Physics 102– Pledged Problem 9

Time allowed: 2 hours at a single sitting Due 5PM Monday, April 16, 2007, in the boxes marked Phys 101-102 in the physics lounge. You may use your own textbook, your notes, and a non-programmed calculator. You may also consult the on-line solutions to the corresponding suggested problems. You should consult no other help. Show how you arrived at your answer; the correct answer by itself may not be sufficient. Further instructions:

(a) Write legibly on one side of 8.5” x 11” white or lightly tinted paper.

(b) Staple all sheets together, including this one, in the upper left corner. Make one vertical fold.

(c) On the outside, print your name in capital letters, your LAST NAME followed by your FIRST NAME.

(d) Below your name, print the phrase “Pledged Problem 9”, followed by the due date.

(e) Write and sign the pledge, with the understanding that you may consult the materials noted above.

(f) Indicate your start time and end time.

I. A conducting rod of mass m and negligible resistance slides without friction along two parallel rails, also of negligible resistance. The rails are separated by a distance l and connected together through a resistor R. The rails rest on a long inclined plane that makes and angle θ to the horizontal. There is a uniform magnetic field B directed vertically upward as shown in the figure below. The rod is released from rest and slides down the incline. Express your answers in terms of B, l, R, θ, m, and possibly other constants.

(a) When the rod has velocity v_{x} down the incline, determine the current induced in the circuit by the changing magnetic flux. Show that this current produces a retarding force up the incline and find the magnitude of that force.

(b) Show that there is a terminal speed v_{t} such that the gravitational force down the incline is balanced by the upward retarding force. Determine the value of v_{t}.

(c) When the rod has reached the terminal velocity v_{t}, what is the I^{2}R power dissipation in the resistor? Compare the power dissipated in the resistor to the change in gravitational potential energy of the rod.

II. In the circuit shown below, the switch is initially opened, then at t = 0 it is closed.

(a) Determine the currents I_{1}, I_{2}, and I_{3} and the potential drop across the inductor E_{L} at t = 0.

(b) Determine the currents I_{1}, I_{2}, and I_{3} and E_{L} at t → ∞.

After the switch has been closed for a long time, it is opened.

(c) Determine the three currents immediately after the switch is opened

(d) Determine the three currents a long time after the switch is opened.

(e) Determine the I(t), the current in the inductor L as a function of time, after the switch is opened.

(f) If L=1H, and R_{1} = R_{2} = R, what value of R is needed so that the time constant for discharging is 30s?