The friction coeficeint betweenteh tow blocks shown in figure is `mu` and the horizontal plance is smooth. A. If the system is slightly diplaced and released find the time period. B. find the magnitude of the frictional force between the blocks when the displacement from the mean 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 oscillation, the blocks move together with time period
`T=2pisqrt((M+m)/(k))`
`omega=(2pi)/(T)=sqrt((k)/(M+m))`
(b) When displacement from mean position is x
`|a|=omega^2x=(kx)/(M+m)`
Resultant force on upper block
`m|a|=(mkx)/(M+m)`
This force is provided by the friction acting on upper block
Friction `f=(mkx)/(M+m)`
(c) Maximum value of friction of SHM of upper block
`=(mkA)/(M+m)`, A: amplitude
Maximum value of friction available `=mumg`
`(mkA)/(M+m)lemumg`
`Ale(mu(M+m)g)/(k)`
`A_(max)=(mu(M+m)g)/(k)`