(Solved): [Practice 6] Small balls 1 and 2 with a mass of 2m and m, respectively, are hung on either side of a ...

[Practice 6] Small balls 1 and 2 with a mass of

and

, respectively, are hung on either side of a pulley (inner ring radius

, outer ring radius

) with the moment of inertia

as shown in the figure. Initially, the difference in height between the two small balls is

and the ball 1 is supported by hand. (a) Once the ball 1 is released, the balls start moving and the pulley starts rotating. The acceleration of ball 1 and 2 is

and

, and tension of thread for ball 1 and 2 is

and

, respectively. Make the equation of motion for ball 1 and 2 , and the equation of rotational motion for the pulley. Also, express

and

using angular acceleration

of the pulley. (b) If

, find

, and

(c) Find the time

until balls 1 and 2 reach the same height.