Question

For a particle executing SHM, the displacement $x$ is given by $x=Acos\omega t$ . Identify the graph which represents the variation of potential energy (PE) as a function of time $t$ and displacement $x$ .

• A. I, III
• B. II, IV
• C. II, III
• D. I, IV

Answer 1

Answer: (A)

Potential energy as a function of displacement :
$PE=\frac{1}{2}k{y}^2$
$\Rightarrow$ PE vs y graph will be an upwards parabola.
$\Rightarrow$ $\text{graph III}$
Potential energy as a function of time:
$PE=\frac{1}{2}k{y}^2=\frac{1}{2}k(A{\left(cos\right)}^2\omega t{)}^2$
$\Rightarrow PE=\frac{1}{2}k{A}^2{\left(cos\right)}^2\omega t=\frac{1}{2}k{A}^2\frac{(1+cos2\omega t)}{2}$
$\Rightarrow$ PE vs time is a cosine wave with angular frequency $2\omega \Rightarrow graph\text{ I}\text{.}$