[Practice 6] Small balls 1 and 2 with a mass of
2m
and
m
, respectively, are hung on either side of a pulley (inner ring radius
2R
, outer ring radius
4R
) with the moment of inertia
I
as shown in the figure. Initially, the difference in height between the two small balls is
L
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
a_(1)
and
a_(2)
, and tension of thread for ball 1 and 2 is
T_(1)
and
T_(2)
, respectively. Make the equation of motion for ball 1 and 2 , and the equation of rotational motion for the pulley. Also, express
a_(1)
and
a_(2)
using angular acceleration
\alpha
of the pulley. (b) If
I=6mR^(2)
, find
a_(1),a_(2),T_(1),T_(2)
, and
\alpha
(c) Find the time
t
until balls 1 and 2 reach the same height.