Exam 3 December 1, 2010

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1 Exam 3 Instructions: You have 60 minutes to complete this exam. This is a closed-book, closed-notes exam. You are allowed to use a calculator during the exam. All work must be shown to receive credit. Usage of mobile phones and other electronic communication devices is NOT permitted during the exam. Problem 1 (20 pts) 2 (20 pts) 3 (20 pts) Total Score Name: Exam 3 Page 1 of 8

2 Problem 1 (20 points): Given: A stepped disk has outer radius R 0, inner radius R i, mass m d and radius of gyration k G about the center of mass G. A rope is wrapped around the inner disk, guided by a pulley and attached to a block of mass m b. The string and the pulley are massless. All bodies are rigid. The disk rolls without slip and the string does not slip with respect to the disk. Find: (a) Draw free body diagrams for the disk and block. (b) Find equations for the angular acceleration of the disk and acceleration of the block. (c) Determine the kinematic relationship between the acceleration of G and B. (d) Solve for the acceleration of point G in terms of the parameters provided in the figure. Exam 3 Page 2 of 8

3 This page is for extra work related to Problem 1. Exam 3 Page 3 of 8

4 Problem 2 (20 points): Given: A homogeneous disk having a mass of m and outer radius R is supported by a cable. This cable is attached to ground at end B and attached to a spring at end E. This spring has a stiffness of k with its upper end attached to ground at D. Let x represent the motion of the disk s center of mass O (measured positively downward) with the spring being unstretched when x = 0. Assume that the cable does not slip on the disk and does not go slack at any time. Find: (a) Draw a free body diagram of the disk. (b) Derive the single differential equation of motion (EOM) for the system in terms of the coordinate x and, at most, the following parameters: g, m, R and k. (c) Determine the natural frequency of the system. Exam 3 Page 4 of 8

5 This page is for extra work related to Problem 2. Exam 3 Page 5 of 8

6 Problem 3a (12 points): Shown below left is a thin ring (of mass m, outer radius R and center A) that is released from rest on a rough incline. Shown below right is a homogeneous disk (of mass m, outer radius R and center B) that is released from rest at the same height of the ring on rough incline. Both the ring and disk roll without slipping on their respective inclines. Let v A2 and v B2 represent the speeds of A and B, respectively, after both A and B have dropped through vertical distance of H. Circle the correct answer. DATUM The initial kinetic energy T 1 for each system is: (a) T 1 > 0 (b) T 1 = 0 (c) T 1 < 0 The initial potential energy V 1 for each system is: (a) V 1 > 0 (b) V 1 = 0 (c) V 1 < 0 The nonconservative work done in moving from position 1 to position 2 U1 2 NC for each system is: (a) U1 2 NC > 0 (b) U1 2 NC = 0 (c) U1 2 NC < 0 The relationship between between the final kinetic energies T 2 of the systems is best described by: (a) T 2A > T 2B (b) T 2A = T 2B (c) T 2A < T 2B Exam 3 Page 6 of 8

7 The final potential energy V 2 for each system is: (a) V 2 > 0 (b) V 2 = 0 (c) V 2 < 0 The relationship between the final speeds of the systems is best described by: (a) v A2 > v B2 (b) v A2 = v B2 (c) v A2 < v B2 Exam 3 Page 7 of 8

8 Problem 3b (8 points): The non-homogenous disk shown below has a center of mass located at G, a geometric center located at O, and an instantaneous center of rotation located a C. Points O and G are a distance e apart, and the disk is of radius r. Experimental analysis reveals that the disk has a centroidal radius of gyration given by k G and mass m. Find: (a) The moment of inertia of the disk about point O. (b) The moment of inertia of the disk about point C. G e 0 C r Exam 3 Page 8 of 8

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