(i) In the shown arrangement, both springs are relaxed. The coefficient of friction between `m_(2) " and " m_(1)` is `mu`. There is no friction between `m_(1)` and the horizontal surface. The blocks are displaced slightly and released. They move together without slipping on each other.
(a) If the small displacement of blocks is x then find the magnitude of acceleration of `m_(2)`. What is time period of oscillations ?
(b) Find the ratio `(m_(1))/(m_(2))` so that the frictional force on `m_(2)` acts in the direction of its displacement from the mean position.
(ii) Two small blocks of same mass m are connected to two identical springs as shown in fig. Both springs have stiffness K and they are in their natural length when the blocks are at point O. Both the blocks are pushed so that one of the springs get compressed by a distance a and the other by `a//2`. Both the blocks are released from this position simultaneously. Find the time period of oscillations of the blocks if - (neglect the dimensions of the blocks)
(a) Collisions between them are elastic.
(b) Collisions between them are perfectly inelastic.

Answer

(a) ` a=((k_(1)+k_(2))/(m_(1)+m_(2)))x ; T=2pisqrt((m_(1)+m_(2))/(k_(1)+k_(2)))`
(b) `(m_(1))/(m_(2)) gt (k_(1))/(k_(2)); T=2pisqrt((m)/(k))` " for both the blocks in both cases".