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2010 Mechanics III A skier of mass m will be pulled up a hill by a rope, as shown above. The of magnitude the magnitude of the acceleration as a function of time t can be modeled by the equations where amax and T are constants. The hill is inclined at an angle Ө above the horizontal, and friction between the skis and the snow is negligible. Express your answers in terms of given quantities and fundamental constants. negligible. (a) Derive an expression for the velocity of the skier as a function of time during the acceleration. Assume the skier starts from rest. (b) Derive an expression for the work done by the net force on the skier from rest until terminal speed is reached. (c) Determine the magnitude of the force exerted by the rope on the skier at terminal speed. (d) Derive an expression for the total impulse imparted to the skier during the acceleration. (e) Suppose that the magnitude of the acceleration is instead modeled as a = amaxe-πt/2T for all t > 0 , where amax and T are the same as in the original model. On the axes below, sketch the graphs of the force exerted by the rope on the skier for the two models., from t = 0 to a time t > T . Label the original model F1 and the new model F2. 2003 Electricity and Magnetism I A spherical cloud of charge of radius R contains a total charge +Q with a nonuniform volume charge density that varies according to the equation where r is the distance from the center of the cloud. Express all algebraic answers in terms of Q, R, and fundamental constants (a) Determine the following as a function of r for r > R. i. The magnitude E of the electric field ii. The electric potential V (b) A proton is placed at point P shown above and released. Describe its motion for a long time after its release. (c) An electron of charge magnitude e is now placed at point P. which is a distance r from the center of the sphere, and released. Determine the kinetic energy of the electron as a function of r as it strikes the cloud. (d) Derive an expression for ρo (e) Determine the magnitude E of the electric field as a function of r for r < R 2002- Mechanics I A crash test car of mass 1,000 kg moving at constant speed of 12 m/s collides completely inelastically with an object of mass M at time t = 0. The object was initially at rest. The speed v in m/s of the car-object system after the collision is given as a function of time t in seconds by the expression (a) Calculate the mass M of the object. (b) Assurning an initial position of x = 0, determine an expression for the position of the car-object system after the collision as a function of time t. (c) Determine an expression for the resisting force on the car-object system after the collision as a function of time t. (d) Determine the impulse delivered to the car-object system from t = 0 to t = 2.0 s Homework Tuesday A ferryboat of mass M 1= 2.0 x 10 5 kilograms moves toward a docking bumper of mass M2 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 A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement; i.e., F = -kx3 This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released. Determine each of the following in terms of k, A, and M a. The potential energy in the spring at the instant the mass is released b. The maximum speed of the mass c. The displacement of the mass at the point where the potential energy of the spring and the kinetic energy of the mass are equal The amplitude of the oscillation is now increased d. State whether the period of the oscillation increases, decreases, or remains the same. Justify your answer A thin plastic rod has uniform linear positive-charge density λ. The rod is bent into a semicircle of radius R as shown above. a. Determine the electric potential Vo at point 0, the center of the semicircle. b. Indicate on the diagram above the direction of the electric field at point O. Explain your reasoning. c. Calculate the magnitude Eo of the electric field at point O. d. Write an approximate expression, in terms of q, V ∞ and Eo, for the work required to bring a positive point charge q from infinity to point P, located a small distance s from point O as shown in the diagram above.