A particle of mass m is moving in a potential well, for which the potential energy is  given by U(x)=U01−cosax where U0  and a are constants. Then for small oscillations,  time period is

U(x)=U0(1−cosax)F=−dUdx=−aU0sin⁡ax For small oscillations, sin⁡ax≈ax⇒ma′=−a2U0x⇒a′=−a2U0mx. Since, a′α−x, oscillations are simple harmonic.  Comparing with the relation a=−ω2x with a′=−a2U0mx We get ω2=a2U0m. Here ω=2πT Therefore 2πT=a2U0m Hence, T=2πma2U0.