Question

A ball suspended by a thread swings in a vertical plane so that its
acceleration values in the extreme and the lowest position are equal.
Find the thread deflection angle in the extreme position.​

Answer 1

Answer:

53 degree

please see the above attachment

The ball has only normal acceleration at the lowest position and only tangential acceleration at any of the extreme position. Let v be the speed of the ball at its lowest position and l be the length of the thread, then according to the problem

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∘