A uniform solid disk of radius R and mass M is free to rotate on a frictionless

Question

A uniform solid disk of radius R and mass M is free to rotate on a frictionless pivot through a point on its rim (see figure below). The disk is released from rest in the position shown by the copper-colored circle.

(a) What is the speed of its center of mass when the disk reaches the position indicated by the dashed circle? (Use any variable or symbol stated above along with the following as necessary: g.)

vCM=2*(gR/3)

(b) What is the speed of the lowest point on the disk in the dashed position? (Use any variable or symbol stated above along with the following as necessary: g.)

vL=4*(gR/3)

(c) Repeat part (a) using a uniform hoop of mass M. (Use any variable or symbol stated above along with the following as necessary: g.)

vCM=?

Please help on part (c).

Explanation / Answer

you want only part C , there it is ,

I = Icm + md2

= mr2 + mr2   = 2mr2  

using energy conservation ,

mghcm = I2 /2

m x g x r = (2mr2 )2 /2

= g/r

velocity of center of masss = r = r x g/r = gr

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