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Question:

A mass m hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass m and radius R. Assuming the pulley to be a perfect uniform circular disc, the acceleration of the mass m, if the string does not slip on the pulley, is :

3/2g

g

g/3

2/3g

Solution:

Equations of motion are:
mg - T = ma (i)
T ⋅ R = 1/2mR2 α (ii)
and
a = Rα (iii)
From (ii) and (iii), we get:
T = 1/2 m a
Substituting this value of T in (i), we get:
mg - 1/2 ma = ma
mg = 3/2 ma
a = 2/3 g