Final Exam Instructions: You have 120 minutes to complete this exam. This is a closed-book, closed-notes exam. You are allowed to use a calculator during the exam. 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) 4 (20 pts) Total Score Name: Instructor: Final Exam Page 1 of 11
Problem 1 (20 points): Given: A rigid body rotates in the horizontal plane about a fixed vertical shaft at point O. This rigid body has a known radius of gyration k O and mass M. Inside this rigid body, a smaller particle of known mass m rides in a smooth slot, attached to the rigid body by a spring with known stiffness k. The spring has a natural length of L. At one instant in time, the particle is not moving with respect to the rigid body and is a distance r 1 = L from the fixed point O. At this instant, the rigid body has an angular velocity ω 1. At a second instant of time, the particle is at a position r 2 = 2L in the slot. Find: The velocity of the particle at the second instant of time. Make sure to write your answer as a vector. Use ω 1 = 10 rad/s, k O = 0.5 m, M = 4 kg, m = 1 kg, k = 3 N/m, and L = 1 m in your analysis. ω r m O θ Final Exam Page 2 of 11
This page is for extra work related to Problem 1. Final Exam Page 3 of 11
Problem 2 (20 points): Given: A homogeneous disk having a mass of m and outer radius R is pinned to ground at its center O and is actuated by an applied torque T (t) = P sin ωt. This drum is in geared-contact with block A. Block A, having a mass of 3m, is able to slide along a smooth horizontal surface and in such a way that the block does not slip in its contact with the drum. Two springs, having stiffnesses of k and 2k, are attached between block A and ground, as shown in the figure below. Let x represent the displacement of the block. When x = 0, the springs are unstretched. Find: (a) The equation of motion for the system in terms of the dynamic variable x; (b) The natural frequency of the system; (c) The system s particular solution x P (t) under the action of the applied torque T (t). T(t) x(t) Final Exam Page 4 of 11
This page is for extra work related to Problem 2. Final Exam Page 5 of 11
Problem 3 (20 points): Part A 4 pts Point P represents a passenger traveling in a automobile. The velocity and acceleration of P, v P and a P, respectively, are shown below at a given instant in time. Circle the item below that most closely describes the motion of P: (a) The speed of P is decreasing, and P is turning to the left. (b) The speed of P is increasing, and P is turning to the left. (c) The speed of P is decreasing, and P is turning to the right. (d) The speed of P is increasing, and P is turning to the right. (e) There is insufficient information to describe the change. Part B 4 pts A polar description with variables r and θ is used to describe the kinematics of point P. For a position with r = 0.5 m and θ = 2 radians, the velocity and acceleration vectors for P are known to be: v P = ( 6ê r + 2ê θ ) m/s a P = (10ê r ) m/s 2 respectively. Circle the item below that most accurately describes the speed of P: (a) The speed of P is increasing. (b) The speed of P is not changing. (c) The speed of P is decreasing. Final Exam Page 6 of 11
Part C 4 pts Blocks A and B are connected by an inextensible cable, as shown in the figure below. Assume that the radius of the pulley is small compared to the other dimensions of the problem. Block A moves along a horizontal path, and block B moves along a vertical path. At the instant shown, B is moving downward with a speed of v B. The distance h > 0. Circle the answer below that most accurately describes the speed of A, v A, as compared to the speed of B: (a) v A > v B (b) v A = v B (c) v A < v B (d) More information is needed about the problem in order to answer this question. Final Exam Page 7 of 11
Part D 8 pts A mechanism is made up of links OA, AB and DE. Pins D and E on link DE are constrained to move along straight guides. Link OA is pinned to ground at O and pinned to link AB at A. Link AB is also pinned to link DE at point B. Pin D moves to the right with a speed of v D. For the position shown: (a) Accurately locate the instant center for link DE. (b) Accurately locate the instant center for link AB. (c) Determine if link AB is rotating counterclockwise, rotating clockwise or is instantaneously at rest. Clearly indicate your results on the drawing below. Final Exam Page 8 of 11
Problem 4 (20 points): Part A 4 pts The force F is lifting block A; however, the speed of A is decreasing. The pulley is non-ideal (it has a non-zero mass moment of inertia). Circle the correct description below. (a) The tension in section C of the cable is larger than the tension in section D. (b) The tension in section C of the cable is smaller than the tension in section D. (c) The tension in section C of the cable is the same as the tension in section D. Part B 4 pts The homogeneous disk shown is moving to the right with its center having a constant speed. Circle the answer below that most accurately describes the friction force on the disk as it moves. (a) The friction force acts to the right. (b) The friction force acts to the left. (c) The friction force is zero. (d) A numerical value for the coefficient of friction is needed in order to answer this question. Final Exam Page 9 of 11
Part C 4 pts Consider Systems A and B shown below. System A is made up of a spring and block with the block moving in pure translation along a smooth horizontal surface. System B is made up of a spring and a homogeneous disk of mass m and outer radius R, with the center of the disk at O and the disk rolling without slipping on a horizontal surface. Each system has the same mass m and same spring stiffness k. Let ω na and ω nb represent the natural frequencies of Systems A and B, respectively. Circle the answer below that most accurately represents the natural frequencies for the two systems: (a) ω na > ω nb (b) ω na = ω nb (c) ω na < ω nb (d) More information is needed on the two systems in order to answer this question. Final Exam Page 10 of 11
Part D 8 pts The following equation of motion (EOM) has been derived for a single-degree-of-freedom system: 2ẍ + 48ẋ + 800x = 200 (a) Determine the undamped natural frequency ω n for the system. (b) Determine the damping ratio ζ for the system. (c) Determine if the system is undamped, underdamped, critically damped, or overdamped. (d) Determine the static deformation x st for the system. Final Exam Page 11 of 11