A uniform rod of length $ l $ is pivoted at point $ A $. It is struck by a horizontal force which delivers an impulse $ J $ at a distance $ x $ from point $ A $ as shown in the figure. The impulse delivered by pivot is zero if $ x $ is equal to,

Question

A uniform rod of length $ l $ is pivoted at point $ A $. It is struck by a horizontal force which delivers an impulse $ J $ at a distance $ x $ from point $ A $ as shown in the figure. The impulse delivered by pivot is zero if $ x $ is equal to,, $ \frac{l}{2} $, $ \frac{\frac{1}{3}}{3} $, $ \frac{2 l}{3} $, $ \frac{3 l}{4} $

MARKS App · Accepted Answer

From impulse-momentum theorem, we can write, Impulse = change in momentum or, J = ∆ p . Let velocity of the centre of mass is v , then, we can write, J = m v - 0 = m v . . . i The rod starts rotating. Let the angular speed of rod about point A is ω . Then, from conservation of angular momentum, we have, J . x = I ω = m l 2 3 ω … i i And from the relation between linear and angular velocity of the centre of mass, we have, v = l 2 ω … iii Solving these three equations, we get, x = 2 3 l .

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