A particle of mass m moves in the potential energy U shoen above. The period of the motion when the particle has total energy E is

Options

A. `2pisqrt((m)/(k))+4sqrt((2E)/(mg^2))`
B. `2pisqrt((m)/(k))`
C. `pisqrt((m)/(k))+2sqrt((2E)/(mg^2))`
D. `2sqrt((2E)/(mg^2))`

Answer

C

Explanation

For `xlt0`
`F=-(dU)/(dx)=-kx`
`ma=-kx`
or `a=-(k)/(m)x`
`-omega_1^2x=-(k)/(m)x`
`omega_1=sqrt((k)/(m))`
`T_1=2pisqrt((m)/(k))`
For `xgt0` `U=mgx`
`F=-(dv)/(dx)=-mg`
But `E=(1)/(2)mv_0^2`
`v_0=sqrt((2E)/(m))`
It is speed at lowest point
`T_2=(2v_0)/(g)=(2)/(g)sqrt((2E)/(m))`
`T=(T_1)/(2)+T_2=pisqrt((m)/(k))+(2)/(g)sqrt((2E)/(m))`