Question

A block of mass m compresses a spring of stiffness k through a distance l/2 as shown in the figure. If the block is not fixed to the spring and released from this position, the period of motion of the block is

• A.
• B.
• C.
• D. $(3+\pi )\sqrt{\frac{m}{k}}$

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 $v=\omega A$

$v=(\sqrt{\frac{k}{m}})(\frac{l}{2})$

Then it moves with constant velocity v for a distance $D=l+l=2l$

$\Rightarrow$ The corresponding time of motion $=t_2=\frac{2l}{v}$

$\Rightarrow t_2=\frac{2l}{\frac{l}{2}\sqrt{\frac{k}{m}}}=4\sqrt{\frac{m}{k}}$

$\therefore$ The time period of motion $=t=t_1+t_2$

$=\pi \sqrt{\frac{m}{k}}+4\sqrt{\frac{m}{k}}=\sqrt{\frac{m}{k}}[\pi +4]$

$\therefore$ the correct option is B.