A string is wrapped around a uniform solid cylinder of radius r , as shown in th

Question

A string is wrapped around a uniform solid cylinder of radius r, as shown in the figure (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. Note that the positive y direction is downward and counterclockwise torques are positive.

Find the magnitude ? 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.

Explanation / Answer

Summing forces,

m g - T = m a   [1]

Summing torques,

T r = I a / r

As I = 1/2 m r^2,

T = 1/2 m a   [2]

Plugging this into equation 1 then solving for a,

a = (2/3) g

Thus,

alpha = a/r = [2g]/[3r]   [ANSWER]

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