clubhead's back, which is then redistributed, most of it very low and towards the toe and heel of the clubhead. This has the general effect of lowering the clubhead's center of mass, placing it underneath that of the ball allowing for a higher launch angle for a given loft. The perimeter weighting also increases the moment of inertia, making the clubhead more resistant to twisting on impact with the ball.
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Indirect evidence for the giant impact scenario comes from rocks collected during the Apollo Moon landings, which show oxygen isotope ratios nearly identical to those of Earth. The highly anorthositic composition of the lunar crust, as well as the existence of KREEP-rich samples, suggest that a large portion of the Moon once was molten; and a giant impact scenario could easily have supplied the energy needed to form such a magma ocean. Several lines of evidence show that if the Moon has an iron-rich core, it must be a small one. In particular, the mean density, moment of inertia, rotational signature, and magnetic induction response of the Moon all suggest that the radius of its core is less than about 25% the radius of the Moon, in contrast to about 50% for most of the other terrestrial bodies.
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Dynamic balance requires that a principal axis of the tire's moment of inertia be aligned with the axis about which the tire rotates, usually the axle on which it is mounted. In the tire factory, the tire and wheel are mounted on a balancing machine test wheel, the assembly is rotated at 100 RPM (10 to 15 mph with recent high sensitivity sensors) or higher, 300 RPM (55 to 60 mph with typical low sensitivity sensors), and forces of unbalance are measured by sensors. These forces are resolved into static and couple values for the inner and outer planes of the wheel, and compared to the unbalance tolerance (the maximum allowable manufacturing limits). If the tire is not checked, it has the potential to cause vibration in the suspension of the vehicle on which it is mounted.
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Discrete calculus is used for modeling either directly or indirectly as a discretization of infinitesimal calculus in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other. Physics makes particular use of calculus; all discrete concepts in classical mechanics and electromagnetism are related through discrete calculus. The mass of an object of known density that varies incrementally, the moment of inertia of such objects, as well as the total energy of an object within a discrete conservative field can be found by the use of discrete calculus.
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The Alfa Romeo Alfetta (Tipo 116, or "Type 116") is a front-engine, five- passenger sedan and fastback coupé manufactured and marketed by Alfa Romeo from 1972–1987 with a production total over 400,000. The Alfetta was noted for the rear position of its transaxle (clutch and transmission) and its De Dion tube rear suspension -- an arrangement designed to optimize handling by balancing front/rear weight distribution, as well as maintaining a low polar moment of inertia and low center of gravity. The interior of Coupé models featured a then unusual central tachometer placement -- by itself, directly in front of the driver. The Alfetta name, which means "little Alfa" in Italian, derived from the nickname of the Alfa Romeo Tipo 159 Alfetta, a successful Formula One car which in its last (1951) iteration paired a transaxle layout to De Dion tube rear suspension -- like its modern namesake.
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Petkevich, p. 144 It is executed in a sitting position, with the knee of the skating leg bent and the free leg held in front. It is difficult to learn, requires a great deal of energy, and is not as exciting to perform as other elements, such as jumps, but it has variations that make it more creative and pleasurable to watch.Petkevich, p. 148 When executing the sit spin, a skater's back should be straight and not curved, his or her hips should be lower than the skating knee, and his or her free leg should be straight. The best sit spin position minimizes the moment of inertia and keeps the heaviest parts of the body as close to the vertical center of gravity as possible. This position is difficult to maintain, however, so skaters will often collapse into a low sit spin position.
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What does this combine with, if anything? There is another vector quantity not often discussed – it is the time-varying moment of mass polar-vector (not the moment of inertia) related to the boost of the centre of mass of the system, and this combines with the classical angular momentum pseudovector to form an antisymmetric tensor of second order, in exactly the same way as the electric field polar-vector combines with the magnetic field pseudovector to form the electromagnetic field antisymmetric tensor. For rotating mass–energy distributions (such as gyroscopes, planets, stars, and black holes) instead of point-like particles, the angular momentum tensor is expressed in terms of the stress–energy tensor of the rotating object. In special relativity alone, in the rest frame of a spinning object, there is an intrinsic angular momentum analogous to the "spin" in quantum mechanics and relativistic quantum mechanics, although for an extended body rather than a point particle.
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Steve Randle, who was the car's dynamicist, was appointed responsible for the design of the suspension system of the McLaren F1. It was decided that the ride should be comfortable yet performance-oriented, but not as stiff and low as that of a true track machine, as that would imply reduction in practical use and comfort as well as increasing noise and vibration, which would be a contradictory design choice in relation to the former set premise – the goal of creating the ultimate road car. From inception, the design of the F1 had a strong focus on adjusting the mass of the car as near the middle as possible by extensive manipulation of placement of, among other things, the engine, fuel and driver, allowing for a low polar moment of inertia in yaw. The F1 has 42% of its weight at the front and 58% at the rear, this figure changes less than 1% with the fuel load.
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Through attempts to lower the center of gravity of the club head, it evolved into a shorter, thicker head slightly curved from front to rear (the so-called "hot dog" putter). The introduction of investment casting for club heads allowed drastically different shapes to be made far more easily and cheaply than with forging, resulting in several design improvements. First of all, the majority of mass behind the clubface was placed as low as possible, resulting in an L-shaped side profile with a thin, flat club face and another thin block along the bottom of the club behind the face. Additionally, peripheral weighting, or the placing of mass as far away from the center of the clubface as possible, increases the moment of inertia of the club head, reducing twisting if the club contacts the ball slightly off-center and thus giving the club a larger "sweet spot" with which to contact the ball.
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In economics, Saari has shown that natural price mechanisms that set the rate of change of the price of a commodity proportional to its excess demand can lead to chaotic behavior rather than converging to an economic equilibrium, and has exhibited alternative price mechanisms that can be guaranteed to converge. However, as he also showed, such mechanisms require that the change in price be determined as a function of the whole system of prices and demands, rather than being reducible to a computation over pairs of commodities. In celestial mechanics, Saari's work on the -body problem "revived the singularity theory" of Henri Poincaré and Paul Painlevé, and proved Littlewood's conjecture that the initial conditions leading to collisions have measure zero. He also formulated the "Saari conjecture", that when a solution to the Newtonian -body problem has an unchanging moment of inertia relative to its center of mass, its bodies must be in relative equilibrium.
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The major advantage of MR - low moment of inertia - is negated somewhat (though still lower than FR), and there is more room for passengers and cargo (though usually less than FR). Furthermore, because both axles are on the same side of the engine, it is technically more straightforward to drive all four wheels, than in a mid-engined configuration (though there have been more high-performance cars with the M4 layout than with R4). Finally, a rear-mounted engine has empty air (often at a lower pressure) behind it when moving, allowing more efficient cooling for air- cooled vehicles (more of which have been RR than liquid-cooled, such as the Volkswagen Beetle, and one of the few production air-cooled turbocharged cars, the Porsche 930). For liquid-cooled vehicles, however, this layout presents a disadvantage, since it requires either increased coolant piping from a front- mounted radiator (meaning more weight and complexity), or relocating the radiator(s) to the sides or rear, and adding air ducting to compensate for the lower airflow at the rear of the car.
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193 Scientist James Richards from the University of Delaware stated that successful jumps depend upon "how much angular momentum do you leave the ice with, how small can you make your moment of inertia in the air, and how much time you can spend in the air". He found that many skaters, although they were able to gain the necessary angular momentum for takeoff, had difficulty gaining enough rotational speed to complete the jump. For example, a skater could successfully complete a jump by making small changes to his or her arm position partway through the rotation, and a small bend in the hips and knees allows a skater "to land with a lower center of mass than they started with, perhaps eking out a few precious degrees of rotation and a better body position for landing". A skater tends to spend the same amount of time in the air regardless of whether he or she completes triple or quadruple jumps, but his or her angular momentum at the start of triples and quadruples is slightly higher than it is for double jumps.
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