Question

A block of mass m compresses a spring of stiffness k through a distance $\frac{l}{2}$ as shown in the figure. If the block is not fixed to the spring, the period of motion of the block is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

The period of oscillation = $2\pi \sqrt{\frac{m}{k}}$
$\Rightarrow$ The period of motion till the block is in contact with the spring is
$t=\frac{T}{2}=\pi \sqrt{\frac{m}{k}}$ then it leaves the spring with a speed $v=\omega A$
$v=\left(\sqrt{\frac{k}{m}}\right)\left(\frac{l}{2}\right)$
Then it moves with constant velocity v for a distance $D=l+l=2l$
$\Rightarrow$ The corresponding time of motion = $t_2=\frac{2l}{v}$
$\Rightarrow t_2=\frac{2l}{\frac{l}{2}\sqrt{\frac{k}{m}}}=4\sqrt{\frac{m}{k}}\therefore Thetimeperiodofmotion=t=t_1+t_2=\pi \sqrt{\frac{m}{k}}+4\sqrt{\frac{m}{k}}=\sqrt{\frac{m}{k}}[\pi +4]\therefore B$