Calculate the time period of the oscillation of a particle of mass m moving in the potential defined as `U(x)={{:((1)/(2) kx^(2)",", x lt 0),(mgx"," , g gt 0):}`

Explanation

In the problem oscillation of particle into 2 cases via `v lt 0` (consider it as SHM where time period is considered as `t_(1))` and another one as `x gt 0` (consider it as motion under gravity, where time period is `t_(2))`
Note :
We want to find total time period, which will be `T=t_(1)+t_(2)`
According to conservation of energy
`(1)/(2)kx^(2)=(1)/(2) mv^(2)`
`:.v^(2)=(kx^(2))/(m)`
`rArr v= sqrt((kx^(2))/(m)) " "...(1)`
We know that,
kinetic
energy (E) }
`(1)/(2) mv^(2)`
`v=sqrt((2E)/(m)) " "....(2)`
In SHM (case : 1)
`T = 2pi sqrt((m)/(k))`
Here , `t_(1)=(T)/(2)`
`= (2pi sqrt((m)/(k)))/(2)`
`= pi sqrt((m)/(k)) " "...(3)`
In case (motion under gravity)
`t_(2)=(2V)/(g)` substituting equation (2) have we get,
`t_(2)=(2)/(g) sqrt((2E)/(m))`
`rArr 2sqrt((2E)/(mg^(2))) " "....(4)`
Adding equation (3) & (4)
Time period of oscillation,
`T=t_(1)+t_(2)`
`= pi sqrt((m)/(k))+2 sqrt((2E)/(mg^(2)))`