In the figure shown a semi-cylinder of radius, `R` is rigidly fixed on a horizontal table of height `l_(0)` from the ground such that there is a common edge of the table and semi-cylinder. A block of mss `m` is attached rigidly to a light spring of stiffnes `k`, whose other end is rigidly attached to the ground. The natural length of the spring is `l_(0)`. A force of constant magnitude `F` is applied on the block to move it on the frictionless surface of the semi-cylinder in a vertical plane such that the path followed by the block is a semicircle of radius `R`.
The force `F` is always tangential to the cylinder. Find the contact force between the cylinder and the block as a function of `theta`, Initial speed of the block is zero and initial length of the spring is `l_(0)`.`0` is the angular displacement of the block w.r.t. the centre of the circular path as shown in the figure.

Explanation

`W_(N)=` work done by normal reaction `=0`
`W_(g)= -mg R sintheta`
`W_("spring")= -(1)/(2)k(R theta)^(2)`
`W_(F)=FR theta`
`SigmaW=DeltaKE`
`rArr -mg R sintheta-(1)/(2)kR^(2)theta^(2)+FR theta=(1)/(2)mv^(2)`
`rArrmv^(2)= -2mg R sin theta-kR^(2)theta^(2)+2FRtheta`......(`i`)
Also, `mg sin theta-N=(mv^(2))/(R )`
`rArr N= mg sin theta-(mv^(2))/(R )`
`=mg sintheta+2mg sintheta+kRtheta^(2)-2Ftheta` using equation (`i`)
`=3mg sintheta+kRtheta^(2)-2Ftheta`