In the shown arrangement, both the spring are in their natural lengths. The coefficient of friction between `m_(2)` and `m_(1)` is `mu`. There is no friction between `m_(1)` and the surface. If the blocks are displaced slightly, they together perform simple harmonic motion. Obtain

(a) Frequency of such oscillations.
(b) The condition if the friction force on clock `m_(2)` is to act in the direction of its displacement from mean position.
( c) If the condition obtained in (b) is met, what can be maximum of their oscillations ?

Answer

A, B, C

Explanation

(a) `f = (1)/(2pi) sqrt ((K_(eff))/("total mass")) = (1)/(2pi) sqrt ((k_(1) + k_(2))/(m_(1) + m_(2)))`
(b) Suppose the system is displaced towards left by a distance `x`.
Restoring force on `m_(1)` :

`F = m_(1)omega^(2)x` (towards right)
`= m_(1)((k_(1) + k_(2))/(m_(1) + m_(2)))x`
Friction `f` on it will be towards right if,
`k_(1)x lt F`
or `k_(1)x lt m_(1)((k_(1) + k_(2))/(m_(1) + m_(2)))x`
or `(k_(1))/(k_(2))lt (m_(1))/(m_(2))`
( c) `k_(1) A_(m) + mu m_(2)g = m_(1)((k_(1) + k_(2))/(m_(1) + m_(2)))A_(m)`
`A_(m)((m_(1)k_(1) + m_(1)k_(2))/(m_(1) + m_(2)) - k_(1)) = mu m_(2)g`
or `A_(m) = (mu(m_(1) + m_(2))m_(2)g)/(m_(1)k_(2) - m_(2)k_(1))`