A solid circular disk of mass M and radius R is supported at its rim by hinge an

Question

A solid circular disk of mass M and radius R is supported at its rim by hinge and hung vertically as shown in the figure below. It is initially rotated so that a line drawn between the hinge and the center of mass makes an angle theta_o with respect to the vertical, and released from rest. Assume the acceleration due to gravity is g and the moment of inertia of the disk about its center of mass is MR^2. Use the principles of conservation of energy to find the angular speed of the disk as it reaches an angle theta. Your answer should be in terms of the given quantities. Explain your calculation for full credit. Find the angular accerelation of the disk as it rotates, at the instant that it is located at the angle theta. Use dynamics to find the components of force, as-a function of the angle theta, that act at the hinge as the disk rotates. Break the vector into two components, a radial force, F_r, that acts along the line connecting the center of mass and the hinge, and a tangential force, F_t, that acts perpendicular to the line connecting the center of mass and the hinge.

Explanation / Answer

a)   Here,   according to conservation of energy

=>    1/2 * I * w2 = mgh * cos(theta)

=>    1/2 * (M * R2)* w2 = MgR * cos(theta)

=>    w = sqrt[2g * cos(theta)/R]           --------------> angular velocity

b)     angular acceleration of disk = (MgRsin(theta))/(M * R2)

                                                      = (gsin(theta))/(R)

c)   Here,    Fr   = mw2R

              Ft = N - mg * cos(theta)

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