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Tilted Rigid Disk Rolling Without Slipping

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A rigid body in the shape of a thumbtack formed from a thin disk of mass M and radius a and a mass-less stem is placed on an inclined plane that makes an angle α with the horizontal. The point of the tack remains stationary at the point α, and the head rolls along a circle of radius b.

Introduce a set of laboratory coordinates whose 3° axis is perpendicular to the inclined plane and whose 2° axis points down the plane, as well as a set of body-associated principal axes with origin at the center of mass, whose 3 axis is perpendicular to the head of the tack pointing outward, whose 2 axis passes through the point of contact with the plane, and whose 1 axis is parallel to the surface and tangent to the circle. Introduce also the set of angles (θ, Φ, γ) that specify the orientation of the tack, as indicated in Fig. P5.4.

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Solution Preview

Angular velocities with respect to three principal axes (1,2,3,) in terms of Euler angles are given as:

w1 = phi_dot * sin(theta)*sin(gamma) + theta_dot * cos(gamma)
w2 = phi_dot * sin(theta)*cos(gamma) - theta_dot * sin(gamma)
w3 = phi_dot * cos(theta) + gamma_dot

Here point to be noted that, these equations are corresponding to theta between 3 and 3o, while in the ...

Solution Summary

A rigid body rotational mechanics problem is solved to help a candidate, unable proceed.