A bead of mass ' m ' slides without friction on the wall of a vertical circular hoop of radius ' R ' as shown in figure. The bead moves under the combined action of gravity and a m

Question

A bead of mass ' m ' slides without friction on the wall of a vertical circular hoop of radius ' R ' as shown in figure. The bead moves under the combined action of gravity and a massless spring ( k ) attached to the bottom of the hoop. The equilibrium length of the spring is ' R '. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes ' R ', would be (spring constant is ' k ', g is accleration due to gravity)

TestHub · Accepted Answer

Work done by gravity ​ = m g ( 2 R − R cos 6 0 ∘ ) = 2 3 m g R ​ ​ Work done by spring ​ = − 2 1 ​ k ( 0 2 − R 2 ) = 2 1 ​ k R 2 ​ Net work = change in kinetic energy i.e. 2 3 m g R ​ + 2 k R 2 ​ = 2 1 ​ m v 2 or v 2 = 3 g R + m k R 2 ​ or v = 3 g R + m k R 2 ​ ​

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