A body of mass m is situated in a potential field `U(x)=U_(0)(1-cosalphax)` when `U_(0)` and `alpha` are constant. Find the time period of small oscillations.

Options

A. `2pisqrt((m)/(U_(0)alpha))`
B. `2pisqrt((m)/(U_(0)alpha^(2)))`
C. `2pisqrt((m)/(2U_(0)alpha))`
D. `2pisqrt((2m)/(U_(0)alpha^(2)))`

Answer

B

Explanation

Given: `U=U_(0)(1-cosalphax)`
`becauseF=-(dU)/(dx)`
`thereforeF=-(d)/(dx)[U_(0)(1-cosalphax)]=-U_(0)alphasinalphax`
As `alphax` is small, so `sinalphax ~ alphax`
`thereforeF=-U_(0)alpha^(2)x` . .. (i)
`Fpropx and -ve` shows that F is directed towards the mean position, hence the body executes SHM.
For SHM, `F=-kx`
Comparing (i) and (ii), we get, `k=U_(0)alpha^(2)`
Time period of oscillation,
`T=2pisqrt((m)/(k))=2pisqrt((m)/(U_(0)alpha^(2)))`