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, $PE$ 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 $PE$ and $KE$ 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.