Question

An object of mass $0.5\text{kg}$ is executing simple harmonic motion. Its amplitude is $5\text{cm}$ and time period $\left(T\right)$ is $0.2\text{s}.$ What will be the potential energy of the object at an instant $t=\frac{T}{4}\text{s}$ starting from the mean position . Assume that the initial phase of the oscillation is zero.

• A. $0.62\text{J}$
• B. $6.2\times {10}^{-3}\text{J}$
• C. $1.2\times {10}^3\text{J}$
• D. $6.2\times {10}^3\text{J}$

Answer 1

The correct option is A $0.62\text{J}$

Time period of an object executing SHM is given as
$T=2\pi \sqrt{\frac{m}{k}}$
$\Rightarrow 0.2=2\pi \sqrt{\frac{0.5}{k}}$
$\Rightarrow k=50{\pi }^2..\left(i\right)$
Expression for SHM is given as
$x=A\sin (\omega t+\varphi )$
$=5\text{cm}\sin (\frac{\omega T}{4}+0)$ [Given: initial phase is zero and $t=\frac{T}{4}$]

$=5\text{cm}\sin \left(\frac{\pi }{2}\right)$
$x=5\text{cm}..\left(ii\right)$
Thus, potential energy of the object at this instant is given as
$P.E=\frac{1}{2}K{x}^2$
from $\left(i\right)\text{\&}\left(ii\right)$
$P.E=\frac{1}{2}\left(50{\pi }^2\right){\left(\frac{5}{100}\right)}^2$
$=0.616\text{J}\approx 0.62\text{J}$