A bob of heavy mass m is suspended by a light string of length l. The bob is given a horizontal velo

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Question

A bob of heavy mass m is suspended by a light string of length l. The bob is given a horizontal velocity v₀. If the string gets slack at some point P making angle θ from the horizontal, the ratio of the speed v of the bob at point P to its initial speed v₀ is:

Accepted Answer

Correct Answer: A — [sinθ/(3sinθ + 2)]^(1/2)Step-by-step Solution:At point P where string becomes slack, the tension T = 0. The centripetal force is provided by the component of gravity along the string:mg sinθ = mv²/l → v² = gl sinθ ...(1)Using conservation of energy (taking the bottom as reference, height gained to P above bottom = l + l sinθ):(1/2)mv₀² = (1/2)mv² + mg(l + l sinθ)v₀² = v² + 2gl(1 + sinθ) ...(2)Substituting v² = gl sinθ from (1) into (2):v₀² = gl sinθ + 2gl(1 + sinθ) = gl sinθ + 2gl + 2gl sinθ = 2gl + 3gl sinθ = gl(3sinθ + 2)Therefore: v²/v₀² = gl sinθ / [gl(3sinθ + 2)] = sinθ/(3sinθ + 2)v/v₀ = [sinθ/(3sinθ + 2)]^(1/2)Key Concepts from Work, Energy and Power (NCERT Class 11, Chapter 6):Conservation of mechanical energy: (1/2)mv₀² = (1/2)mv² + mgh. Total mechanical energy is conserved when only conservative forces do work.When the string becomes slack, tension T = 0. At that instant, the net radial force equals the centripetal force needed: mg sinθ = mv²/l.Height gained from bottom to point P at angle θ from horizontal: h = l + l sinθ.These types of problems combine energy conservation with force analysis at a specific point.

A bob of heavy mass m is suspended by a light string of length l. The bob is given a horizontal velocity v₀. If the string gets slack at some point P making angle θ from the horizontal, the ratio of the speed v of the bob at point P to its initial speed v₀ is:

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A bob of heavy mass m is suspended by a light string of length l. The bob is given a horizontal velocity v₀. If the string gets slack at some point P making angle θ from the horizontal, the ratio of the speed v of the bob at point P to its initial speed v₀ is:

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