Sample Exam Questions ME274 – Basic Mechanics II

1 1 Sample Exam Questions ME274 Basic Mechanics II Attached are copies of exam Questions from past semesters in ME 274. These problems cover material from throughout the course (both midterm exams and final exam). These problems are provided to you as a study guide for your exams . The level of difficulty may or may not be reflective of the Questions that you will encounter in your exams this semester. 2 3 Problem 1 Given: A particle P travels on a path described by the polar equation: ! R=2" (R in feet and ! in radians) with a constant speed of ! v=3ft/sec. Find: For the position of P at point A shown below: a) determine the values of ! R and ! " . b) make a sketch of path unit vectors ! ut and ! un, and the polar unit vectors ! uR and ! u" in the figure below.

2 C) determine the values of ! R and ! " . d) determine the radius of curvature of the path. Answers for c) and d): ! R = " = #= ! R P A 4 Problem 2 At the instant shown below, particle P has velocity and acceleration vectors, ! vP and ! aP that point directly down (vertical) and directly to the right (horizontal), respectively. Furthermore at this instant, R = 4 meters, ! = and ! aP=2m/sec2. The speed of P is known to be constant of 10 m/sec for all time. For the instant shown: a) show the polar unit vectors ! eR and ! e" in the figure below. b) find the values of ! R and " . c) find the values of ! R and " . d) determine the value of the radius of curvature of the path for point P. R!PvPaP Answer for c) and d): !

3 R =6m/sec " =2rad/sec ! R = " =# 5 Problem 3 Given: Particle P travels on a path given in terms of its polar coordinates: ! R=12"2 (feet) where ! is in radians and! " =4rad/sec = constant. Find: For ! = "/2, a) show the polar unit vectors uRand u! in the figure below at ! = "/2. b) find the velocity and acceleration in terms of their polar coordinates. c) show the path unit vectors utand un in the figure below at ! = "/2. d) determine the speed vP and the rate of change of speed of P. Answers for d): ! vP= ! v P= R ! path of P P 6 Problem 4 Given: A cord is being unwrapped from a spool with the cord having a CONSTANT speed of ! vcord=5ft/sec as it leaves the spool. The cord is then wrapped over a fixed peg O. At the free end of the cord is a particle P.

4 The angular rotation ! " for the free end of the cord is related to the radial distance R from O to P by: ! R2 " =constant=#10ft2/sec where R is in feet and ! " is in rad/sec. At the position shown below, R = 2 feet and ! "=#3rads. Find: For the position of P shown in the figure: a) Sketch the polar unit vectors ! eR and ! e" on the figure. b) Determine the velocity ! vP and acceleration ! aP of the particle P. Write your answers as vectors. c) Sketch ! vP and ! aP on the figure below. Include the path description unit vectors ! et and ! en in your sketch. d) Based on your sketch in c) above, is the rate of change of speed of P positive, negative or zero? Provide an explanation with your answer. Answers to part b): ! vP=5eR"5e#()ft/secaP=" ()ft/sec2 O ( fixed peg)vcordR!

5 Pspool 7 Problem 5 Given: Point P is traveling on a curved path, as shown below. The tangential component of acceleration of P, ! at, and the radius of curvature of the path # of P are known to be the following functions of the position s on the path: ! at=dvdt=3s2"= +1 where s is in meters and time is in seconds. When s = 0, the speed of P is 15 meters/sec. Find: At the position of P when s = 4 meters, a) determine the acceleration vector of point P, ! aP. Write your answer as a vector. b) make a sketch ! aP on the figure below. Include the path description unit vectors ! ut and ! un in your sketch. Answer to a): ! aP=48et+ ()m/sec2 8 Problem 6 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 = sA2 (meters/sec2) with the speed of A being zero when sA = 0.

6 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. Answers: ! vA= B A SA 3 meters C cable SB 9 Problem 7 Given: A cylinder with an outer radius of R = 3 ft rolls without slipping on a horizontal surface. The center of the disk O has an acceleration that is known in terms of the position x of: ! aO=32x2 (x in feet and aO in ft/sec2) At position 1 (x = 0), point O has a velocity of 1 ft/sec to the right. Find: At position 2, where x = 2 ft, a) determine the velocity of point O. b) determine the angular velocity and angular acceleration of the cylinder.

7 Write your answers as vectors. c) determine the velocity and acceleration of point A (at this position A is on the same horizontal line as O). Write your answers as vectors. d) sketch the velocity vector for A (from c) above) in the Position 2 figure below. Answers to part c): ! vA=3i+3j()ft/secaA=9i+6j()ft/sec2 no slip O O R aO A x Position 1 Position 2 10 Problem 8 Given: A mechanism is made up of two links AB and BC pinned together at point B, and with link AB pinned to ground at point A. Link BC is pinned to a block that is constrained to move on a horizontal plane. At the instant shown, AB is vertical, and point C is moving to the right with a speed of 10 m/sec and slowing down at a rate of 3 m/sec2. Find: At the instant shown, a) determine the angular velocity of links AB and BC.

8 Write your answers as vectors. b) determine the angular acceleration of links AB and BC. Write your answers as vectors. Answers to part b): ! "BC= ()rad/sec2"AB= ()rad/sec2 2m 1m A B C vC 11 Problem 9 Given: A disk having a radius of r = ft is rolling without slipping on a rough horizontal surface to the right with its center O moving at a CONSTANT speed of ! vO=20ft/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.

9 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. Answers for b): ! "BD=# rad/sec2 (CW)"AB= (CCW) A B D vO O 3 ft no slip ft C 12 Problem 10 Given: A mechanism is made up of links AB and BC and a wheel pinned to BC at the wheel s center C. The wheel rolls without slipping on a horizontal surface. Link AB rotates counterclockwise with a constant rate of 3 rad/sec. At the instant shown, link AB is vertical. Find: At the instant shown, a) determine the angular velocities of link BC and the wheel. b) determine the angular accelerations of link BC and the wheel. c) determine the velocity and acceleration of point E on the perimeter of the wheel (at the instant shown, E is on the same horizontal line as C).

10 Write your answers as vectors. 2m 1m A B C m no slip E $AB 13 Problem 11 Given: A disk having a radius of r = 2 ft is rolling without slipping on a rough horizontal surface to the right with its center O moving at a CONSTANT speed of ! vO=20ft/sec. A rigid bar having a length of L = 4 ft is attached to point A on the circumference of the disk. The other end is attached to a block at pin B with the block constrained to moving on a vertical guide. At the position shown, point A is on the same horizontal line as point O, and % = . Find: At the position shown, a) determine the velocity and acceleration of point B. Express your answers as vectors. b) show the location of the instant center of bar AB in the figure below.