A particle of mass `m` oscillates inside the smooth surface a fixed pipe of radius `R`. The axis of the pipe is horizontal and the particle moves from `B` to `A` and back. At an instant the kinetic energy of the particle is `K` (say at position of the particle shown in figure ). Then choose the correct value of force applied by particle on the pipe at this instant.

Options

A. `(K)/(R)`
B. `(5K)/(R)`
C. `(3K)/(R)`
D. `(2K)/(R)`

Answer

C

Explanation

Let velocity of particle at point `P` be `v`.
From conservation of mechanical energy
`(1)/(2).mv^(2) = K = mgh`
Let `N` be the normal reaction between the particle and the pipe at this instant.
Then `N-mg sintheta = (mv^(2))/(R)`
But, `(mv^(2))/(R) = (2K)/(R)` and `sintheta = (h)/(R)`
Hence, `N = mg ((h)/(R)) + (2K)/(R) = (K)/(R) + (2K)/(R)`
`(':. K = mgh)`
Hence, `N = (3K)/(R) =` force on the pipe.