A rod of length l ⁡ = 2 m is maintained to rotate with a constant angular velocity ω = 1 rad s - 1 about a vertical axis passing through one end (diagram). There is a spring constant k = 1 N m - 1 which just encloses rod inside it in natural length. One end of the spring is attached to the axis of rotation. S is a sleeve of mass m = 1 kg which can just fit on rod. All surfaces are smooth. With what minimum kinetic energy (in j) sleeve should be projected so that it enters on the rod without jerk and completely compresses the spring.

Question

A rod of length l ⁡ = 2 m is maintained to rotate with a constant angular velocity ω = 1 rad s - 1 about a vertical axis passing through one end (diagram). There is a spring constant k = 1 N m - 1 which just encloses rod inside it in natural length. One end of the spring is attached to the axis of rotation. S is a sleeve of mass m = 1 kg which can just fit on rod. All surfaces are smooth. With what minimum kinetic energy (in j) sleeve should be projected so that it enters on the rod without jerk and completely compresses the spring.,

MARKS App · Accepted Answer

For entering without jerk v 2 = l ⁡ ω 0 = 2 m s - 1 Using work-energy theorem on the sleeve after entering in the frame of rod W spring + W centrifugal = Δ K - 1 2 k l ⁡ 2 - 1 2 m ω 2 l ⁡ 2 = 0 - 1 2 mv 1 2 ⇒ v 1 2 = 8 Now, K = 1 2 m v 1 2 + v 2 2 = 6

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