Question

A particle of mass $m$ is acted upon by a force $F=t^2-kx$ . Initially, the particle is at rest at the origin. Then

• A. its displacement will be in simple harmonic
• B. its velocity will be in simple harmonic
• C. its acceleration will be in simple harmonic
• D. particle will move with constant velocity

Answer 1

Answer: (C)

$a=\frac{t^2}{m}-\frac{k}{m}x$
$\frac{da}{dt}=\frac{1}{2}2t-\frac{k}{m}.\frac{dx}{dt}$
$\frac{d^{2}a}{d{t}^2}=\frac{2}{m}-\frac{k}{m}a$
$\frac{d^{2}a}{d{t}^2}+\frac{k}{m}a-\frac{2}{m}=0$
This is second order differential equation for acceleration, hence $a$ will be in simple harmonic.