Question

A heavy particle hanging from a fixed point by a light inextensible string of length l is projected horizonally with speed sqrt(gh) . Find the speed of the particle and the inclination of the string to the vertical at the instant of the motion when the tension in the string is equal to the weight of the particle.

Answer 1

Hence the speed of the particle is v = √gl/3

Explanation:

Let T = mg when angle is θ

h = l (1−cosθ)

Using conservation of energy at A and B

1 / 2 m(u^2 − v^2) = mgh

u = gl, v = speed of particle at B

v^2 = u^2 − 2gh

T − mg cosθ = mv^2 / l

mg − mgcosθ = mv^2 / l

v^2 = gl( 1 − cosθ)    -----(1)

gl ( 1 − cosθ) = gl − 2gl(1 − cosθ)  

cosθ = 2 / 3 , Substituting in (1)

Therefore v = √gl/3

Hence the speed of the particle is v = √gl/3