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Homework Due 5-8-2013 1. A stick of length 2L and negligible mass has a point mass m affixed to each end. The stick is arranged so that it pivots in a horizontal plane about a frictionless vertical axis through its center. A spring of force constant k is connected to one of the masses as shown above. The system is in equilibrium when the spring and stick are perpendicular. The stick is displaced through a small angle θ, as shown and then released from rest at t = 0 a. Determine the restoring torque when the stick is displaced from equilibrium through the small angle θ, b. Determine the magnitude of the angular acceleration of the stick just after it has been released. c. Write the differential equation whose solution gives the behavior of the system after it has been released. d. Write the expression for the angular displacement θ of the stick as a function of time t after it has been released from rest 2. A ferryboat of mass M1= 2.0 x 105 kilograms moves toward a docking bumper of mass M 2 that is attached to a shock absorber. Shown below is a speed v vs. time t graph of the ferryboat from the time it cuts off its engines to the time it first comes to rest after colliding with the bumper. At the instant it hits the bumper, t = 0 and v = 3 meters per second After colliding inelastically with the bumper, the ferryboat and bumper move together with an initial speed of 2 meters per second. Calculate the mass of the bumper M2. a. After colliding, the ferryboat and bumper move with a speed given by the expression v = 2e -4t. Although the boat never comes precisely to rest, it travels only a finite distance. Calculate that distance. b. While the ferryboat was being slowed by water resistance before hitting the bumper, its speed was given by 1/v = 1/3 + βt, where β is a constant. Find an expression for the retarding force of the water on the boat as a function of speed Homework Due 5-8-2013 3. The figure above left shows a hollow, infinite, cylindrical, uncharged conducting shell of inner radius r1, and outer radius r2 . An infinite line charge of linear charge density +λ is parallel to its axis but off center. An enlarged cross section of the cylindrical shell is shown above right. (a) On the cross section above right, i. sketch the electric field lines, if any, in each of regions I, II, and III and ii. use + and - signs to indicate any charge induced on the conductor. (b) In the spaces below, rank the electric potentials at points a, b, c, d, and e from highest to lowest (I = highest potential). If two points are at the same potential, give them the same number. ____Va ____Vb _____Vc _____Vd _____Ve (c) The shell is replaced by another cylindrical shell that has the same dimensions but is nonconducting and carries a uniform volume charge density +ρ. The infinite line charge, still of charge density +λ is located at the center of the shell as shown above. Using Gauss's law, calculate the magnitude of the electric field as a function of the distance r from the center of the shell for each of the following regions. Express your answers in terms of the given quantities and fundamental constants. i. r < r1 ii. r1 < r < r2 iii. r>r2