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Assign9 - PHY 53 Summer 2010 Assignment 9 1 A heavy...

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Assignment 9 1. A heavy cylindrical container is being rolled up an incline as shown, by applying a force parallel to the incline. The static friction coefficient is μ s . The cylinder has radius R , mass m and moment of inertia about its symmetry axis I . a. Draw the cylinder and the incline and make a free-body diagram showing all the forces on the cylinder and where they are applied. b. Assume the motion up the incline is at constant speed, and that d > R . What direction is the static friction force? How do you know? c. Assuming μ s is large enough, or θ is small enough, to allow the cylinder to roll without slipping, how large must F be? Ans : F = R d mg sin θ . 2. In the previous problem: a. What is the best value for d , i.e., the value that lets F be as small as possible? b. Show that the minimum value of F is 1/2 the force one would need to push the cylinder up a frictionless incline of the same angle θ . c. Suppose the person applies this minimum force to roll the cylinder up the incline. What is the largest angle of incline that can be used, in terms of μ s ? Ans : tan θ = 2 μ s . d F θ PHY 53 Summer 2010 1
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3. This situation is like one in Assignment 3, except that now the pulley’s inertia must be taken into account. The incline is frictionless, the string is ideal, but the pulley has radius R and moment of inertia I about its frictionless axle. Call T 1 the tension in the string attached to M , and T 2 the tension in the string attached to m .
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