The potential energy of a peticle of mass `'m'` situated in a unidimensional potential field varies as `U(x) = 0 [1 - cos ax]`, where `U_(0)` and a are constants. The time period of small oscillations of the particle about the mean positions is :

Options

A. `2pisqrt((m)/(aU_(0)))`
B. `2pisqrt((am)/(U_(0)))`
C. `2pisqrt((m)/(a^(2)U_(0)))`
D. `2pisqrt((3^(2)m)/(U_(0)))`

Answer

C

Explanation

Restoring Force `F = (-du)/(dx) = (-d)/(dx) (u_(0) (1-cos ax)`
`F(x) = -u_(0)asinax`
for small angle `sin ax approx ax`
`F = -u_(0)a^(2)x rArr acc. (-u_(0)a^(2)x)/(m) = -omega^(2)x = ((2pi)/(T))^(2) xx x`
So, Time speed period `T = 2pisqrt((m)/(u_(0)a^(2)))`