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(a) Applying the principle of conservation of energy, derive a third differential equation for the general motion of the top of Prob. 18.137(b) Eliminating the derivatives $\phi$ and $\dot{\psi}$ from the equation obtained and from the two equations of Prob. 18.137 , show that the rate of nutation $\dot{\theta}$ is defined by the differential equation $\dot{\theta}^{2}=f(\theta)$ where\[f(\theta)=\frac{1}{I^{\prime}}\left(2 E-\frac{\beta^{2}}{I}-2 m g c \cos \theta\right)-\left(\frac{\alpha-\beta \cos \theta}{I^{\prime} \sin \theta}\right)^{2}\]$(c)$ Further show, by introducing the auxiliary variable $x=\cos \theta$ that the maximum and minimum values of $\theta$ can be obtained by solving the cubic equation for $x$\[\left(2 E-\frac{\beta^{2}}{I}-2 m g c x\right)\left(1-x^{2}\right)-\frac{1}{I^{\prime}}(\alpha-\beta x)^{2}=0\]

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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

Cornell University

Simon Fraser University

Hope College

McMaster 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.

03:40

Consider the damped forced…

04:15

The solution for damped si…

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