Question

A uniform disc of radius r rolls without slipping inside a circular track determine the natural frequency of the oscillation

Answer 1

Answer:

uniform disk of radius 'r' rolls without slipping inside a circular track of radoius, R, as shown in the figure below. Note that 'm' is the mass of the disk, I=1/2mr^2 is the mass moment of inertia of the disk about its mass center, 'v' is the translational velocity of the disk center, and 'w' is the angular velocity of the disk.

1)Calculate the total energy 'E' of the disk. Assume that the zero potential energy surface is at the lowest point of the circular track.

2)Derive the dynamic equation of motion using the energy method for small angular motion in the coordinate θ.

3)Calculate the natural frequency of the system

4)Derive the dynamic equations of motion using Newton's laws of motion in the coordinate θ

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