A particle of mass `m` is located in a uni-dimensional potential field where potential energy of the particle depends on the coordinates `x as: U (x) = U_(0) (1 - cos Ax), U_(0)` and A constants.
Find the period of small oscillation that the particle performs about the equilibrium position.

Explanation

As `U(x)=U_(0)(1-cos a x )`
So, `F_(x)=-(dU)/(dx)=-U_(0)a sin ax ...(1)`
or, `F_(x)=-U_(0)a a x (` because for small angle of oscillations ` sin a x ~=a x)`
or, `F_(x)=-U_(0)a^(2)x …..(1)`
But we know `F_(x)=-m omega_(0)^(2)x`, for small oscillation
Thus `omega_(0)^(2)=(U_(0)a^(2))/(m)` or ` omega_(0)=s sqrt((U_(0))/(m))`
Hence the sought time period
`T=(2pi)/(omega_(0))=(2pi)/(a ) sqrt((m)/(U_(0)))=2pisqrt(m)/(a^(2)U_(0))`