The potential energy of a particle of mass `1kg` in motion along the x- axis is given by: `U = 4(1 - cos 2x)`, where `x` in meters. The period of small oscillation (in sec) is

Explanation

`F=-(dU)/(dx)=(-d)/(dx)[4(1-cos2x)]=-8sin2x`
or `F=-8xx2x`
[When x is small, `sin2x~~2x]`
or `F=-16x`
As `F prop x` and `-ve` sign that x is directed towards equilibrium position, hence the particle will execute linear SHM.
Here, spring factor, `k=16N//m`
`:.` time period, `T=2pisqrt((m)/(k))=2pisqrt((1)/(16))`
`=(pi)/(2)s`