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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 acceleration due to gravity)

Asked in JEE Main 28th Jan 1st Shift 2025 · Energy stored and maximum compression

• (1) √2Rg+(4kR²)/m
• (2) √2Rg+(kR²)/m
• (3) 2√gR+(kR²)/m
• (4) √3Rg+(kR²)/m

Step-by-step solution

Why the other options are wrong

• (1) Comes from taking the spring's initial length 2R as its extension and the drop as R.
• (2) Comes from taking the drop as R, i.e. placing the bead at the centre level instead of R/2 above the bottom.
• (3) Comes from a drop of 2R (all the way to the bottom) and an extension of 2R.

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