Question

A block of mass m is pushed against α spring of spring constant k fixed at one end to a wall. The block can slide on α frictionless table as shown in figure. The natural length of the spring is L₀ and it is compressed to half its natural length when the block is released. Find the velocity of the block as α function of its distance x from the wall (for $x>L_0$).

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

When the block is released, the spring pushes it towards right. The velocity of the block increases till the spring acquires its natural length. Thereafter, the block loses contact with the spring and moves with constant velocity.

Initially, the compression of the spring is $\frac{L_0}{2}$. When the distance of the block from the wall becomes x, where $x<L_0$, the compression is ($L_0$ - x). Using the principle of conservation of energy,

$\frac{1}{2}k(\frac{L_0}{2}{)}^2=\frac{1}{2}k(L_0-x{)}^2+\frac{1}{2}m{v}^2$.

$v=\sqrt{\frac{k}{m}}{[\frac{L_0{}^2}{4}-(L_0-x{)}^2]}^{\frac{1}{2}}$.

Solving this,

When the spring acquires its natural length, x = $L_0$ and $v=\sqrt{\frac{k}{m}}\frac{L_0}{2}$. Thereafter, the block continues with this velocity.