Question

A block of mass m is connected rigidly with a smooth plank by a light spring of stiffness K. If the plank is moved with constant velocity ${\upsilon }_0$, find the work done by the external agent till the maximum compression of the spring.

• A. $\frac{1}{2}{mu}_0^2$
• B. $\frac{3}{4}{mu}_0^2$
• C. ${mu}_0^2$
• D. $\frac{3}{2}{mu}_0^2$

Answer 1

The correct option is C ${mu}_0^2$

Let work done by spring and external agent be $W_s$ and $W_e$ respectively.

At the time of maximum compression, the velocity of block will be equal to velocity of plank i.e $v_0$.

So, $\Delta K=\frac{1}{2}m{v}_0^2$ (because plank's velocity did not change)

Now, from the reference of plank

Work done by Spring $\left(W_s\right)$= Change in Kinetic Energy of block

$W_s=0-\frac{1}{2}m{v}_0^2$

$\Rightarrow W_s=-\frac{1}{2}m{v}_0^2$

On Applying Work-Energy theorem on whole system from the reference of ground, we have

$W_e+W_s=\Delta K$

$\Rightarrow W_e-\frac{1}{2}m{v}_0^2=\frac{1}{2}m{v}_0^2-0$

$\Rightarrow W_e=\frac{1}{2}m{v}_0^2+\frac{1}{2}m{v}_0^2$

$=m{v}_0^2$

So, the correct option will be $\left(C\right)$