The friction coefficient between the two blocks shown in figure is `mu` and the horizontal plane 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 amplitude the tow blocks oscillate together. The angular frequency is
`omega=sqrt(k/(M+m))`
and so the time period `T=2pisqrt((M+m)/k)`.
b. The acceleration of the blocks at displacement x from the mean position is
`a=-omega^2x=(-kx)/(M+m)` ltbr. the resultant force on teh upper block is therefore
`ma=(-mkx)/(M+m)`
this force is provided by friction of the lower block. Hence the magnitude of the frictioN/Al force is `(mk|x|)/(M+m)`
c. Maximum force of friction required for simple harmonic motion of the upper block is `(mkA)/(M+m)` at the extreme positions. But the madximum frictioN/Al force can only be `mumg`. Hence
`(mkA)/(M+m)=mumg or A=(mu(M+m)g)/k`.