Question

The potential energy of a particle of mass $0.02\, kg$ moving along $x$-axis is given by $V=A x(x-4) J$ 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 $x=2\, m$
• D. The period of oscillation of the particle is $\frac{\pi}{5} sec$

Answer 1

Answer: (B), (C)

$V=A x(x-4) J_{m}=0.02=A x^{2}-4 A x$
$F=-\frac{d V}{d x}=-2 A x+4 A $ (not constant)
$F=-2 A(x-2)$
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{2 A}{m} $
$T=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{m}{2 A}}$
$=2 \pi \sqrt{\frac{1}{100 A}}=\frac{\pi}{5} \sqrt{\frac{1}{A}}$