A puck of mass m is attached to a cord passing through a small hole in a frictio

Question

A puck of mass m is attached to a cord passing through a small hole in a frictionless table. The cord is pulled downward with a tension T. The puck is initially orbiting with speed vi in a circle of radius ri . The tension is then slowly increased so that the radius of the circle starts decreasing. It is increased sufficiently slowly so that you may assume that the mass on the table is in perfect circular motion at all times.. This decreases the radius of the circle to r. (a) The process just described conserves angular momentum. Explain why. (b) Use angular momentum conservation to determine the speed of the puck when the radius is r. (c) Find the tension in the cord as a function of r. (d) How much work is done in moving m from ri to r? Note: The tension changes with r.

Explanation / Answer

a) Here force(tension) acts on the puck always towards the center.

so, torque produced by the tension is zero.

when net torque acting on body is zero. Its angular momentum is constant.

Torqe = dL/dt

when , torque is zero, dL/dt = 0

==> L = constant.

b) Apply, m*vi*ri = m*v*r

==> v = vi*ri/r

c) T = m*a_rad

= m*v^2/r

d) Workdone = change in kinetic energy

= 0.5*m*v^2 - 0.5*m*vi^2

= 0.5*m*(vi*ri/r)^2 - 0.5*m*vi^2

= 0.5*m*vi^2*(ri/r)^2 - 0.5*m*vi^2

= 0.5*m*vi^2*((ri/r)^2 - 1)

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