A ball is hanging vertically by a light inextensible string of length L from fixed point O.The ball of mass m is given a speed u at the lowest position such that it completes a vertical circle with ce

Question

A ball is hanging vertically by a light inextensible string of length L from fixed point O.The ball of mass m is given a speed u at the lowest position such that it completes a vertical circle with centre at O as shown. Let AB be a diameter of circular path of ball making an angle 60° with vertical as shown, then :

TestHub · Accepted Answer

Points A and B are ends of a diameter; their height difference is L.

$K_{A} - K_{B} = m gL \Rightarrow v_{A}^{2} - v_{B}^{2} = 2 gL$

Resolve radial force at A and B, with their radial vertical components opposite.

$T_{A} - T_{B} = \frac{m ( v _{A}^{2} - v _{B}^{2} )}{L} + m g = 3 m g$

$F_{c, A} - F_{c, B} = \frac{m ( v _{A}^{2} - v _{B}^{2} )}{L} = 2 m g$

The sum of radial acceleration vectors has magnitude 2g; their tangential acceleration vectors sum to a perpendicular √3g.

$∣ a_{A} + a_{B} ∣ = (2 g)^{2} + (3 g)^{2} = 7 g$

A, B, C and D are correct.

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