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Unformatted text preview: PHY 7A  Review for Final Exam Dimitri DounasFrazer Spring 2008 In all affairs it’s a healthy thing now and then to hang a question mark on the things you have long taken for granted.Bertrand Russell Problem 1 Suppose a small tunnel is drilled through the earth at some latitude θ , as shown in the figure below. A ball placed in one opening of the tunnel rolls back and forth through the earth. For this problem, assume that the earth is a sphere of uniform density and that the size of the tunnel is negligible compared to the size of the earth. Assume further that one of the openings of the tunnel is located in Berkeley. The radius of the earth is 6 . 4 × 10 3 km and the city of Berkeley is located at a latitude of 38 ◦ north of the equator. R θ North Pole 1. Find the period of oscillation if the ball rolls without slipping. Express your answer in hours. 2. What is the maximum speed of the ball? Express your answer in miles per hour. (Hint: Use the WorkEnergy Theorem to avoid confusion about reference points.) 3. How do your answers to Parts (1) and (2) change if the opening of the tunnel is in Anchorage instead of Berkeley? The latitude of Anchorage is 61 ◦ north of the equator. 4. Repeat Parts (1) and (2) for a cylinder and a hoop. Which object has the smallest period of oscillation? Which has the largest maximum speed? 1 Problem 2 A cube with sidelength L and density ρ cube = αρ water floats in a calm pond. At equilibrium, the cube floats upright and is submerged to a depth L = αL . Your task is to investigate the stability of the cube’s equilibrium position; that is, you will determine whether the cube will wobble or roll over. Suppose the cube is tipped slightly from its equilibrium position so that its bottom surface is rotated by an angle θ relative to the surface of the water, as shown in the figure below. y x B θ P L’ O L C D A 1. The buoyant force acts at the center of mass of the displaced water. Hence, in order to compute the torque due to the buoyant force we must first find the water’s center of mass. The displaced water consists of the triangle ABC and the rectangle BCDO. Show that the center of mass of the triangle ABC is r ABC = ( L/ 3) ˆ i + ( L/ 3) tan θ ˆ j , relative to an origin placed at B. The x and y axes are oriented along the sides of the cube....
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This note was uploaded on 10/28/2009 for the course PHYSICS 7A/7B taught by Professor All during the Fall '08 term at Berkeley.
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