Question

Two blocks A and B of mass $m$ and $2m$ are connected together by a light spring of stiffness $k$. The system is lying on a smooth horizontal surface with the block $A$ in contact with a fixed vertical wall as shown in the figure. The block $B$ is pressed towards the wall by a distance $x_0$ and then released. There is no friction anywhere. If spring takes time $\Delta t$ to acquire its natural length then average force on the block $A$ by the wall is :

• A. $Zero$
• B. $\frac{\sqrt{2mk}}{\Delta t}{x}_0$
• C. $\frac{\sqrt{mk}}{\Delta t}{x}_0$
• D. $\frac{\sqrt{3mk}}{\Delta t}{x}_0$

Answer 1

Answer: (B)

$\frac{\sqrt{2mk}}{\Delta t}{x}_0$

Let $v$ is the velocity of mass $2m$ in natural length of spring.
When maximum compression of the spring $x_0$ has occured, velocity of mass $2m$ is zero.
Then from conservation of energy we have,

$\frac{1}{2}k{x}_0^2=\frac{1}{2}\left(2m\right)v^2$

$\therefore v=\sqrt{\frac{k}{2m}}{x}_0$;

Velocity of the CM $v_{cm}=\frac{2mv+m.0}{2m+m}=\frac{2}{3}v=\frac{1}{3}\sqrt{\frac{2k}{m}}{x}_0$
$\frac{\Delta p}{\Delta t}=F_{avg}$;

$\therefore F_{avg}=\left(3m\right)v_{cm}\times \Delta t$...............(1)

Substituting $v_{cm}$ in eqn (1), $F_{avg}=\frac{\sqrt{2mk}{x}_0}{\Delta t}$