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Question

A heavy particles is suspended by a string of length l from a fixed point O. The particle is given a horizontal velocity v0 the string slack at some angle and the particle proceeds on a parabola. Find the value of v0 if the particle passes through the point of suspension.

A
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B
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C
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D
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Solution

The correct option is A
Let at P, the string slacks when it makes an angle θ with the vertical, hence at the point P the centripetal force is only due to the component of the gravitational force.
mgcosθ=mv2l where v = velocity of the particle at P.
v2=gl.cosθ (i)
Conserving energy at initial point and at P we get
12mv20=12mv2+mg(1+cosθ) (ii)
From (i) and (ii) we get 12mv20=12mglcosθ+mgl(1+cosθ)
v20=2gl+3gcosθv20=gl[2+3cosθ] (iii)
Now particle will pass point of suspension if lsinθ=(vcosθ)t (iv)
And lcosθ=(vsinθ12gt2) (v) Eliminating t from (iv) and (v) we get.
lcosθ=(vsinθ)[lsinθvcosθ]12g[lsinθvcosθ]2 Substituting v2=glcosθ, and simplifying tanθ=2
Therefore cosθ=13 (vi)
Substituting the value of cosθ in equation (iii), we get v=gl(2+3)1/2

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