A small block of mass `m` is tied to the top of a smooth inclined plane with the help of a string of length `L` as shown in the figure. The inclined plane of inclination `theta` with horizontal is rotated with an angular velocity `omega` about a verticle axis passing through the end of the string fixed to the plane.
(a) Find the maximum value of `omega` so that the block maintains contact with the inclined plane.
(b) Find the ratio of the tension in the string and the normal reaction between the block and the plane.

Explanation

We will solve this problem in rotating fram of reference by applying pseudo force. As the block is rest in this frame, hence the net force on each direction is zero.
`r=L cos theta`
Resolving force `bot^(ar)` to the plane and along the plane:
`bot^(ar)` to the plane:
`N+momega^(2)r sin theta-mg cos theta=0`
`N=mg cos theta-m omega^(2)r sin theta`
For no loss of contact:
`Nge0`
`mg cos theta-m omega^(2)rsin thetage0`
`omeegalesqrt((gcostheta)/(r sin theta)=(g cos theta)/(L costhetasintheta)=(g)/(Lsintheta))`
`omega_(max)=sqrt((g)/(Lsintheta))`
Along the plane:
`T-mg sin theta-m omega^(2)rcos theta=0`
`T=mg sin theta+m omega^(2)rcos theta`
`N=mg cos theta-m omega^(2)rsin theta`
`(T)/(N)=(g sintheta+omega^(2)r costheta)/(g costheta-omega^(2)rsin theta)`
`=(gsintheta+omega^(2)Lcos^(2)theta)/(g cos theta-omega^(2)Lsin theta costheta`