Sample Exams

1

2 Problem No. 1 Given: A particle P is traveling in the xy-plane on a path given by:

xy = 6

; x and y are given in ft

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.

Sample Exam Questions Full-Length Problems

y

ME 274

P

x

Answer:

( (

)

v P = 20 i ! 30 j ft / sec

)

a P = 600 j ft / sec 2

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:

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 circle’s center C

aA = 0.3 sA2 (meters/sec2)

Find:

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:

For the position of P shown: a) draw the path and polar unit vectors for P. b) determine numerical values for r˙ and !˙ . c) determine numerical values for ˙r˙ and !˙˙ .

When sA = 4 meters, a)

v

find the speed of block A.

b) find the speed of block B.

P

r

!

O

Answers: vA = 3.58 m / sec vB = 2.86 m / sec

A SB

SA

cable

C B

3 meters

C 3m

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.

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.

Express your answers as vectors.

b) find the angular accelerations of bars AB and BD. c)

vA

show (or describe in words) the location of the instant center for link AB.

2m

A

3m

B

O 1m

slot

B

A

C

O

vO

1.5 ft no slip

3 ft Answers:

C

! AB = 0

(

)

D

! BC = 120k rad / sec 2

Answers: ! BD = 122.2 rad / sec 2 (CW )

! AB = 33.3rad / sec 2

(CCW )

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 plate’s 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.

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:

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.

Determine the acceleration of P when it reaches point A on the guide. Express your answer in vector form.

HINT: In solving this problem, use an observer that is attached to the rotating plate. y A r O

circular guide attached to rotating plate

P

u

x

"

TOP VIEW

Answer:

(

)

! w = 13.33k rad / sec

9

10

Problem No. 8

Problem No. 9

Given:

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.

Find:

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 ˙˙ = !2.5 m /sec 2 . An observer is R = 2 meters , R˙ = 4 m /sec and R 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.

Determine a)

the speed of A and B after impact.

b)

the coefficient of restitution, e, for the impact of A and B.

x B x

Y

y

R

P

B

P

Y R

!

! 0.5 m

A

O

X

"

Answer: a P = "# !20.5i + 36.5 j + 60k $% m / sec 2

y

A

O

X

"

Answer:

e = 0.5

11

12

Problem No. 10

Problem No. 11

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

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.

i)

the normal force acting on P by the circular path, and

ii)

the friction force acting on P. v

g

Find the velocity of particle B after it has moved 1 meters outward on the bar. Write your answer as a vector. HINT:

P

30°

Use both the work-energy equation and the angular impulse-momentum equation in your solution.

2 meters

a

F

Answer: f = 7448 newtons

(

)

v B2 = 33.6 u R + 36.8 u ! m / sec

13

14

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).

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, µk = 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.

D

C

A

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.

B

E

36.87° F

P

Answer: vB2 = 7.20 m / sec smooth

R

! O

HORIZONTAL PLANE

15

16

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:

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

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.

A Position 1 L B smooth

36.87°

53.13°

# vf v0

B Position 2

$

e = 0.3 B

Answers: ! = 13.5°

v f = 4.93 m / sec

A

A 53.13°

e = 0.6

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 bar’s 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.

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.

B

C

F

2m 3m

A

T = 1840 newtons F = ! 1588 newtons

vA

no slip

g

B G

O

!

g

53.13°

K

R

A

19

20

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.

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:

Determine the angular acceleration of the bar immediately after release. Write your answer as a vector. A

L

B

smooth

36.87°=!

(

Find: a) b)

Determine the velocity of point O at position 2. Determine the velocity of point O at position 3.

Use the following parameter values in your calculations: k = 7000N /m , L0 = 0.5 m , m = 100kg , r = 0.1m and F = 800 N .

)

! = 4.30 k rad / sec 2

A 0.3 m

r

F

O

O

O

0.4 m 1

vO3 = 5.40m / sec

NO SLIP

0.4 m 2

SMOOTH

3

21

22

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:

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:

Find the SPEED of block B after it has dropped 0.1 meters.

(

Determine the acceleration of the center of mass G of the bar on release. Write your answer as a vector.

)

aG = !6.45 m / sec 2 j vB = 1.083 m / sec

R

m

C O M

B

K

L

m

G g

!

smooth

A

A

23 Problem No. 22 Given: A stepped drum has a mass of m, inner radius r, outer radius 2r and centroidal

24 Problem No. 23

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:

Given:

A sinusoidal forcing f ( t ) = f 0 sin !t acts on particle A of the springmass system shown below. x A

For this system:

m

5k

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.

K "1 + ( k / r )2 $ m O # %

!

A

O

a)

B

+

D

k

Find:

m

!n = 2

f(t)

K

Draw a free body diagram of particle A and derive the differential equation of motion for the system. b) Derive the particular solution x p ( t ) of the equation of motion found in a) above. c) Determine the amplitude of response for x p ( t ) corresponding to: k = 10,000 N /m , m = 600 kg , f 0 = 300 N and ! = 15 rad /sec . d) Make a sketch of x p ( t ) on the axes provided below. Be careful in indicating the phase of x p ( t ) relative to f ( t ) in your plot.

X = ! 4 mm

25

26

Problem No. 24 Given: A homogeneous disk having an outer radius of R and mass m rolls without slipping on a horizontal surface. A spring of stiffness k connects the disk center O to a moving support A, and a second spring of stiffness 4k connects O to a second moving support B. Both supports A and B have the same prescribed motion of y ( t ) = y 0 sin!t . Let x describe the motion of point O. Find:

For this problem: a) Draw a free body diagram (FBD) for the disk. b) From your FBD above, derive the differential equation of motion for the disk. c) Based on your equation of motion in b), what is the natural frequency for this system? Use m = 10 kg and k = 300 N/m. d) DERIVE the particular solution x P ( t ) of the equation of motion in b) using the parameters in c) and y 0 = 0.05 meters and ! = 15 rad /sec . e) Make a sketch of x P ( t ) . Carefully indicate the amplitude, period and phase of this response. y(t)

x

y(t)

k A

no slip

x P (t) = ! 0.04 sin 15t

4k O

R

B

27

28 Problem No. 1 A polar description with variables r and ! (shown below) is used to describe the kinematics of point P. For a position with r = 2 meters and ! = 0.5 radians, the following information is known about the time derivatives of r and !:

r˙ = 8 m /sec !˙ = "3 rad /sec ˙r˙ = !˙˙ = 0 For this position, determine the rate of change of speed of P. Is the speed increasing, decreasing or constant? Justify your response. You should include sketches of the velocity and acceleration vectors of P in the figure below.

Sample Exam Questions Short Answer Problems ME 274

P

r

! O

29

30 Problem No. 3

Problem No. 2 Blocks B and C are connected by a single inextensible cable, with this cable being wrapped around pulleys at D and E. In addition, the cable is wrapped around a pulley attached to block A as shown. Assume the radii of the pulleys to be small. Blocks B and C move downward with speeds of vB = 6 ft/sec and vC = 18 ft/sec,

Point P moves on a path described by y P = x P2 /2 with x P ( t ) = 3 sin !t . These coordinates have units of meters with t being given in seconds. Determine the acceleration of P at t = 0. Write your answer as a vector. y

respectively. Determine the velocity of block A when sA = 4 ft,

P

yP

3 ft

3 ft

D

E

sA

sC

sB

C

A vC B

vB

O

xP

x

31 Problem No. 4 The inner surface of a stepped drum (having inner and outer radii of R and 2R, respectively) rolls without slipping on a horizontal surface. The center of the drum moves to the left with a constant speed of vO, as shown below. Points A and B lie on the inner and outer surfaces, respectively, of the drum. At the instant shown, A is directly to the right of O, and B is directly below O. For this position, a) draw the velocity vectors for points A and B in the figure. b) draw the acceleration vectors for points A and B in the figure.

32 Problem No. 5 Link AB of the mechanism shown below is rotating counterclockwise at the configuration shown where AB and DE are vertical and BD is horizontal. At this configuration: a) link BD is rotating counterclockwise. b) link BD has zero angular speed at this instant. c) link BD is rotating clockwise.

B

D

You are not required to show any work for this problem. However, without supporting calculations or descriptions, no partial credit can be given. E A

2R vO R

B

O

A no slip

33 Problem No. 6 A particle P of mass m traveling on a horizontal surface strikes a smooth wall with a speed of v1 at an angle of !. The coefficient of restitution between the wall and the particle is 0 < e < 1. Circle the answer below that most closely describes the angle # at which the ball rebounds from the wall: a) ! < " d) ! = " y e) ! > " $ ! f) ! = 0 g) ! = 90°

P

34 Problem No. 7 Sphere A, having a mass of M, initially moves to the right with a speed of vA1. Sphere A then strikes sphere B, having a mass of 2M, which is initially at rest. Sphere A has zero velocity after impacting B. What is the coefficient of restitution between A and B?

vA1

A v1

v2

(initially moving to right)

B (initially stationary)

35 Problem No. 8 A particle moving in a horizontal plane approaches a wall with a speed of 10 ft/sec at the direction shown. After impact the particle leaves the wall with a speed of 7.5 ft/sec at the direction shown. The coefficient of restitution for this impact is: a) b) c) d)

e = 0.375 e = 0.75 e = 0.5625 e = 1.0

36.87 ° 7.5 ft/sec 53.13 °

y

36 Problem No. 9 A thin bar of length L and mass m is pinned to ground at A. Circle the point listed below that corresponds to the smallest mass moment of inertia: a) Pinned end A b) Center of mass G A c) Center of percussion B d) Free end D

x

G

10 ft/sec

B

D

37 Problem No. 10 A homogeneous disk of mass m rolls without slipping with its center moving to the right at a constant speed of vO on a horizontal surface. Circle the answer below that most accurately describes the friction force acting on the disk by the horizontal surface on which it rolls. a) The friction force acts to the right. b) The friction force acts to the left. c) The friction force is zero. d) The direction of the friction force cannot be determined without knowing the coefficient of friction.

O

vO

no slip

Problem No. 11 A homogeneous disk of mass m is acted upon by a torque T at its center O. Assume that the disk rolls without slipping on a horizontal surface. Circle the answer below that most accurately describes the friction force acting on the disk by the horizontal surface on which it rolls: a) The friction force acts to the right. T b) The friction force acts to the left. c) The friction force is zero. d) The direction of the friction force cannot be determined without knowing the coefficient of friction.

no slip

O

38 Problem No. 12 A rope is wrapped around a homogeneous disk having an outer radius of R and mass m. A force F acts on the right end of the rope. The center of the disk is known to have a downward acceleration. Assume that the rope does not slip on the disk. Circle the answer below which most accurately describes the tension in the rope: a) The tension in section AB is smaller C than the tension in section CD. b) The tension in section AB is equal to F the tension in section CD. c) The tension in section AB is larger than the tension in section CD. A d) Cannot answer question without knowing the radius R of the disk. g D B R

39 Problem No. 13 A force F acts on the end of a cable that is wrapped around pulley B. As a result, block A is being lifted; however, the speed of A is decreasing. The mass of the pulley is NOT negligible. Assume that the cable does not slip on the pulley. a) The tension in section C of the cable is larger than the tension in section D of the cable. b) The tension in section C of the cable is smaller than the tension in section D of the cable. c) The tension in section C of the cable is the same as the tension in section D of the cable. F

B (non-ideal pulley)

C

D

40 Problem No. 14 A solid, homogeneous disk of mass M and outer radius R is released from rest on a rough surface that is inclined at an angle of ! from the horizontal, as shown in Figure (a) below. A thin ring, also having a mass of M and outer radius of R is released from rest on a rough surface that is inclined at an angle of ! from the horizontal, as shown in Figure (b) below. Both the solid disk and the thin ring roll without slipping on the inclined surface. Circle the answer below that most accurately describes the translational speeds of the centroids of the disk and ring after each has dropped an elevation of h: 1. 2. 3. 4.

The disk is traveling faster than the ring. The ring is traveling faster than the disk. Both are traveling at the same speed. Not enough information provided to answer question.

Provide a justification of your answer using the work-energy equation.

A vA

aA solid, homogenous disk of mass M

no slip

thin ring of mass M

no slip h

!

Figure (a)

!

Figure (b)

41 Problem No. 15 The mass moments of inertia for three points A, B and D on a rigid body are given below: IA = 20 kg-m IB = 50 kg-m

2

2

ID = 30 kg-m

42 Problem No. 16 A homogeneous block slides to the left on a horizontal, rough surface. Circle the answer below which most accurately describes the location of the normal force on the block due to the ground as the block slides: 1. The normal force acts at a point that lies to the left of G.

2

where A, B and D are three arbitrary points lying on the body. The mass of the body is known to be 15 kg. Which of these three points (A, B or D) is physically closest to the center of mass of the rigid body? Provide a justification of your answer using the parallel axis theorem.

2. The normal force acts at a point that lies to the right of G. 3. The normal force acts directly under G. 4. Not enough information to determine the location of the normal force relative to G. G is the center of mass of the block. Provide a justification of your answer based on an FBD of the block and the NewtonEuler equations.

A D

sliding to the left

B G

43 Problem No. 17 Consider points A, B and G on a rigid body where G is the body’s center of mass with A and B being arbitrary points with known distances from G. Circle the correct answer below related to the relative sizes of the mass of moment of inertia of the body about these three points. a) The mass moment of inertia for point A is the largest of that for these three points. b) The mass moment of inertia for point B is the largest of that for these three points. c) The mass moment of inertia for point G is the largest of that for these three points. d) The mass moments of inertia for all three points are the same.

A

1 ft

44 Problem No. 18 A block of mass m is released from rest on a smooth incline, as shown in Figure I. A sphere of the same mass m is released from rest on an incline at the same angle with the sphere able to roll without slipping on the incline, as shown in Figure II. At a second position, both the block and sphere have traveled a distance d down their inclines. At this position, a) the block is traveling faster than the sphere. b) the sphere is traveling faster than the block. c) the block and sphere are traveling with the same speed.

d G

m

d

m no slip

smooth

0.3 ft B

! Part (ii) FIGURE I

!

FIGURE II

45

46

Problem No. 19 A REARWHEEL drive automobile is experiencing a forward acceleration on a horizontal roadway. Circle the figure below that correctly shows the direction of the friction forces between the tires and the roadway.

acceleration

front

acceleration

rear

friction

friction

front

rear

friction

acceleration

Problem No. 20 A force F acts at the end of a cable that is wrapped around an inner radius of a spool. The outer radius of the spool is able to roll without slipping on a horizontal surface. The figure shown below has been drawn to scale. a) The center of the spool O will accelerate to the left as a result of force F. b) The center of the spool O will accelerate to the right as a result of force F. c) The center of the spool O will not move as a result of force F.

friction

F O no slip

acceleration

C front

friction

rear

friction

front

friction

rear

friction

47

48

Problem No. 21 Consider the spring-mass system shown below where the stiffness parameter k is in N/m and the mass m is in kg. The natural frequency for this system is given by: a) p = 4k /m rad /sec b) p = 6k /m rad /sec c) p = 2! 4k /m rad /sec d) p = k /m rad /sec 3k k 2k

m