Question

The potential energy of a particle of mass $0.02kg$ moving along $x$-axis is given by $V=Ax(x-4)J$ where $x$ is in metres and $A$ is a constant. Which of the following is/are correct statement(s)?

• A. The particle is acted upon by a constant force
• B. The particle executes simple harmonic motion
• C. The speed of the particle is maximum at $x=2m$
• D. The period of oscillation of the particle is $\frac{\pi }{5}sec$

Answer 1

Answer: (B), (C)

$V=Ax(x-4)J_m=0.02=A{x}^2-4Ax$
$F=-\frac{dV}{dx}=-2Ax+4A$ (not constant)
$F=-2A(x-2)$
Hence position execute S.H.M
mean position is when force $=0$ i.e. $x=2$
and speed is maximum at mean position
${\omega }^2=\frac{2A}{m}$
$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{2A}}$
$=2\pi \sqrt{\frac{1}{100A}}=\frac{\pi }{5}\sqrt{\frac{1}{A}}$