The friction coefficient between the two blocks shown in figure is `mu` and the horizontal plane is smoth. (a) if the systeam is slightly displacement and released, find the time period. (b) Find the magnitude of the fricational force between the blocks when the displacement from the fmean position is `x`. (c) What can be the maximum amplitude if the upper block does not slip relative to the lower block ?

Explanation

(a) For small amplitude, the two blocks oscillate together. The angular requency is `omega = sqrt((k)/(M + m))` and so the time period `T = 2pi sqrt((M + m)/(k))`
(b) The acceleration of the blocks at displacement `x` from the mean position is
`a = -omega^(2)x = ((-kx)/(M + m))`
The resultant force on the upper block is, therefore, `ma = ((-mkx)/(M + m))`
This force is provided by the frication of the lower block. Hence, the magnitude of the frication force is `(mk |x|)/(M + m)`
(c) Maximum force of frication required for simple harmonic motion of the upper block is `(mk_(A))/(M + m)` as the extreme positions. But the maximum frication force can only be `mu mg`. Hence, `(mkA)/(M + m) = mu mg` or `A = (mu(M + m)g)/(k)`