# Sample Exams

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<p>1</p>
<p>2 Problem No. 1 Given: A particle P is traveling in the xy-plane on a path given by:</p>
<p>xy = 6</p>
<p>; x and y are given in ft</p>
<p>and in such a way that the x-component of the velocity is a constant 20 ft/sec. Find: At position of x = 2 feet: a) determine the Cartesian components of the velocity and acceleration of P. b) make an accurate sketch of the velocity and acceleration vectors found in a) as well as the path unit vectors e t and e n on the figure provided below. c) determine the rate of change of speed and the radius of curvature for the path of P. d) is the speed of P increasing or decreasing? Provide an explanation for your response.</p>
<p>Sample Exam QuestionsFull-Length Problems ME 274</p>
<p>y P</p>
<p>x</p>
<p>Answer:</p>
<p>v P = 20 i ! 30 j ft / sec a P = 600 j ft / sec 2</p>
<p>( (</p>
<p>)</p>
<p>)</p>
<p>3 Problem No. 2 Given: Blocks A and B are connected by a cable that has a length of L = 10 meters with the cable being pulled over a pulley at C. Block A is constrained to move along a guide in such a way that the its acceleration is a function of the position sA as: aA = 0.3 sA2 (meters/sec2) with the speed of A being zero when sA = 0. Block B is constrained to move along a surface that is perpendicular to the guide for A. Assume that the cable does not stretch or go slack during the motion of the system. Also assume that the pulley C is small compared to the other dimensions of the problem. Find: When sA = 4 meters, a) find the speed of block A. b) find the speed of block B.</p>
<p>4 Problem No. 3 Given: A particle P travels on a circular path with a constant speed of v = 20 m/sec. A observer at point O watches the motion of P in terms of the radial distance r and rotation angle of !. At the position shown ! = 36.87 and P is directly above the circles center C Find: For the position of P shown: a) draw the path and polar unit vectors for P. b) determine numerical values for r and ! . r c) determine numerical values for and ! .</p>
<p>vr</p>
<p>P</p>
<p>!</p>
<p>C3m</p>
<p>Answers: vA = 3.58 m / sec vB = 2.86 m / sec</p>
<p>O ASB</p>
<p>cable</p>
<p>SA</p>
<p>C B</p>
<p>3 meters</p>
<p>5 Problem No. 4 Given: Bar BC is pinned to ground at pin C and is pinned to bar AB at pin B. Pin A at the left end of bar AB is constrained to move within a circular slot having a radius of 2 meters and center at point O. Pin A is known to be traveling with a constant speed of 12 meters/sec. At the instant shown, bar BC is vertical, AB is horizontal and point A is on the same horizontal line as point O. Find: At the position shown: a) determine the angular velocity of bars AB and BC. b) determine the angular acceleration of bars AB and BC. Express your answers as vectors.vA</p>
<p>6 Problem No. 5 Given: A disk having a radius of r = 1.5 ft is rolling without slipping on a rough horizontal surface to the right with its center O moving at a CONSTANT speed of vO = 20 ft / sec . A rigid bar AB having a length of 4 ft is attached to point A on the circumference of the disk. The other end of AB is attached a second rigid bar, BD (having a length of 3 ft), at pin B with point D pinned to ground. At the position shown, bar AB has a horizontal orientation, bar BD has a vertical orientation and point A is on the same horizontal line as point O. Find: At the position shown, a) find the angular velocities of bars AB and BD. b) find the angular accelerations of bars AB and BD. c) show (or describe in words) the location of the instant center for link AB.</p>
<p>2m</p>
<p>A O</p>
<p>3m</p>
<p>B</p>
<p>1m</p>
<p>B</p>
<p>A</p>
<p>O 1.5 ft</p>
<p>vO</p>
<p>slot</p>
<p>C</p>
<p>no slip</p>
<p>3 ft Answers: C</p>
<p>! AB = 0</p>
<p>! BC = 120k rad / sec 2</p>
<p>(</p>
<p>)</p>
<p>D</p>
<p>Answers: ! BD = 122.2 rad / sec 2 (CW )</p>
<p>! AB = 33.3rad / sec 2</p>
<p>(CCW )</p>
<p>7 Problem No. 6 Given: A particle P is constrained to move along a circular guide of radius r = 0.5 meters, as shown in the figure below. This guide is attached to a square plate (having dimensions of 2r x 2r) with the plate rotating CW about a vertical shaft passing through the plates center O with a constant rate of " = 4 rad/sec. The speed of P relative to the guide is known to be a constant u = 10 m/sec. Find: Determine the acceleration of P when it reaches point A on the guide. Express your answer in vector form.</p>
<p>8 Problem No. 7 Given: Arm AB is pinned to ground at pin A and is rotating CW at a rate of 4 rad/sec. Pin P is constrained to slide in a slot that is cut into arm AB. Pin P is also attached to the outer circumference of a wheel that rolls without slipping on a horizontal surface. At the instant shown, arm AB is horizontal, P is at an angular position that is 36.87 from the vertical and the distance from A to P is 3 feet, all as shown in the figure below. Find: At the position shown: a) determine the angular velocity of the wheel. b) determine the velocity of pin P as seen by an observer riding along on bar AB. Express your answers as vectors in terms of components xy, where the xy coordinate system is attached to bar AB as shown below.</p>
<p>HINT: In solving this problem, use an observer that is attached to the rotating plate. y A r O P u</p>
<p>circular guide attached to rotating plate</p>
<p>x</p>
<p>"</p>
<p>TOP VIEW</p>
<p>Answer:</p>
<p>! w = 13.33k rad / sec</p>
<p>(</p>
<p>)</p>
<p>9 Problem No. 8 Given: An L-shaped arm AO rotates about a fixed vertical axis with a constant rate of ! = 5 rad /sec . Particle P slides on a straight bar OB which is pinned to OA at O. Bar OB is raised at a constant rate of ! = 3 rad /sec . When ! = 90 (position shown below right), it is known that R = 2 meters , R = 4 m /sec and R = !2.5 m /sec 2 . An observer is attached to arm OB along with coordinate axes xyz. Coordinate axes XYZ are fixed. For the position with ! = 90 , find the acceleration of particle P. x B Y x y R !0.5 m</p>
<p>10 Problem No. 9 Particles A and B (having masses of 4 kg and 8 kg, respectively) are constrained to move on a smooth horizontal surface. Particle A moves directly to the right with a speed of vA1 = 20 m / sec when it strikes the stationary particle B. After impact, A is known to move in a direction that is parallel to the contact surface with B. Assume that the contact surface of A and B during impact is smooth. Determine a) b) the speed of A and B after impact. the coefficient of restitution, e, for the impact of A and B.</p>
<p>Find:</p>
<p>P O X</p>
<p>B</p>
<p>Y R y " O</p>
<p>P ! X</p>
<p>A</p>
<p>"</p>
<p>A</p>
<p>Answer: a P = " !20.5i + 36.5 j + 60k $ m / sec 2 # %</p>
<p>Answer:</p>
<p>e = 0.5</p>
<p>11 Problem No. 10 Particle P, having a mass of 40 kg, slides along a rough circular path with a radius of 2 meters. A constant vertical force F = 300 newtons acts in the downward direction on P. At the instant shown, the P has a speed of 15 m/sec in the direction shown and has an acceleration pointing horizontally to the right. At the position shown, find i) ii) the normal force acting on P by the circular path, and the friction force acting on P.v g</p>
<p>12 Problem No. 11 Particle A is rigidly attached to bar OA (having a length of 3 meters), with OA being pinned to ground at point O such that OA moves in a HORIZONTAL plane. A second particle B is able to slide without friction on OA, and is attached to A with a spring having a stiffness of k = 500 N/m and an unstretched length of 1.5 meters. Particles A and B each have a mass of 10 kg, and bar OA has a mass that is negligible compared to A and B. At the instant shown, bar OA is rotating CW with a speed of !1 = 20 rad / sec , particle B is not moving relative to bar OA and the spring is unstretched. Find the velocity of particle B after it has moved 1 meters outward on the bar. Write your answer as a vector. HINT: Use both the work-energy equation and the angular impulse-momentum equation in your solution.</p>
<p>P</p>
<p>30</p>
<p>2 meters</p>
<p>a</p>
<p>F</p>
<p>Answer:f = 7448 newtons</p>
<p>v B2 = 33.6 u R + 36.8 u ! m / sec</p>
<p>(</p>
<p>)</p>
<p>13 Problem No. 12 Pellet P having a mass of m = 10 kg is pushed through a barrel (having negligible mass) by means of compressed air such that the force on the pellet by the compressed air is a constant F = 900 newtons. The barrel is constrained to move in a HORIZONTAL plane by rotating about point O. The system is released with R = 1.5 meters, ! = 10 rad/sec (CCW) and with the pellet stationary with respect to the barrel. When the pellet is at a position with R = 2 meters, a) find the angular velocity of the barrel (using the angular impulse-momentum equation). b) find the velocity of the pellet (using the results from a) and the work-energy equation). Write your answers as vectors. Include an accurate free body diagram of the pellet and a sketch of the coordinate axes used in determining your solution.</p>
<p>14 Problem No. 13 Given: An inextensible cord connects particle B (having a mass of 30 kg) to ground with the cord being pulled over smooth pulleys at D and C. Pulley D is connected to particle A (having a mass of 10 kg). The coefficients of static friction between A and ground and between B and ground are identical, k = 0.3 . Initially B is moving down the incline with a speed of 5 m/sec. Find: a) Determine the work done on A and B by friction after B has moved 4 meters down the incline. b) Find the speed of B after it has traveled 4 meters down the incline.</p>
<p>D A</p>
<p>C</p>
<p>E</p>
<p>B</p>
<p>36.87 P F Answer: vB2 = 7.20 m / secsmooth</p>
<p>R</p>
<p>! O</p>
<p>HORIZONTAL PLANE</p>
<p>15 Problem No. 14 Given: A billiard ball having a mass of m = 0.3 kg strikes a bumper at A with a speed of v 0 = 10 m /sec at an angle of ! = 53.13. Following this, the ball strikes a second bumper at B. The coefficients of restitution between the ball and bumpers at A and B are known to be 0.5 and 0.3, respectively. Assume that the ball moves on a horizontal plane at all times and that all surfaces are smooth. Find: a) Determine the rebound angle # of the ball after its impact with bumper B. b) Determine the speed of the ball v f after its impact with bumper B.</p>
<p>16 Problem No. 15 Given: The motion of a thin, homogeneous bar having a mass of m =30 kg and length L = 2 meters is constrained such that ends A and B move along two smooth guides. The bar is released from rest at Position 1 shown below. Ignore the mass of the rollers at A and B. Find: a) Draw a free body diagram (FBD) of the bar. b) Determine which forces in the FBD not included in the potential energy do work. c) Find the velocity of the center of mass when the bar is in Position 2 shown below where the bar is horizontal. Write your answer as a vector.smooth</p>
<p>A Position 1 L Bsmooth 36.87 53.13</p>
<p># vf v0 $ B Position 2 e = 0.3 B Answers: ! = 13.5 A e = 0.6 A53.13</p>
<p>v f = 4.93 m / sec</p>
<p>17 Problem No. 16 Given: A thin, homogeneous bar AB (having a length of 3 meters and a mass of 100 kg) is suspended by a cable BC that has a length of 2 meters and has negligible mass. End A of the bar is constrained to move along a smooth inclined plane. A force F acts a parallel to the incline in such a way that the speed of A is v A = 15 m /sec = constant. At the instant shown bar AB is horizontal and cable BC is vertical. Find: For the position shown: a) Using kinematics, find the acceleration of the bars center of mass G and the angular acceleration of the bar. Express your answers as vectors. b) Find the force F and the tension in cable BC.</p>
<p>18 Problem No. 17 A thin homogeneous bar of length L = 2 meters and mass m = 40 kg is pinned to ground at point O. A spring having an unstretched length of 2.5 meters and stiffness of K = 1000 N/m is attached between end A and pin B. A homogeneous disk with a mass of M = 120 kg and radius R = 0.8 meters is PINNED to end A of the bar. The disk rolls without slipping on the inside of a circular surface. The system is released from rest with ! = 0. Find the angular velocity of the bar when ! = 90. You need to include an appropriate FBD for the system used as well as an indication of the DATUM line(s) used in your analysis.</p>
<p>B</p>
<p>C 2m 3m B G</p>
<p>g F A g53.13</p>
<p>!no slip</p>
<p>O</p>
<p>R</p>
<p>K A</p>
<p>vA</p>
<p>T = 1840 newtonsF = ! 1588 newtons</p>
<p>19 Problem No. 18 Given: Homogeneous, thin bar AB (having a length of L = 3 meters and a mass of m = 50 kg) is released from rest at a horizontal orientation with end B in contact with a smooth, inclined surface. Find: Determine the angular acceleration of the bar immediately after release. Write your answer as a vector. AL</p>
<p>20 Problem No. 19 Given: A spring of stiffness k and unstretched length L0 is attached between fixed point A and the center O of a homogeneous disk (disk has a mass of m and outer radius of r). At position 1, when the disk is at rest, a constant force F is applied to the right at O. Between positions 1 and 2 the disk rolls without slipping, and between positions 2 and 3 the surface on which the disk moves is smooth. Find: a) b) Determine the velocity of point O at position 2. Determine the velocity of point O at position 3.</p>
<p>B</p>
<p>smooth</p>
<p>36.87=!</p>
<p>Use the following parameter values in your calculations: k = 7000N /m , L0 = 0.5 m , m = 100kg , r = 0.1m and F = 800 N .</p>
<p>! = 4.30 k rad / sec 20.3 m</p>
<p>(</p>
<p>)</p>
<p>A</p>
<p>r O</p>
<p>F O O</p>
<p>0.4 m 1</p>
<p>0.4 m 2</p>
<p>NO SLIP</p>
<p>SMOOTH</p>
<p>3</p>
<p>vO3 = 5.40m / sec</p>
<p>21 Problem No. 20 Given: A homogeneous drum (having a mass of m = 50 kg, outer radius of R = 0.5 meters and centroidal radius of gyration of kO = 0.4 meters) is pinned to block B (having a mass of M =150 kg) at O. The drum and block are suspended by a cable and spring (of stiffness K = 1000 N/meter) as shown in the figure below. This system is released from rest with the spring being stretched by an amount of 0.2 meters. Assume that the cable does not slip on the drum and that the bearing at O is smooth. Find: Find the SPEED of block B after it has dropped 0.1 meters.</p>
<p>22 Problem No. 21 Given: A thin, homogeneous bar of length L = 2 meters and mass m = 40 kg is released from rest at an angle ! = 36.87 from a smooth horizontal surface, as shown. Find: Determine the acceleration of the center of mass G of the bar on release. Write your answer as a vector.</p>
<p>aG = !6.45 m / sec 2 jvB = 1.083 m / sec</p>
<p>(</p>
<p>)</p>
<p>m</p>
<p>L G</p>
<p>R</p>
<p>K</p>
<p>!A</p>
<p>gsmooth</p>
<p>C OM</p>
<p>m</p>
<p>A</p>
<p>B</p>
<p>23 Problem No. 22 Given: A stepped drum has a mass of m, inner radius r, outer radius 2r and centroidal radius of gyration of kO. A cable wrapped around the inner radius of the drum is connected to block D, also having a mass of m. A second cable is wrapped around the outer radius and connected to a spring of stiffness K, as shown in the figure below. Let ! represent the rotation of the drum as measured from the position for which the spring is unstretched. Find: For this system: a) Determine the differential equation of motion in terms of the coordinate !. b) Determine the natural frequency of free vibrations. Leave answer in terms of m, r, kO and K. m</p>
<p>24 Problem No. 23 Given: A sinusoidal forcing f ( t ) = f 0 sin !t acts on particle A of the springmass system shown below.x Am</p>
<p>f(t)</p>
<p>5k</p>
<p>k</p>
<p>Find: a) Draw a free body diagram of particle A and derive the differential equation of motion for the system. b) Derive the particular so...</p>

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