Along a thin ring of radius R and mass 2 m , two beads, each of mass m , can slide. The beads are connected by a spring with stiffness k. Initially, the ring is held against a vertical wall on a horiz

Question

Along a thin ring of radius R and mass 2 m , two beads, each of mass m , can slide. The beads are connected by a spring with stiffness k. Initially, the ring is held against a vertical wall on a horizontal surface, and the beads are in unstable equilibrium along the ring's diameter. The diagram shows a top view. After a small push, the beads start moving away from the wall. At the moment when the distance from the beads to the wall reaches a maximum value of 1.6 R , the ring is released. (Neglect gravity)

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At maximum bead distance 1.6R, their speed is zero.

$cos θ = - 0.6, sin θ = 0.8$

(A) Compare the spring energies:

$(2 R - ℓ)^{2} = (1.6 R - ℓ)^{2}$

$ℓ = 1.8 R (A)$

(B) Speed peaks at spring length ℓ.

$m v^{2} = \frac{1}{2} k (0.2 R)^{2}$

$v = 0.2 R \frac{k}{2 m} (B)$

(C) After release, horizontal centre of mass stays fixed.

$y_{CM} = \frac{2 m R + 2 m ( 1.6 R )}{4 m} = 1.3 R$

At the diameter, both beads lie at the ring centre, 1.3R from the wall: (C).

Answer: A, B, C.

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JEE Advanced 2027 (Class 12+)