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angular velocity and work done by the tension

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The 0.2 kg ball ( ball is sliding not rotating) and the supporting cord are revolving about the vertical axis on the fixed smooth conical surface with an angular velocity of W = 4 radians/sec. The green ball is held in position b = .3 m by the tension T in the yellow cord.
If b is reduced to the constant value of 0.2 m by increasing T in the cord, compute the new angular velocity W and the work U, done on the system by T.

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Rotation and Angular Momentum are investigated. The solution is detailed and was rated '5/5' by the student who originally posted the questions.

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angular velocity of the particle and work done

A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string, whose other end passes through a hole in the table, where I'm holding it. Initially, the particle is moving in a circle of radius r0 with angular velocity w0, but I now pull the string down through the whole until at length r remains between the hole in the particle.

(a) What is angular velocity of the particle now?
(b) Assuming that I pull the string so slowly that we can approximate the particles that my circle of slowly shrinking the radius, calculate the work I did pulling the string.
(c) Compare your answer with part (b) with the particles gain in kinetic energy.

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