Question

A string is wrapped around a uniform solid cylinder of radius r, as shown in (Figure 1). The cylinder can rotate freely about its axis. The loose end of the string is attached to a block. The block and cylinder each have mass m. Find the magnitude alpha of the angular acceleration of the cylinder as the block descends. Express your answer in terms of the cylinder's radius r and the magnitude of the acceleration due to gravity g.

Answer 1

Force balance on block

mg - T = m*a

T = tension in rope

a = acceleration of block

Torque balance around cylinder

Torque = I*alpha = T*r

Icylinder = 0.5*m*r^2

alpha = a/r

0.5*m*r^2*a/r = T*r

T = 0.5*m*a

Using above value

mg - T = m*a

mg - 0.5*m*a = m*a

m*g = 1.5*m*a

a = 2*g/3

Angular acceleration will be

alpha = a/r = 2*g/(3*r)