The correct option is C π2
For a conservative force, we can say that
→F=−(∂U∂x^i+∂U∂y^j+∂U∂z^k)
Since the given potential energy is only a function x we can write the above equation as
F=−dUdx
From the data given in the question,
F=−ddx(4(1+cos2x)=−8sin2x ....(1)
Let m be the mass and a be the acceleration of the particle.
From Newton's second law, F=ma
Using(1) in the above equation we get,
a=−8sin2x
For small values of x, sin2x is approximately equal to 2x.
Thus,
a=−16x
Comparing this with a=−ω2x
we get, ω2=16⇒ω=4
We know that,
The time period of oscillation is given by T=2πω
⇒T=2π4=π2 s
Thus, option (c) is the correct answer.