Question

The potential energy of the particle of mass 1 kg in motion along x axis is given by:
$U = 4(1-cos(2x))$. Here ‘x’ is in meters. The period of small oscillation in seconds is _______.
$\text{A}. \ 2\pi$
$\text{B}. \ \pi$
$\text{C}. \ \dfrac{\pi}2$
$\text{D}. \ \sqrt{2}\pi$

Answer 1

In a conservative field, negative of potential energy is defined as the net force on the particle i.e. $F = -\dfrac{dU}{dx}$, where x is the direction in which particle is moving. Further, small oscillations can be calculated by comparing the equation of motion of the particle with the differential equation of S.H.M.

Formula used:
$F = -\dfrac{dU}{dx}, T = \dfrac{2\pi}{\omega}$

Complete answer:
The net force acting on the particle along the x-axis can be calculated by differentiating the potential energy expression with respect to ‘x’.
i.e. $F = -\dfrac{dU}{dx}$
or $F = -\dfrac{d}{dx}(4(1-cos2x))$
or $F = -4\left( \dfrac{d(1)}{dx}-\dfrac{d(cos2x)}{dx}\right) = -4 (0 -(-2sin2x)) = -8 sin2x$
[$as, \dfrac{d}{dx} (cos(ax+b)) =-asin(ax+b)$]
Hence, $F_{net}=-8sin2x$
Now, in the question, we’re supposed to calculate the oscillation for smaller amplitudes. Hence, taking sin(2x)=2x, for smaller values of ‘x’.
Hence $F_{net}= -8(2x)=-16x$
As F = ma,
Hence $a=\dfrac{-16x}{m} = \dfrac{-16x}1 = -16x$ [ as m = 1kg]
Thus, $a = -16x$, is the equation of motion of the particle.
Now comparing it with differential equation of S.H.M: $a=-\omega^2x$, we get
$\omega^2=16$
Or $\omega = 4$
Hence $T=\dfrac{2\pi}{4} = \dfrac{\pi}{2}$

So, the correct answer is “Option C”.

Additional Information:
A conservative force is that force in which the work done by the force does not depend upon the path taken i.e. independent of path, called state function.

Note:
In such type of questions, where we are supposed to find the time period of oscillation of a particle, first of all we need to get the net force acting on the particle and then somehow finding the equation of motion of particle and then comparing it with the differential equation of S.H.M can be used to find time period in any type of question.