Question

A small ball is attached to one end of a string swings in a vertical plane such that its acceleration at extreme point is equal tovthe aceleration at mean position the angle theta is

Answer 1

v^2/l=gsinα (1)

wehre α is the maximum deflection angle

From Newton's law in projection form: Ft=mwt

−mgsinθ=mvdv/ldθ

or, −glsinθdθ=vdv

On integrating both the sides within their limits.

−gl∫α0sinθdθ=∫0vvdv

or, v2=2gl(1−cosα) (2)

Note: Eq. (2) can easily be obtained by the conservation of mechanical energy of the ball in the uniform field of gravity.

From Eqs. (1) and (2) with θ=α

2gl(1−cosα)=lgcosα

or cosα=23 so, α=53∘