Question

A block of mass $m$ starts moving with an initial velocity $v_0$ at a distance $L$ towards a stationary spring of stiffness $K$ attached to the wall as shown in figure.

Find the distance travelled by block before coming to rest.
Given that surface is rough only for distance $L$ and friction coefficient $\mu$ is such that $(\frac{1}{2}m{v}_0^2>\mu mgL)$.

• A. $v_0\sqrt{\frac{m}{K}}$
• B. $\sqrt{\frac{\frac{1}{2}m{v}_0-\mu mL}{K}}$
• C. $\sqrt{\frac{\frac{1}{2}m{v}_0-\mu mL}{2K}}$
• D. $\sqrt{\frac{m{v}_0^2-2\mu mgL}{K}}$

Answer 1

The correct option is D $\sqrt{\frac{m{v}_0^2-2\mu mgL}{K}}$

Let block covers distance$(L+X_0)$ before coming to rest.
Work done by all forces $=\Delta KE$
$(W{)}_{\text{friction}}$ + $(W{)}_{\text{spring}}=\Delta KE$

$-\mu mgL-\frac{1}{2}K{x}_0^2=-\frac{1}{2}m{v}_0^2$

$\mu mgL+\frac{1}{2}K{x}_0^2=\frac{1}{2}m{v}_0^2$
$\frac{1}{2}K{x}_0^2=\frac{1}{2}m{v}_0^2-\mu mgL$
$x_0=\sqrt{\frac{m{v}_0^2-2\mu mgL}{K}}$

Hence option D is the correct answer