Question

Two blocks of masses $m_1$ and $m_2$ are connected with a massless unstretched spring and placed over a plank moving with an acceleration $a$ as shown in the figure. The coefficient of friction between the blocks and the platform is $\mu$.

• A. Spring will be stretched if $a>\mu g$
• B. Spring will be compressed if $a\le g$
• C. Spring will neither be stretched nor be compressed for $a\le g$
• D. Spring will be in its natural length under all conditions

Answer 1

The correct option is D Spring will be in its natural length under all conditions

FBD of the planck


FBD of the blocks (from the frame of reference of the planck)


$m_{1}a$ and $m_{2}a$ are the pseudoforces acting on the two blocks.

Case (i) $a>\mu g$
Then, $m_{1}a-\mu m_{1}g=m_1{a}^{\prime }--\left(I\right)m_{2}a-\mu m_{2}g=m_2{a}^{\prime \prime }--\left(II\right)$

Solving (I) and (II),
$a^{\prime }=a^{\prime \prime }=a-\mu g$

Since both the blocks move with the same acceleration w.r.t the planck, the spring will be at its natural length.

Case (ii) $0<a<\mu g$

In this case, friction force will balance the pseudo-force and hence the blocks will remain stationary w.r.t each other. So, again the spring will be at its natural length

Thus, we can conclude that the spring will be at its natural length under all conditions.