A wheel is mounted on frictionless central pivot and it can rotate freely in the vertical plane. There is a horizontal light rod fixed to the wheel below the pivot. There is a small sleeve of mass m which can slide along the rod without friction. The sleeve is connected to a light spring. The other end of the spring is fixed to the rim as shown. The sleeve is at the centre of the rod and the spring is relaxed. Now the wheel is held at rest and the sleeve is moved towards left so as to compress the sprig by some distance. The sleeve and the wheel are released simultaneously from this position.
(a) Is it possible that the wheel does not rotate as the sleeve perform SHM on the rod ?
(b) Find the value of spring constant `k` for situation described in (a) to be possible. The distance of rod from centre of the wheel is `d`.

Answer

(a) yes
(b) `k = (mg)/(d)`