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Unformatted text preview: wolz (cmw2833) HW #10 Antoniewicz (57420) 1 This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Consider the problem of the solid sphere rolling down an incline without slipping. The incline has an angle , the spheres length up the incline is , and its height is h . At the beginning, the sphere of mass M and radius R rests on the very top of the incline. M h What is the minimum coefficient of friction such that the sphere rolls without slipping? The moment of inertia of a sphere with respect to an axis through its center is 2 5 M R 2 . 1. = 5 7 cos 2. = 3 7 tan 3. = 2 7 sin 4. = 3 7 sin 5. = 5 7 tan 6. = 2 7 tan correct 7. = 3 5 cos 8. = 2 7 cos Explanation: Consider the forces acting on the sphere: mg cos N f mg sin Using the parallel-axis theorem I = 2 5 M R 2 + M R 2 = 7 5 M R 2 , and because the sphere rolls without slipping, = a R . With the origin at the point of contact between the sphere and the incline surface, summationdisplay : M g R sin = I M g R sin = 7 5 M R 2 a R a = 5 7 g sin . The net force along the direction of the incline is summationdisplay F = M g sin - f = M parenleftbigg 5 7 g sin parenrightbigg , where f N = M g cos is the minimum no slipping criterion. Then M g sin - M g cos = M parenleftbigg 5 7 g sin parenrightbigg cos = parenleftbigg 1- 5 7 parenrightbigg sin = 2 7 tan . 002 10.0 points A solid cylinder of mass M = 14 kg, radius R = 0 . 42 m and uniform density is pivoted on a frictionless axle coaxial with its symmetry axis. A particle of mass m = 3 . 2 kg and initial velocity v = 14 m / s (perpendicular to the cylinders axis) flies too close to the wolz (cmw2833) HW #10 Antoniewicz (57420) 2 cylinders edge, collides with the cylinder and sticks to it. Before the collision, the cylinder was not ro- tating. What is the magnitude of its angular velocity after the collision? Correct answer: 10 . 4575 rad / s. Explanation: Basic Concept: Conservation of Angu- lar Momentum, L particle z + L cylinder z = const . The axle allows the cylinder to rotate without friction around a fixed axis but it keeps this axis fixed. Let the z coordinate axis run along this axis of rotation; then the axle may exert arbitrary torques in x and y directions but z 0. Consequently, the z componenent of the angular momentum must be conserved, L z = const, hence when the particle collides with the cylinder L before z, part + L before z, cyl = L z, net = L after z, part + L after z, cyl ....
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- Spring '08