Two blocks of masses m and M, $$(M \gt m)$$, are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released then ($$\mu$$ = coefficient of friction between the two blocks)

(A) The time period of small oscillation of the two blocks is $$T = 2\pi\sqrt{\frac{m+M}{k}}$$
(B) The acceleration of the blocks is $$a = \frac{kx}{M+m}$$
(C) The magnitude of the frictional force on upper block is $$\frac{m\mu|x|}{M+m}$$
(D) The maximum amplitude of the upper block, if it does not slip, is $$\frac{\mu(M+m)g}{k}$$
(E) Maximum frictional force can be $$\mu(M+m)g$$.

Choose the correct answer from the options given below :

Treat both blocks as one system (if no slipping occurs).

Total mass

M+m

Restoring force by spring:

F=−kx

So equation of motion is

$$(M+m)a=-kx$$

Thus

$$a=-\frac{kx}{M+m}$$

So (B) is correct.

For SHM,

$$\omega=\sqrt{\frac{k}{M+m}}$$

Hence

$$T=2\pi\sqrt{\frac{M+m}{k}}$$

So (A) is correct.

For upper block m, only horizontal force is friction.

It must provide acceleration

$$a=\frac{kx}{M+m}$$

Therefore friction needed is

$$f=ma$$

$$f=\frac{mkx}{M+m}$$

So (C) is wrong 

For no slipping,

required friction must not exceed maximum static friction:

$$\frac{mkA}{M+m}\le\mu mg$$

Cancel m:

$$\frac{kA}{M+m}\le\mu g$$

Thus maximum amplitude

$$A_{\max}=\frac{\mu(M+m)g}{k}$$

So (D) is correct.

Now (E):

Maximum possible friction between blocks is

$$f_{\max}=\mu N$$

Normal reaction is only due to upper block’s weight:

N=mg

So

$$f_{\max}=\mu mg$$

So (E) is false.