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ME240_CA2_W08_Solution

# ME240_CA2_W08_Solution - ME 240 Introduction to Dynamics...

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ME 240: Introduction to Dynamics and Vibrations Mechanical Engineering Department The University of Michigan Computer Assignment #2 Solution Assigned: 14 March 2008. Due: 28 March 2008 Introduction Dr. Perkins and his students have developed a 6 degree of freedom wireless inertial sensor for mounting on bowling balls; see Figure 1 . This sensor can be used to measure the the velocity of the center of mass and the angular velocity of the bowling ball as it is released from the bowler’s hand. Your job is to help Dr. Perkins predict the dynamics of the ball as it travels down the lane. Consider a bowling ball of mass m , moment of inertia about its center I = 2 / 5 mr 2 and radius r traveling along a bowling lane with length L and coefficient of friction μ ; see Figure 2 . The distance traveled by the ball is described by the coordinate x , and the rotation of the ball is described by the angle θ . The bowler releases the ball at x 0 and θ 0 , with initial speed ˙ x 0 and angular velocity ˙ θ 0 . The ball will slip while also rolling along the lane until it begins to roll without slipping. To describe the frictional behavior we will use the equation: F f = μN ( 2 e - v C /v f 1 + e - v C /v f - 1) (1) in which F f is the friction force, N

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