Figure shows a pulley of mass m and radius r with two blocks of masses `m_(1)` and `m_(2)` attached with a light and unstretchable string. Find the acceleration of the blocks, tensions in the string and the force exerted by the pulley on the celling from which it is hanging. Assume no slipping between the string and the wheel.

Explanation

From the free - body diagram of the mass `m_(1)` [Fig. (a)]
`T_(1)-m_(1)g=m_(1)a " "` ….(i)
From the free - body diagram of the mass `m_(2)`, [Fig. (b)]
`m_(2)g-T_(2)=m_(2)a " "` ….(ii)
Considering the rotational motional of the pulley, the torque on it is.
`T_(2)r - T_(1)r=l(a)/(r ) " "` [Fig (c )]
or, `T_(2)-T_(3)=l(a)/(r ) " "` ...(iii)

Adding (i), (ii) and (iii), we get
`a=((m_(2)-m_(1))/(m_(1)+m_(2)+(l)/(r^(2))))g " "` ...(iv)
and hence from (i)
`T_(1)=(m_(1)(2m_(2)+(l)/(r^(2)))g)/(m_(1)+m_(2)+(l)/(r^(2))) " "` ...(v)

`T_(2)=(m_(2)(2m_(1)(l)/(r^(2)))g)/(m_(1)+m_(2)+(l)/(r^(2))) " "` ...(vi)
For a pulley, take `l = (1)/(2)mr^(2)` or `(l)/(r^(2))=(m)/(2)`.