Motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower point is

Options

A. simple harmonic motion
B. non-periodic motion
C. periodic motion
D. periodic but not SHM

Explanation

Consider the motion of the ball inside a smooth curved bowl.

For small angular displacement or slightly released motion, it can be considered as angular SHM.
When the ball is at an angle of 0 the restoring force (g sin 0) m acts as shown.

`:'ma=mgsin0`
`rArra=mgsin0`
`rArr(d^(2)x)/(dt^(2))=-gsin0=-gxx(x)/(R)` `[:.sin0~~0=x//R]`
Hence, motion is SHM
`rArrd2pi//dt^(2)prop(-x)`
`rArromega=sqrt(g//R)` `(":.aprop-x)`
Time period `T=(2pi)/(omega)=2pisqrt((R)/(g))`
As motion is SHM, hence it must be periodic.