Final Exam December 15, 2014

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1 Final Exam Instructions: You have 120 minutes to complete this exam. This is a closed-book, closed-notes exam. You are allowed to use the ME approved calculator only 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: Berger (9:30am-10:20am) Cappelleri (1:30pm-2:20pm) Exam 3 Page 1 of 12

2 Problem 1 (20 points): Given: Particle B having a mass of m is rigidly attached to arm OB, where OB has negligible mass and a length of L. OB is pinned to ground at end O and is able to rotate about O without any frictional resistance. A second particle A (also having a mass of m) is able to slide along arm OB. When OB is rotating with a rotational speed of ω = ω 1, particle A is released from rest with respect to OB at the midpoint of the arm. Particle A slides outward on arm OB, eventually impacting particle B. The coefficient of restitution for the impact of A with B is e. Find: Determine the following in terms of the known variables (m, L, ω 1, e): (a) ω 2, the angular velocity of arm OB immediately before A impacts B using Angular Impulse- Momentum (b) Ṙ 2, the velocity of A along the ê R direction immediately before A impacts B using Work-Energy (c) V A3, the velocity of A immediately after its impact with B using your knowledge of central impacts. Write your answer as a vector in terms of its ê R and ê φ components. Exam 3 Page 2 of 12

3 This page is for extra work related to Problem 1. Exam 3 Page 3 of 12

4 Problem 2 (20 points): Given: The 2-kg square plate is pinned to the 5-kg collar at A, and the collar slides without friction on the horizontal slideway. A force of P = 100 N is applied to the collar as shown, and the system starts from rest. Find: Immediately after the force is applied to the collar, determine the following: (a) a C, the acceleration of the collar (b) α, the angular acceleration of the plate Exam 3 Page 4 of 12

5 This page is for extra work related to Problem 2. Exam 3 Page 5 of 12

6 Problem 3 (20 points): Given: A homogeneous disk (of mass m and outer radius r) rolls without slipping on a rough, horizontal surface. A spring (of stiffness 3k) is attached between the center O of the disk and a moveable base B. A second spring (of stiffness k) is attached between point B and ground. Base B is given a prescribed motion of x B (t) = b sin ωt. The coordinates x and x B are both zero when the springs are unstretched. Find: For this problem: (a) Draw a FBD for the disk assuming x B (t) > x (b) Derive the differential equation of motion for the disk in terms of coordinate x (c) Determine the numerical value for the natural frequency of the system (d) Derive the particular solution of the system, x p (e) Determine if the disk is moving in-phase or out-of-phase with the base B. Use the following parameters in your analysis: m = 20 kg, k= 1440 N/m, r = 0.25 m, b = 0.16 m, and ω = 10 rad/s.. Exam 3 Page 6 of 12

7 This page is for extra work related to Problem 3. Exam 3 Page 7 of 12

8 Problem 4 (20 points): Part A (5 points): A block is pushed along a rough horizontal surface as shown. For each force acting on the block (listed below), circle the best description of the role that force plays in this problem: is it a conservative force, a non-conservative force, or does it do no work? (+1 point each) (a) 500 N force: conservative non-conservative does no work (b) spring force: conservative non-conservative does no work (c) gravity: conservative non-conservative does no work (d) normal force: conservative non-conservative does no work (e) friction force: conservative non-conservative does no work Exam 3 Page 8 of 12

9 Part B (6 points): Consider systems A and B shown below. System A is made up of a spring of stiffness k and block of mass m, with the block sliding on a smooth horizontal surface. System B is made up of a spring of stiffness k and homogeneous disk of mass m and radius R. The center of the disk is at point O and it rolls without slipping on the horizontal surface. Let ω na and ω nb represent the natural frequencies of systems A and B, respectively. (a) (+2 points) draw a complete FBD for each system (b) (+2 points) derive the equation of motion for each system (c) (+2 points) Circle the correct natural frequency relationship: ω na > ω nb ω na = ω nb ω na < ω nb Exam 3 Page 9 of 12

10 Part C (5 points): Arm AB rotates about a fixed vertical axis with a constant rate of ω 1. A ring, with its center at O and of radius r, rotates about arm AB with a constraint rate of ω 2. A particle P moves along the right with θ = constant. Let the XY Z axes be fixed, and the xyz axes and the observer are attached to the ring. At the position shown, θ = 90 o and the xyz axes are aligned with the XY Z axes. We wish to use the following equation to determine the acceleration of P for the position shown: a P = a O + ( a P/O )rel + 2 ω ( v P/O ) rel + ω ( ω r P/O ) + α r P/O Provide expressions for each term in this equation, expressed in the XY Z coordinate system. (+1 point each) (a) a O = (b) ( a P/O )rel = (c) ω = (d) α = (e) ( v P/O )rel = Exam 3 Page 10 of 12

11 Part D (2 points): An inhomogeneous disk has its center of mass at G and is rotating with a constant angular velocity ω about the geometric center of the disk O. At position I, G is directly above O; in position II, G is directly below O. Let (F O ) I and (F O ) II represent the magnitudes of the reaction force on the disk at O at positions I and II, respectively. Circle the correct answer below. (a) (F O ) I > (F O ) II (b) (F O ) I = (F O ) II (c) (F O ) I < (F O ) II Exam 3 Page 11 of 12

12 Part E (2 points): The mechanism shown below is made up of rigid links AB, BD, and DE. Link AB is rotating in the counterclockwise direction with angular speed ω AB. (a) On the figure, clearly identify the location of the instant center for link BD. (b) Circle the correct answer for the rotational direction for DE: clockwise counterclockwise Exam 3 Page 12 of 12

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