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Consider a rigid body of arbitrary shape that is attached at its mass center $O$ and subjected to no force other than its weight and the reaction of the support at $O$(a) Prove that the angular momentum $\mathbf{H}_{o}$ of the body about the fixed point $O$ is constant in magnitude and direction, that the kinetic energy $T$ of the body is constant, and that the projection along $\mathbf{H}_{O}$ of the angular velocity $\omega$ of the body is constant.(b) Show that the tip of the vector $\omega$ describes a curve on a fixed plane in space (called the invariable plane), which is perpendicular to $\mathbf{H}_{o}$ and at a distance $2 T / H_{o}$ from $O$$(c)$ Show that with respect to a frame of reference attached to the body and coinciding with its principal axes of inertia, the tip of the vector $\omega$ appears to describe a curve on an ellipsoid of equation\[I_{x} \omega_{x}^{2}+I_{y} \omega_{y}^{2}+I_{z} \omega_{z}^{2}=2 T=\mathrm{constant}\]The ellipsoid (called the Poinsot ellipsoid) is rigidly attached to the body and is of the same shape as the ellipsoid of inertia, but of a different size.

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Physics 101 Mechanics

Chapter 18

Kinetics of Rigid Bodies in Three Dimensions

Section 3

Motion of a Gyroscope

Rotation of Rigid Bodies

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

University of Washington

Simon Fraser University

Lectures

02:34

In physics, a rigid body is an object that is not deformed by the stress of external forces. The term "rigid body" is used in the context of classical mechanics, where it refers to a body that has no degrees of freedom and is completely described by its position and the forces applied to it. A rigid body is a special case of a solid body, and is one type of spatial body. The term "rigid body" is also used in the context of continuum mechanics, where it refers to a solid body that is deformed by external forces, but does not change in volume. In continuum mechanics, a rigid body is a continuous body that has no internal degrees of freedom. The term "rigid body" is also used in the context of quantum mechanics, where it refers to a body that cannot be squeezed into a smaller volume without changing its shape.

06:08

The angular velocity of a rigid body is the rate of change of the angular displacement relative to time. In SI units, it is measured in radians per second.

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04:39

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