Question

During SHM, a particle has displacement $x$ form mean position. If acceleration. Kinetic energy and excess potential energy are represented by a $K$ and $U$ respectively, then choose the appropriate graph

• A.
• B.
• C.
• D.

Answer 1

Answer: (D)

The KE of a particle at a distance $x$ from its mean position is given by
$K=\frac{1}{2} m\left(A^{2}-x^{2}\right) \Omega^{2}$
It means curve between $K$ and $x$ will be parabola. Hence options (a) and (b) wrong. The PE of a particle is given by
$U=\frac{1}{2} m \omega^{2} x^{2}$
But acceleration $a=-\omega^{2} x$. Hence, $P E$ is proportional to $a^{2}$.
At mean position, if will be equal to zero and maximum at extreme position. Hence (c option is correct). The sum of $P E$ and $K E$ is always constant. It is equal to $\frac{1}{2} m A^{2} \omega^{2}$. Hence,
$U+K =\frac{1}{2} m \omega^{2} A^{2}$
$U =-K+\frac{1}{2} m \omega^{2} A^{2}$
It means curve between $U$ and $K$ will be straight line having positive intercept on $y$-axis and negative slop.