Question

A block of mass $1\text{kg}$ is connected to a spring of spring constant ${\pi }^2\text{N/m}$ fixed at the other end and kept on smooth level ground. The block is pulled by a distance of $1\text{cm}$ from its natural length position and released. After what time does the block compress the spring by $\frac{1}{2}\text{cm}$?

• A. $\frac{2}{3}\text{sec}$
• B. $\frac{1}{3}\text{sec}$
• C. $\frac{1}{6}\text{sec}$
• D. $\frac{1}{12}\text{sec}$

Answer 1

The correct option is A $\frac{2}{3}\text{sec}$

We can write the equation of SHM performed by the spring-block system as
$x=1\cos \left(\omega t\right)$

When spring is compressed by $\frac{1}{2}\text{cm}$,
$-\frac{1}{2}=1\cos \left(\omega t\right)$
$\Rightarrow \omega t=\frac{2\pi }{3}$
$\Rightarrow t=\frac{2\pi }{3\omega }=\frac{T}{3}$
$\Rightarrow t=\frac{2\pi }{3}\sqrt{\frac{m}{k}}=\frac{2\pi }{3}\sqrt{\frac{1}{{\pi }^2}}=\frac{2}{3}\text{sec}$