Missing energy in a rope and a capacitor

Consider a uniform rope of mass density \lambda coiled on a smooth horizontal table. One end is pulled straight up with a constant speed v_0 as shown.

  1. Find the force exerted on the end of the rope as function of the height y.
  2. Compare the power delivered to the rope with the rate of change of the rope’s mechanical energy.

(This is a problem from chapter 5 of Kleppner and Kolenkow)

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To find the force exerted at the top end, note that if we were to pull up a fixed mass with constant velocity v_0, the total force on the mass should …

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A spherical capacitor

A spherical conducting shell with radius b is concentric with a conducting ball with radius a, with a<b.

  1. Compute the capacitance C = Q / \Delta \phi when the shell is grounded and the ball has charge Q.
  2. Compute the capacitance when the ball is grounded and the shell has charge Q.
  3. Compute the full matrix of coefficients of capacitance for the two conductors.
  4. Considering these conductors as a capacitor, determine its capacitance. That is, assign equal and opposite charges \pm Q to the shell and the ball, and compute C = Q / \Delta \phi.

Related Problem: Insulating spherical shell with a hole


(a) First, we ground the shell and give the …

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