Question

What is the maximum compression in the spring, if the lower block is shifted to rightward with acceleration $ a $ . (Given that all surfaces are smooth as shown).

(A) $ \dfrac{{ma}}{{2k}} $
(B) $ \dfrac{{2ma}}{k} $
(C) $ \dfrac{{ma}}{k} $
(D) $ \dfrac{{4ma}}{k} $

Answer 1

Hint : To solve this question, we need to consider the free body diagram of the smaller block. Then, from the equilibrium of the block at the maximum compression of the spring, we will get the required maximum compression.

Formula Used: The formula which is used in solving this question is given as
 $\Rightarrow F = kx $ , here $ F $ is the force applied by a spring having the spring constant $ k $ due to an extension $ x $ in the spring.

Complete step by step answer
Since the wall with which the left end of the spring is connected is attached to the lower block, so the wall starts moving towards the smaller block with the acceleration $ a $ . With respect to the wall, the smaller block is moving leftward with acceleration $ a $ . So the spring will start compressing. This compression will be continued until the spring force generated makes the smaller block in equilibrium. The free body diagram of the smaller block at the maximum compression is as shown in the below figure

So, at the maximum compression of the spring, we have
 $\Rightarrow k{x_{\max }} = ma $
 $\Rightarrow {x_{\max }} = \dfrac{{ma}}{k} $
Thus the maximum compression of the spring comes out to be equal to $ \dfrac{{ma}}{k} $ .
Hence, the correct answer is option C.

Note
This question could also be solved by using the concept of pseudo force. For that, we have to consider the motion of the smaller block with respect to the lower block. Also, we cannot apply the work energy theorem to find out the maximum compression in the spring. This is due to the fact that the kinetic energy of the block is non zero at the extreme position. In fact, the work energy is applied to find out the velocity of the block at the extreme position.