A child is swinging a ball of mass `M` around on a light spring which has spring constant `k`. The ball describes a horizontal circle. The stretched spring has a length a horizontal circle. The stretched spring has a length `l` and makes an angle `theta` with the vertical as shown in fig. the acceleration due to gravity is g neglect air resistance.

a. In terms of only the given quantities , what is the magnitude of force `F` that the spring exerts on man `M`?
b.In terms of `F,k`and `l`what is the natural length `l_(0)` of the spring i.e. the length of the spring when it is not steched?
c. In terms of `F, l,M` and `B` what is the speed v of the ball?

Explanation

The particle move in a horizontal circle . The force along y- direction are balanced . Towards of horizontal circle centripetal acceleration acts therefore that must be a resultant force . Note that along the length of spring forces are not balanced because this direction has component of acceleration

`a. F = cos theta = Mg rArr F = (Mg)/(cos theta)`

b.`F = K Delta l`
`Deltal = (F)/(K) = (Mg)/(K cos theta)`
`l_(0) = 1 - Delta l rArr l_(0) = l - (Mg)/(K cos theta)`
c. `F sin theta = (mv^(2))/(r )`
`(mg)/(cos theta) sin theta = (mv^(2))/(l sin theta) rArr v = sqrt ((g 1 sin^(2) theta)/(cos theta))`