A metal bar has mass m a length L and a circular cross section of radius b and is made of a material of resistivity ? The bar is mounted so that its length is horizontal parallel to the x-axis and its ends are free to move without friction on a conducting vertical track composed of the two vertical conducting rods that are parallel to the y-axis and are separated by a distance L. At the bottom of the track at the ground the two rods are connected by a second stationary conducting rod of length L. Current can thus make a complete circuit and is free to flow across the bottom bar, up one side of the track, across the moveable bar, and down the other side of the track to the original starting position. Assume the acceleration due to gravity is the constant g in the downward direction. A constant magnetic field B is present in the direction which is into the page. The bar is dropped from rest at a height H at a time t = 0. Compute the following in terms of quantities given or a subset thereof, and relevant physical constants. Assume the electrical resistance of the track and the bottom bar is zero. Also assume that H is very large and that the bar reaches its terminal velocity long before it reaches the ground. As a choice of coordinates make the downward direction the positive y-direction and the positive x-direction to the right.This makes the positive z-direction into the page.
(a) First draw the system with the x-y plane as the page you are drawing on and the y-axis as the vertical axis. Make the downward direction the positive y-direction and the positive x-direction to the right. Indicate the direction of the magnetic field and the induced current in the loop as the bar drops.
(b) As the bar drops what is the induced current around the loop as a function of time. Give your answer in terms of quantities given and the instantaneous speed of the bar v(t).....you do not need to solve for v(t)
(c) As the bar drops and attains a speed v(t) there is a Lorentz force on the bar. Write down an expression for the lorentz force and indicate its direction as either up or down.
(d) Now assume that the initial height H is large enough so that there is plenty of time for the bar to reach a maximum terminal speed. Using Newton's 2nd law what is the terminal speed of the bar? Draw a free body diagram.
(e) Now at Terminal speed what is the power input to the bar as positive work per second done by gravity?
(f) If gravity is doing positive work per second and the bar is a a constant speed where is all the energy going?