First slide

Simple harmonic motion

Question

A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is $U (x) = k | x |^{3}$ , where k is a positive constant. If the amplitude of oscillation is a, then its time period T is

Difficult

A

$Proportional to \frac{1}{\sqrt{a}}$
B

$Independent of a$
C

$Proportional to \sqrt{a}$
D

$Proportional to a^{3 / 2}$

Solution

$U = k | x |^{3} \Rightarrow F = - \frac{d U}{d x} = - 3 k | x |^{2} . . . . . . . . (i)$
$Also, for SHM, x = a \sin ω t and \frac{d^{2} x}{d t^{2}} + ω^{2} x = 0$
$\begin{array}{l} \Rightarrow Acceleration a = \frac{d^{2} x}{d t^{2}} = - ω^{2} x \\ \Rightarrow F = m a = m \frac{d^{2} x}{d t^{2}} = - m ω^{2} x . . . . . . . (i i) \end{array}$
$From eqs. (i) and (ii), we get ω = \sqrt{\frac{3 k x}{m}}$
$\Rightarrow T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{3 k x}} = 2 π \sqrt{\frac{m}{3 k (a \sin ω t)}} \Rightarrow T \propto \frac{1}{\sqrt{a}}$

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