Introduction to

STATICSand

DYNAMICS

Problem BookRudra Pratap and Andy Ruina

Spring 2001

c Rudra Pratap and Andy Ruina, 1994-2001. All rights reserved. No part ofthis book may be reproduced, stored in a retrieval system, or transmitted, inany form or by any means, electronic, mechanical, photocopying, or otherwise,without prior written permission of the authors.

This book is a pre-release version of a book in progress for Oxford UniversityPress.

The following are amongst those who have helped with this book as editors,artists, advisors, or critics: Alexa Barnes, Joseph Burns, Jason Cortell, IvanDobrianov, Gabor Domokos, Thu Dong, Gail Fish, John Gibson, Saptarsi Hal-dar, Dave Heimstra, Theresa Howley, Herbert Hui, Michael Marder, Elaina Mc-Cartney, Arthur Ogawa, Kalpana Pratap, Richard Rand, Dane Quinn, PhoebusRosakis, Les Schaeffer, David Shipman, Jill Startzell, Saskya van Nouhuys, BillZobrist. Mike Coleman worked extensively on the text, wrote many of the ex-amples and homework problems and created many of the figures. David Hohas brought almost all of the artwork to its present state. Some of the home-work problems are modifications from the Cornells Theoretical and AppliedMechanics archives and thus are due to T&AM faculty or their libraries in waysthat we do not know how to give proper attribution. Many unlisted friends,colleagues, relatives, students, and anonymous reviewers have also made helpfulsuggestions.

Software used to prepare this book includes TeXtures, BLUESKYs implemen-tation of LaTeX, Adobe Illustrator and MATLAB.

Most recent text modifications on January 21, 2001.

Contents

Problems for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 0Problems for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 2Problems for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 10Problems for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 15Problems for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 18Problems for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 31Problems for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 41Problems for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 60Problems for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 74Problems for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 83Problems for Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 88Problems for Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 100Answers to *d questions

Problems for Chapter 1 1

Problems for Chapter 1

Introduction to mechanics Because nomathematical skills have been taught so far, thequestions below just demonstrate the ideas andvocabulary you should have gained from thereading.

1.1 What is mechanics?

1.2 Briefly define each of the words below (us-ing rough English, not precise mathematicallanguage):

a) Statics,

b) Dynamics,

c) Kinematics,

d) Strength of materials,

e) Force,

f) Motion,

g) Linear momentum,

h) Angular momentum,

i) A rigid body.

1.3 This chapter says there are three pillarsof mechanics of which the third is Newtonslaws, what are the other two?

1.4 This book orgainzes the laws of mechanicsinto 4 basic laws numberred 0-III, not the stan-dard Newtons three laws. What are thesefour laws (in English, no equations needed)?

1.5 Describe, as precisely as possible, a prob-lem that is not mentionned in the book butwhich is a mechanics problem. State whichquantities are given and what is to be deter-mined by the mechanics solution.

1.6 Describe an engineering problem which isnot a mechanics problem.

1.7 About how old are Newtons laws?

1.8 Relativity and quantum mechanics haveoverthrown Newtons laws. Why are engineersstill using them?

1.9 Computation is part of modern engineering.

a) What are the three primary computerskills you will need for doing problemsin this book?

b) Give examples of each (different thatnthe examples given).

c) (optional) Do an example of each on acomputer.

2 CONTENTS

Problems for Chapter 2

Vector skills for mechanics

2.1 Vector notation and vec-tor addition

2.1 Represent the vector r = 5 m 2 m inthree different ways.

2.2 Which one of the following representationsof the same vector

F is wrong and why?

2N

3N -3 N + 2 N

13 N

13 N

a) b)

c) d)

23

23

problem 2.2:(Filename:pfigure2.vec1.2)

2.3 There are exactly two representations thatdescribe the same vector in the following pic-tures. Match the correct pictures into pairs.

a) b)

e) f)

c) d)

30o 30o

4N

2N

4N

2

3 N2 N(- + 3 )

3 N + 1 N 3 N( 13 + )problem 2.3:

(Filename:pfigure2.vec1.3)

2.4 Find the sum of forcesF1 = 20 N

2 N ,F2 = 30 N( 12 +

12 ), and

F3 =

20 N( +3 ).

2.5 In the figure shown below, the position

vectors are rAB = 3 ftk,rBC = 2 ft , and

rCD = 2 ft( + k). Find the position vectorrAD.

A

B C

D

k

rAB

rBC

rCD

problem 2.5:(Filename:pfigure2.vec1.5)

2.6 The forces acting on a block of massm = 5 kg are shown in the figure, whereF1 = 20 N, F2 = 50 N, and W = mg. Findthe sum

F (= F 1 +

F 2 +

W )?

34

43

W

F 1

F 2

problem 2.6:(Filename:pfigure2.vec1.6)

2.7 Three position vectors are shown in thefigure below. Given that rB/A = 3 m( 12 +

32 ) and

rC/B = 1 m 2 m , find

rA/C.

A

B

C

problem 2.7:(Filename:pfigure2.vec1.7)

2.8 Given that the sum of four vectorsFi , i =

1 to 4, is zero, whereF1 = 20 N,

F2 =

50 N , F3 = 10 N( + ), findF4.

2.9 Three forcesF = 2 N 5 N , R =

10 N(cos +sin ) and W = 20 N , sumup to zero. Determine the angle and draw theforce vector

R clearly showing its direction.

2.10 Given thatR1 = 1 N+1.5 N and

R2 =

3.2 N 0.4 N , find 2 R1 + 5R2.

2.11 For the unit vectors 1 and 2 shownbelow, find the scalars and such that

2 32 = .

x

y

60o

1

1

1

2

problem 2.11:(Filename:pfigure2.vec1.11)

2.12 In the figure shown, T1 = 20

2 N, T2 =40 N, and W is such that the sum of the threeforces equals zero. If W is doubled, find and such that

T1,

T2, and 2

W still sum up to

zero.

x

y

60o 45o

T1T2

Wproblem 2.12:

(Filename:pfigure2.vec1.12)

2.13 In the figure shown, rods AB and BC areeach 4 cm long and lie along y and x axes,respectively. Rod CD is in the xz plane andmakes an angle = 30o with the x-axis.

(a) Find rAD in terms of the variable length`.

(b) Find ` and such that

rAD =

rAB

rBC + k.

y

x

z

4 cm

4 cm

A B

D

C

`

30o

problem 2.13:(Filename:pfigure2.vec1.13)

2.14 Find the magnitudes of the forcesF1 =

30 N40 N and F2 = 30 N+40 N . Drawthe two forces, representing them with theirmagnitudes.

Problems for Chapter 2 3

2.15 Two forcesR = 2 N(0.16 +

0.80 ) andW = 36 N act on a particle.

Find the magnitude of the net force. What isthe direction of this force?

2.16 In Problem 2.13, find ` such that the lengthof the position vector rAD is 6 cm.

2.17 In the figure shown, F1 = 100 N andF2 = 300 N. Find the magnitude and directionof

F2

F1.

x

y

F1F2

30o45o

F 2-

F 1

problem 2.17:(Filename:pfigure2.vec1.17)

2.18 Let two forcesP and

Q act in the direc-tion shown in the figure. You are allowed tochange the direction of the forces by changingthe angles and while keeping the magni-itudes fixed. What should be the values of and if the magnitude of

P + Q has to be the

maximum?

x

y

P

Q

problem 2.18:(Filename:pfigure2.vec1.18)

2.19 Two points A and B are located in the xyplane. The coordinates of A and B are (4 mm,8 mm) and (90 mm, 6 mm), respectively.

(a) Draw position vectors rA andrB.

(b) Find the magnitude of rA andrB.

(c) How far is A from B?

2.20 In the figure shown, a ball is suspendedwith a 0.8 mlong cord from a 2 mlong hoist OA.

(a) Find the position vector rB of the ball.

(b) Find the distance of the ball from theorigin.

x

y

45o

0.8 m2m

O

A

B

problem 2.20:(Filename:pfigure2.vec1.20)

2.21 A 1 m 1 m square board is supportedby two strings AE and BF. The tension in thestring BF is 20 N. Express this tension as avector.

x

y

E

F

2 m2m

2.5 m

1 m

11

1 mBA

CD

plate

problem 2.21:(Filename:pfigure2.vec1.21)

2.22 The top of an L-shaped bar, shown in thefigure, is to be tied by strings AD and BD tothe points A and B in the yz plane. Find thelength of the strings AD and BD using vectorsrAD and

rBD.

A

B

1 m

2 m

30o

y

x

zproblem 2.22:

(Filename:pfigure2.vec1.22)

2.23 A cube of side 6 inis shown in the figure.

(a) Find the position vector of point F, rF,from the vector sum rF =

rD+

rC/D+

rF/c.

(b) Calculate |rF|.(c) Find rG using

rF.

y

x

z

A B

E F

GHCD

problem 2.23:(Filename:pfigure2.vec1.23)

2.24 A circular disk of radius 6 inis mountedon axle x-x at the end an L-shaped bar as shownin the figure. The disk is tipped 45o with thehorizontal bar AC. Two points, P and Q, aremarked on the rim of the plate; P directly par-allel to the center C into the page, and Q at thehighest point above the center C. Taking the

base vectors , , and k as shown in the figure,find

(a) the relative position vector rQ/P,

(b) the magnitude |rQ/P|.

A

12"

x x

45o

6"

O

Q

Q

D

DC

CP

6"

k

problem 2.24:(Filename:pfigure2.vec1.24)

2.25 Find the unit vector AB, directed frompoint A to point B shown in the figure.

x

y

1 m

1 m

2 m

3 m

A

B

problem 2.25:(Filename:pfigure2.vec1.25)

2.26 Find a unit vector along string BA andexpress the position vector of A with respect toB, rA/B, in terms of the unit vector.

x

z

y

A

B3 m

2.5 m

1.5 m

1 m

problem 2.26:(Filename:pfigure2.vec1.26)

2.27 In the structure shown in the figure, ` =2 ft, h = 1.5 ft. The force in the spring is F =krAB, where k = 100 lbf/ ft. Find a unit vectorAB along AB and calculate the spring forceF = FAB.

4 CONTENTS

x

y

h `

B

O

C

30o

problem 2.27:(Filename:pfigure2.vec1.27)

2.28 Express the vector rA = 2 m 3 m +5 mk in terms of its magnitude and a unit vectorindicating its direction.

2.29 LetF = 10 lbf + 30 lbf and W =

20 lbf . Find a unit vector in the direction ofthe net force

F + W , and express the the net

force in terms of the unit vector.

2.30 Let 1 = 0.80 + 0.60 and 2 = 0.5 +0.866 .

(a) Show that 1 and 2 are unit vectors.

(b) Is the sum of these two unit vectors alsoa unit vector? If not, then find a unitvector along the sum of 1 and 2.

2.31 If a mass slides from point A towards pointB along a straight path and the coordinates ofpoints A and B are (0 in, 5 in, 0 in) and (10 in,0 in, 10 in), respectively, find the unit vector

AB directed from A to B along the path.

2.32 Write the vectorsF1 = 30 N + 40 N

10 Nk,F2 = 20 N + 2 Nk, and

F3 =

10 N 100 Nk as a list of numbers (rowsor columns). Find the sum of the forces usinga computer.

2.2 The dot product of twovectors

2.33 Express the unit vectors n and in termsof and shown in the figure. What are the x

and y components of r = 3.0 ftn 1.5 ft?

n

x

y

problem 2.33:(Filename:efig1.2.27)

2.34 Find the dot product of two vectorsF =

10 lbf20 lbf and = 0.8+0.6 . SketchF and and show what their dot product rep-resents.

2.35 The position vector of a point A is rA =30 cm. Find the dot product of rA with =

32 + 12 .

2.36 From the figure below, find the componentof force

F in the direction of .

x

y

30o 10o

F = 100 N

problem 2.36:(Filename:pfigure2.vec1.33)

2.37 Find the angle betweenF1 = 2 N +

5 N andF2 = 2 N + 6 N .

2.38 A forceF is directed from point A(3,2,0)

to point B(0,2,4). If the x-component of theforce is 120 N, find the y- and z-componentsof

F .

2.39 A force acting on a bead of mass m is given

asF = 20 lbf+22 lbf +12 lbfk. What is

the angle between the force and the z-axis?

2.40 Given = 2 rad/s + 3 rad/s , H 1 =(20+30 ) kg m2/ s and H 2 = (10+15+6k) kg m2/ s, find (a) the angle between andH 1 and (b) the angle between

and

H 2.

2.41 The unit normal to a surface is given asn = 0.74 + 0.67 . If the weight of a blockon this surface acts in the direction, findthe angle that a 1000 N normal force makeswith the direction of weight of the block.

2.42 Vector algebra. For each equation belowstate whether:

(a) The equation is nonsense. If so, why?

(b) Is always true. Why? Give an example.

(c) Is never true. Why? Give an example.

(d) Is sometimes true. Give examples bothways.

You may use trivial examples.

a)

A+ B = B + Ab)

A+ b = b + Ac)

A B = B Ad)

B/

C = B/Ce) b/

A = b/Af)

A = ( A B) B+( A C) C+( A D) D

2.43 Use the dot product to show the law ofcosines; i. e.,

c2 = a2 + b2 + 2ab cos .(Hint: c = a+

b ; also, c c = c c )

bc

a

problem 2.43:(Filename:pfigure.blue.2.1)

2.44 (a) Draw the vector r = 3.5 in +3.5 in 4.95 ink. (b) Find the angle this vec-tor makes with the z-axis. (c) Find the anglethis vector makes with the x-y plane.

2.45 In the figure shown, and n are unit vec-tors parallel and perpendicular to the surfaceAB, respectively. A force

W = 50 N acts

on the block. Find the components ofW along

and n.

30o

A

BO

n

W

problem 2.45:(Filename:pfigure2.vec1.41)

2.46 From the figure shown, find the compo-nents of vector rAB (you have to first find thisposition vector) along

(a) the y-axis, and

(b) along .

y

x

z

AB

3 m30o

2 m

2 m 1 m

problem 2.46:(Filename:pfigure2.vec1.42)

2.47 The net force acting on a particle isF =

2 N + 10 N . Find the components of thisforce in another coordinate system with ba-sis vectors = cos + sin and = sin cos . For = 30o, sketch thevector

F and show its components in the two

coordinate systems.

2.48 Find the unit vectors eR and e in termsof and with the geometry shown in figure.

Problems for Chapter 2 5

What are the componets ofW along eR and

e ?

e

eR

`

W

problem 2.48:(Filename:pfigure2.vec1.44)

2.49 Write the position vector of point P interms of 1 and 2 and

(a) find the y-component of rP,

(b) find the component of rP alon 1.

x

y

1

2

`1

`2P

1

2

problem 2.49:(Filename:pfigure2.vec1.45)

2.50 What is the distance between the pointA and the diagonal BC of the parallelepipedshown? (Use vector methods.)

A 1

3

4

CB

problem 2.50:(Filename:pfigure.blue.2.3)

2.51 LetF1 = 30 N + 40 N 10 Nk,

F2 =

20 N + 2 Nk, and F3 = F3x + F3y F3z k. If the sum of all these forces must equalzero, find the required scalar equations to solvefor the components of

F3.

2.52 A vector equation for the sum of forcesresults into the following equation:

F

2(

3 )+ R5(3 + 6 ) = 25 N

where = 0.30 0.954 . Find the scalarequations parallel and perpendicular to .

2.53 Let F1 +

F2 +

F3 =

0 , where

F1,

F2, and

F3 are as given in Problem 2.32.

Solve for , , and using a computer.

2.54 Write a computer program (or use acanned program) to find the dot product oftwo 3-D vectors. Test the program by com-

puting the dot products , , and k.Now use the program to find the components

ofF = (2 + 2 3k)N along the line

rAB = (0.5 0.2 + 0.1k)m.2.55 Let rn = 1 m(cos n + sin n ), wheren = 0 n1 . Using a computer generatethe required vectors and find the sum

44n=0

ri , with 1 = 1o and 0 = 45o.

2.3 Cross product, moment,and moment about an axis

2.56 Find the cross product of the two vectorsshown in the figures below from the informa-tion given in the figures.

x

y

x

y

x

y

x

y

x

y

x

y

4

4

4

33

22

33

22

4

260o

30o45o

45o30o

43

x

y

x

y

(-1,2) (2,2)

(-1,-1) (2,-1)

a

b

b

b

a

a

b

a

a

b

b

a

a

b

a = 3 +

b = 4

(a) (b)

(c) (d)

(e) (f)

(g) (h)

105o

5

problem 2.56:(Filename:pfigure2.vec2.1)

2.57 Vector algebra. For each equation belowstate whether:

(a) The equation is nonsense. If so, why?

(b) Is always true. Why? Give an example.

(c) Is never true. Why? Give an example.

(d) Is sometimes true. Give examples bothways.

You may use trivial examples.

a)B C = C B

b)B C = C B

c)

C ( A B) = B ( C A)d)

A( B C) = ( A C) B( A B) C

2.58 What is the momentM produced by a 20

N force F acting in the x direction with a leverarm of r = (16 mm)?

2.59 Find the moment of the force shown onthe rod about point O.

x

y

O

F = 20 N

2 m45o

problem 2.59:(Filename:pfigure2.vec2.2)

2.60 Find the sum of moments of forcesW and

T about the origin, given that W =

100 N, T = 120 N, ` = 4 m, and = 30o.

x

y

O

T

W

`/2

`/2

problem 2.60:(Filename:pfigure2.vec2.3)

2.61 Find the moment of the force

a) about point A

b) about point O.

= 30o

F = 50 N

O2 m

1.5 m

A

problem 2.61:(Filename:pfigure2.vec2.4)

2.62 In the figure shown, OA = AB = 2 m. Theforce F = 40 N acts perpendicular to the armAB. Find the moment of

F about O, given that

= 45o. If F always acts normal to the armAB, would increasing increase the magnitudeof the moment? In particular, what value of will give the largest moment?

6 CONTENTS

x

y

O

F

AB

`

`

problem 2.62:(Filename:pfigure2.vec2.5)

2.63 Calculate the moment of the 2 kNpayloadon the robot arm about (i) joint A, and (ii) jointB, if `1 = 0.8 m, `2 = 0.4 m, and `3 = 0.1 m.

x

y

2 kN

A

B

O

30o

45oC

`1 `2

`3

problem 2.63:(Filename:pfigure2.vec2.6)

2.64 During a slam-dunk, a basketball playerpulls on the hoop with a 250 lbf at point C of thering as shown in the figure. Find the momentof the force about

a) the point of the ring attachment to theboard (point B), and

b) the root of the pole, point O.

O

3'

250 lbf15o

6"

1.5'

10'

board

B

Abasketball hoop

problem 2.64:(Filename:pfigure2.vec2.7)

2.65 During weight training, an athelete pullsa weight of 500 Nwith his arms pulling on ahadlebar connected to a universal machine bya cable. Find the moment of the force about theshoulder joint O in the configuration shown.

problem 2.65:(Filename:pfigure2.vec2.8)

2.66 Find the sum of moments due to thetwo weights of the teeter-totter when the teeter-totter is tipped at an angle from its verticalposition. Give your answer in terms of the vari-ables shown in the figure.

h

OB

OA = hAB = AC = `

W

W

C

A

`

`

problem 2.66:(Filename:pfigure2.vec2.9)

2.67 Find the percentage error in computingthe moment of

W about the pivot point O as

a function of , if the weight is assumed to actnormal to the arm OA (a good approximationwhen is very small).

O

A

W

`

problem 2.67:(Filename:pfigure2.vec2.10)

2.68 What do you get when you cross a vectorand a scalar?

2.69 Why did the chicken cross the road?

2.70 Carry out the following cross products indifferent ways and determine which methodtakes the least amount of time for you.

a) r = 2.0 ft + 3.0 ft 1.5 ftk; F =0.3 lbf 1.0 lbfk; r F =?

b) r = ( + 2.0 + 0.4k)m; L =(3.5 2.0k) kg m/s; r L =?

c) = ( 1.5 ) rad/s; r = (10 2 + 3k) in; r =?

2.71 A forceF = 20 N 5 Nk acts through

a point A with coordinates (200 mm, 300 mm,-100 mm). What is the moment

M(= r F )

of the force about the origin?

2.72 Cross Product program Write a programthat will calculate cross products. The input tothe function should be the components of thetwo vectors and the output should be the com-ponents of the cross product. As a model, hereis a function file that calculates dot products inpseudo code.

%program definitionz(1)=a(1)*b(1);z(2)=a(2)*b(2);z(3)=a(3)*b(3);w=z(1)+z(2)+z(3);

2.73 Find a unit vector normal to the surfaceABCD shown in the figure.

4"

5"5"

DA

C

B

x

z

y

problem 2.73:(Filename:efig1.2.11)

2.74 If the magnitude of a forceN normal

to the surface ABCD in the figure is 1000 N,write

N as a vector.

x

z

y

AB

D1m 1m

1m1mC

1m

problem 2.74:(Filename:efig1.2.12)

2.75 The equation of a surface is given as z =2x y. Find a unit vector n normal to thesurface.

2.76 In the figure, a triangular plate ACB, at-tached to rod AB, rotates about the z-axis. Atthe instant shown, the plate makes an angle of60o with the x-axis. Find and draw a vectornormal to the surface ACB.

x

z

y

60o

45o

45o

1mA

B

C

problem 2.76:(Filename:efig1.2.14)

2.77 What is the distance d between the originand the line AB shown? (You may write yoursolution in terms of

A andB before doing any

arithmetic).

Problems for Chapter 2 7

y

z

1B

dO

A

1

1

x

k

A

B

problem 2.77:(Filename:pfigure.blue.1.3)

2.78 What is the perpendicular distance be-tween the point A and the line BC shown?(There are at least 3 ways to do this using var-ious vector products, how many ways can youfind?)

x

y

B A

320

3

C

problem 2.78:(Filename:pfigure.blue.2.2)

2.79 Given a force,F 1 = (3+ 2 + 5k)N

acting at a point P whose position is given byr P/O = (4 2 + 7k)m, what is the mo-ment about an axis through the origin O with

direction = 25 + 1

5?

2.80 Drawing vectors and computing withvectors. The point O is the origin. Point A hasxyz coordinates (0, 5, 12)m. Point B has xyzcoordinates (4, 5, 12)m.

a) Make a neat sketch of the vectors OA,OB, and AB.

b) Find a unit vector in the direction ofOA, call it O A .

c) Find the forceF which is 5N in size

and is in the direction of OA.

d) What is the angle between OA and OB?

e) What is r BO F ?

f) What is the moment ofF about a line

parallel to the z axis that goes throughthe point B?

2.81 Vector Calculations and Geometry.The 5 N force

F 1 is along the line OA. The

7 N forceF 2 is along the line OB.

a) Find a unit vector in the direction OB.

b) Find a unit vector in the direction OA.

c) Write bothF 1 and

F 2 as the product

of their magnitudes and unit vectors intheir directions.

d) What is the angle AOB?

e) What is the component ofF 1 in the

x-direction?

f) What is r DO F 1? (

r DO r O/D

is the position of O relative to D.)

g) What is the moment ofF 2 about the

axis DC? (The moment of a force aboutan axis parallel to the unit vector is

defined as M = (rF )where r is

the position of the point of applicationof the force relative to some point onthe axis. The result does not dependon which point on the axis is used orwhich point on the line of action of

F

is used.).

h) Repeat the last problem using either adifferent reference point on the axis DCor the line of action OB. Does the solu-tion agree? [Hint: it should.]

y

z

x

F2

F1

4m

3m

5m

O

AB

CD

problem 2.81:(Filename:p1sp92)

2.82 A, B, and C are located by positionvectors r A = (1, 2, 3), r B = (4, 5, 6), andr C = (7, 8, 9).

a) Use the vector dot product to find theangle B AC (A is at the vertex of thisangle).

b) Use the vector cross product to find theangle BC A (C is at the vertex of thisangle).

c) Find a unit vector perpendicular to theplane ABC .

d) How far is the infinite line defined byAB from the origin? (That is, how closeis the closest point on this line to theorigin?)

e) Is the origin co-planar with the pointsA, B, and C?

2.83 Points A, B, and C in the figure define aplane.

a) Find a unit normal vector to the plane.

b) Find the distance from this infiniteplane to the point D.

c) What are the coordinates of the pointon the plane closest to point D?

1

2

3 4

4

5

5

7

(3, 2, 5) (0, 7, 4)

(5, 2, 1) (3, 4, 1)

B

D

AC

x y

z

problem 2.83:(Filename:pfigure.s95q2)

2.4 Equivalant force sys-tems and couples

2.84 Find the net force on the particle shownin the figure.

43

6 N

P

8 N

10 N

problem 2.84:(Filename:pfigure2.3.rp1)

2.85 Replace the forces acting on the parti-cle of mass m shown in the figure by a singleequivalent force.

30o45o

T

mg

m

2T

T

problem 2.85:(Filename:pfigure2.3.rp2)

2.86 Find the net force on the pulley due to thebelt tensions shown in the figure.

8 CONTENTS

30o

50 N

50 N

problem 2.86:(Filename:pfigure2.3.rp3)

2.87 Replace the forces shown on the rectan-gular plate by a single equivalent force. Whereshould this equivalent force act on the plate andwhy?

300 mm

200 mm

4 N

6 N

5 N

A D

B C

problem 2.87:(Filename:pfigure2.3.rp4)

2.88 Three forces act on a Z-section ABCDE asshown in the figure. Point C lies in the middleof the vertical section BD. Find an equivalentforce-couple system acting on the structure andmake a sketch to show where it acts.

C

D E

BA

60 N 0.5 m

0.5 m

0.6 m40 N

100 N

problem 2.88:(Filename:pfigure2.3.rp5)

2.89 The three forces acting on the circularplate shown in the figure are equidistant fromthe center C. Find an equivalent force-couplesystem acting at point C.

C

RF

F

32 F

problem 2.89:(Filename:pfigure2.3.rp6)

2.90 The forces and the moment acting on pointC of the frame ABC shown in the figure areCx = 48 N, Cy = 40 N, and Mc = 20 Nm.Find an equivalent force couple system at pointB.

Cx

Cy

CB

1.5 m

1.2 m

A

MC

problem 2.90:(Filename:pfigure2.3.rp7)

2.91 Find an equivalent force-couple systemfor the forces acting on the beam in Fig. ??, ifthe equivalent system is to act at

a) point B,

b) point D.

C

A B D

2 m

1 kN 2 kN

2 kN

2 m

problem 2.91:(Filename:pfigure2.3.rp8)

2.92 In Fig. ??, three different force-couplesystems are shown acting on a square plate.Identify which force-couple systems are equiv-alent.

0.2 m20 N

20 N

20 N

40 N

30 N

30 N

30 N

30 N

0.2 m

6 Nm

problem 2.92:(Filename:pfigure2.3.rp9)

2.93 The force and moment acting at point Cof a machine part are shown in the figure whereMc is not known. It is found that if the givenforce-couple system is replaced by a single hor-izontal force of magnitude 10 N acting at pointA then the net effect on the machine part is thesame. What is the magnitude of the momentMc?

20 cm

30 cm10 N

C

MC

A

B

problem 2.93:(Filename:pfigure2.3.rp10)

2.5 Center of mass and cen-ter of gravity

2.94 An otherwise massless structure is madeof four point masses, m, 2m, 3m and 4m, lo-cated at coordinates (0, 1 m), (1 m, 1 m), (1 m,1 m), and (0,1 m), respectively. Locate thecenter of mass of the structure.

2.95 3-D: The following data is given fora structural system modeled with five pointmasses in 3-D-space:

mass coordinates (in m)0.4 kg (1,0,0)0.4 kg (1,1,0)0.4 kg (2,1,0)0.4 kg (2,0,0)1.0 kg (1.5,1.5,3)

Locate the center of mass of the system.

2.96 Write a computer program to find the cen-ter of mass of a point-mass-system. The inputto the program should be a table (or matrix)containing individual masses and their coordi-nates. (It is possible to write a single programfor both 2-D and 3-D cases, write separate pro-grams for the two cases if that is easier foryou.) Check your program on Problems 2.94and 2.95.

2.97 Find the center of mass of the followingcomposite bars. Each composite shape is madeof two or more uniform bars of length 0.2 m andmass 0.5 kg.

(a) (b)

(c)

problem 2.97:(Filename:pfigure3.cm.rp7)

Problems for Chapter 2 9

2.98 Find the center of mass of the follow-ing two objects [Hint: set up and evaluate theneeded integrals.]

y

y

xO

m = 2 kg(a)

xO

(b)

r = 0.5 m

r = 0.5 m

m = 2 kg

problem 2.98:(Filename:pfigure3.cm.rp8)

2.99 Find the center of mass of the followingplates obtained from cutting out a small sec-tion from a uniform circular plate of mass 1 kg(prior to removing the cutout) and radius 1/4 m.

(a)

(b)

200 mm x 200 mm

r = 100 mm

100 mmproblem 2.99:

(Filename:pfigure3.cm.rp9)

10 CONTENTS

Problems for Chapter 3

Free body diagrams

3.1 Free body diagrams

3.1 How does one know what forces and mo-ments to use in

a) the statics force balance and momentbalance equations?

b) the dynamics linear momentum balanceand angular momentum balance equa-tions?

3.2 A point mass m is attached to a pistonby two inextensible cables. There is gravity.Draw a free body diagram of the mass with alittle bit of the cables.

A9a

6a

5a

GB

C

problem 3.2:(Filename:pfigure2.1.suspended.mass)

3.3 Simple pendulum. For the simple pendu-lum shown the body the system of interest is the mass and a little bit of the string. Drawa free body diagram of the system.

L

problem 3.3:(Filename:pfigure.s94h2p1)

3.4 Draw a free body diagram of mass m atthe instant shown in the figure. Evaluate theleft hand side of the linear momentum balanceequation (

F = ma) as explicitly as possi-

ble. Identify the unknowns in the expression.

hm = 10 kg

frictionless

F = 50 Nx

x

y

problem 3.4:(Filename:pfig2.2.rp1)

3.5 A 1000 kg satellite is in orbit. Its speed is vand its distance from the center of the earth is R.Draw a free body diagram of the satellite. Drawanother that takes account of the slight dragforce of the earths atmosphere on the satellite.

R

mv

problem 3.5:(Filename:pfigure.s94h2p5)

3.6 The uniform rigid rod shown in the figurehangs in the vertical plane with the support ofthe spring shown. Draw a free body diagramof the rod.

`

m k`/3

problem 3.6:(Filename:pfig2.1.rp1)

3.7 FBD of rigid body pendulum. The rigidbody pendulum in the figure is a uniform rodof mass m. Draw a free body diagram of therod.

uniform rigid bar, mass m

g

`

problem 3.7:(Filename:pfigure2.rod.pend.fbd)

3.8 A thin rod of mass m rests against a fric-tionless wall and on a frictionless floor. Thereis gravity. Draw a free body diagram of therod.

A

B

L

G

problem 3.8:(Filename:ch2.6)

3.9 A uniform rod of mass m rests in the backof a flatbed truck as shown in the figure. Drawa free body diagram of the rod, set up a suitablecoordinate system, and evaluate

F for the

rod.frictionless

m

problem 3.9:(Filename:pfig2.2.rp5)

3.10 A disc of mass m sits in a wedge shapedgroove. There is gravity and negligible friction.The groove that the disk sits in is part of anassembly that is still. Draw a free body diagramof the disk. (See also problems 4.15 and 6.47.)

1 2

x

y

r

problem 3.10:(Filename:ch2.5)

3.11 A pendulum, made up of a mass m at-tached at the end of a rigid massless rod oflength `, hangs in the vertical plane from ahinge. The pendulum is attached to a springand a dashpot on each side at a point `/4 fromthe hinge point. Draw a free body diagramof the pendulum (mass and rod system) whenthe pendulum is slightly away from the verticalequilibrium position.

Problems for Chapter 3 11

m

k kc c

1/4 `

3/4 `

problem 3.11:(Filename:pfig2.1.rp5)

3.12 The left hand side of the angular momen-tum balance (Torque balance in statics) equa-tion requires the evaluation of the sum of mo-ments about some point. Draw a free body di-agram of the rod shown in the figure and com-pute

MO as explicitly as possible. Now

compute MC. How many unknown forces

does each equation contain?

C O

m = 5 kgL/2

L

1

3

problem 3.12:(Filename:pfig2.2.rp3)

3.13 A block of mass m is sitting on a friction-less surface at points A and B and acted uponat point E by the force P . There is gravity.Draw a free body diagram of the block.

b2b

2d

d

C D

A B

G

PE

problem 3.13:(Filename:ch2.1)

3.14 A mass-spring system sits on a conveyerbelt. The spring is fixed to the wall on oneend. The belt moves to the right at a constantspeed v0. The coefficient of friction betweenthe mass and the belt is . Draw a free bodydiagram of the mass assuming it is moving tothe left at the time of interest.

mk

problem 3.14:(Filename:pfig2.1.rp6)

3.15 A small block of mass m slides down anincline with coefficient of friction . At aninstant in time t during the motion, the blockhas speed v. Draw a free body diagram of theblock.

m

problem 3.15:(Filename:pfig2.3.rp5)

3.16 Assume that the wheel shown in the fig-ure rolls without slipping. Draw a free bodydiagram of the wheel and evaluate

F and

MC. What would be different in the ex-pressions obtained if the wheel were slipping?

C

P

F = 10 N

m = 20 kg

rR

problem 3.16:(Filename:pfig2.2.rp4)

3.17 A compound wheel with inner radius rand outer radius R is pulled to the right bya 10 N force applied through a string woundaround the inner wheel. Assume that the wheelrolls to the right without slipping. Draw a freebody diagram of the wheel.

C

P

r

R

F = 10 N

m = 20 kg

problem 3.17:(Filename:pfig2.1.rp8)

3.18 A block of mass m is sitting on a fric-tional surface and acted upon at point E by thehorizontal force P through the center of mass.The block is resting on sharp edge at point Band is supported by a small ideal wheel at pointA. There is gravity. Draw a free body diagramof the block including the wheel, assuming theblock is sliding to the right with coefficient offriction at point B.

b2b

2d

d

C D

A B

G

PE

problem 3.18:(Filename:ch2.2)

3.19 A spring-mass model of a mechanicalsystem consists of a mass connected to threesprings and a dashpot as shown in the figure.The wheels against the wall are in tracks (notshown) that do not let the wheels lift off the wallso the mass is constrained to move only in thevertical direction. Draw a free body diagramof the system.

kk

kc

m

problem 3.19:(Filename:pfig2.1.rp2)

3.20 A point mass of mass m moves on a fric-tionless surface and is connected to a springwith constant k and unstretched length `. Thereis gravity. At the instant of interest, the masshas just been released at a distance x to theright from its position where the spring is un-stretched.

a) Draw a free body diagram of the of themass and spring together at the instantof interest.

b) Draw free body diagrams of the massand spring separately at the instant ofinterest.

(See also problem 5.32.)x

`

m

problem 3.20:(Filename:ch2.10)

3.21 FBD of a block. The block of mass 10 kgis pulled by an inextensible cable over the pul-ley.

a) Assuming the block remains on thefloor, draw a free diagram of the block.

b) Draw a free body diagram of the pulleyand a little bit of the cable that ridesover it.

12 CONTENTS

hm = 10 kg

frictionless

F = 50 Nx

x

y

problem 3.21:(Filename:pfigure2.1.block.pulley)

3.22 A pair of falling masses. Two masses A& B are spinning around each other and fallingtowards the ground. A string, which you canassume to be taught, connects the two masses.A snapshot of the system is shown in the figure.Draw free body diagrams of

a) mass A with a little bit of string,

b) mass B with a little bit of string, and

c) the whole system.

A

B

m

m 30o

problem 3.22:(Filename:pfigure.s94h2p4)

3.23 A two-degree of freedom spring-masssystem is shown in the figure. Draw free bodydiagrams of each mass separately and then thetwo masses together.

m1

x1k1

k2k4

k3

x2

m2

problem 3.23:(Filename:pfig2.1.rp4)

3.24 The figure shows a spring-mass model ofa structure. Assume that the three masses aredisplaced to the right by x1, x2 and x3 from thestatic equilibrium configuration such that x1 0, x < 0)

F = Ax + Bx2 + CxAssume x(0) = 0.Using numerical solution, find values ofA, B,C,m, and x0 so that

(a) the mass never crosses the origin,

(b) the mass crosses the origin once,

(c) the mass crosses the origin many times.

Problems for Chapter 5 19

5.21 A car accelerates to the right with constantacceleration starting from a stop. There is windresistance force proportional to the square ofthe speed of the car. Define all constants thatyou use.

a) What is its position as a function oftime?

b) What is the total force (sum of allforces) on the car as a function of time?

c) How much power P is required of theengine to accelerate the car in this man-ner (as a function of time)?

problem 5.21: Car.(Filename:s97p1.2)

5.22 A ball of mass m is dropped verticallyfrom a height h. The only force acting on theball in its flight is gravity. The ball strikesthe ground with speed v and after collisionit rebounds vertically with reduced speed v+directly proportional to the incoming speed,v+ = ev, where 0 < e < 1. What is the max-imum height the ball reaches after one bounce,in terms of h, e, and g.

a) Do this problem using linear momen-tum balance and setting up and solvingthe related differential equations andjump conditions at collision.

b) Do this problem again using energy bal-ance.

5.23 A ball is dropped from a height of h0 =10 m onto a hard surface. After the first bounce,it reaches a height of h1 = 6.4 m. What is thevertical coefficient of restitution, assuming itis decoupled from tangential motion? What isthe height of the second bounce, h2?

h0h1

h2g

problem 5.23:(Filename:Danef94s1q7)

5.24 In problem 5.23, show that the number ofbounces goes to infinity in finite time, assum-ing that the vertical coefficient is fixed. Findthe time in terms of the initial height h0, the co-efficient of restitution, e, and the gravitationalconstant, g.

5.2 Energy methods in 1D

5.25 The power available to a very strong ac-celerating cyclist is about 1 horsepower. As-sume a rider starts from rest and uses thisconstant power. Assume a mass (bike +rider) of 150 lbm, a realistic drag force of.006 lbf/( ft/ s)2v2. Neglect other drag forces.

(a) What is the peak speed of the cyclist?

(b) Using analytic or numerical methodsmake a plot of speed vs. time.

(c) What is the acceleration as t inthis solution?

(d) What is the acceleration as t 0 inyour solution?

5.26 Given v = gR2r2

, where g and R are

constants and v = drdt . Solve for v as a functionof r if v(r = R) = v0. [Hint: Use the chainrule of differentiation to eliminate t , i.e., dvdt =dvdr drdt = dvdr v. Or find a related dynamicsproblem and use conservation of energy.]

Also see several problems in the harmonic os-cillator section.

5.3 The harmonic oscillatorThe first set of problems are entirely aboutthe harmonic oscillator governing differentialequation, with no mechanics content or con-text.

5.27 Given that x = (1/s2)x , x(0) = 1 m,and x(0) = 0 find:

a) x( s) =?b) x( s) =?

5.28 Given that x + x = 0, x(0) = 1, andx(0) = 0, find the value of x at t = /2 s.5.29 Given that x +2x = C0, x(0) = x0, andx(0) = 0, find the value of x at t = / s.

The next set of problems concern one mass con-nected to one or more springs and possibly witha constant force applied.

5.30 Consider a mass m on frictionless rollers.The mass is held in place by a spring with stiff-ness k and rest length `. When the spring isrelaxed the position of the mass is x = 0. Attimes t = 0 the mass is at x = d and is let gowith no velocity. The gravitational constant isg. In terms of the quantities above,

a) What is the acceleration of the block att = 0+?

b) What is the differential equation gov-erning x(t)?

c) What is the position of the mass at anarbitrary time t?

d) What is the speed of the mass when itpasses through x = 0?

x

m

d`

problem 5.30:(Filename:pfigure.blue.25.1)

5.31 Spring and mass. A spring with restlength `0 is attached to a mass m which slidesfrictionlessly on a horizontal ground as shown.At time t = 0 the mass is released with noinitial speed with the spring stretched a distanced. [Remember to define any coordinates orbase vectors you use.]

a) What is the acceleration of the mass justafter release?

b) Find a differential equation which de-scribes the horizontal motion of themass.

c) What is the position of the mass at anarbitrary time t?

d) What is the speed of the mass when itpasses through the position where thespring is relaxed?

m

d`0

problem 5.31:(Filename:s97f1)

5.32 Reconsider the spring-mass system inproblem 3.20. Let m = 2 kg and k = 5 N/m.The mass is pulled to the right a distancex = x0 = 0.5 m from the unstretched posi-tion and released from rest. At the instant ofrelease, no external forces act on the mass otherthan the spring force and gravity.

a) What is the initial potential and kineticenergy of the system?

b) What is the potential and kinetic en-ergy of the system as the mass passesthrough the static equilibrium (un-stretched spring) position?

x`

m

problem 5.32:(Filename:ch2.10.a)

5.33 Reconsider the spring-mass system fromproblem 5.30.

a) Find the potential and kinetic energy ofthe spring mass system as functions oftime.

20 CONTENTS

b) Using the computer, make a plot of thepotential and kinetic energy as a func-tion of time for several periods of os-cillation. Are the potential and kineticenergy ever equal at the same time? Ifso, at what position x(t)?

c) Make a plot of kinetic energy versuspotential energy. What is the phase re-lationship between the kinetic and po-tential energy?

5.34 For the three spring-mass systems shownin the figure, find the equation of motion of themass in each case. All springs are massless andare shown in their relaxed states. Ignore grav-ity. (In problem (c) assume vertical motion.)

mk, `0 k, `0

mk, `0

mk, `0

k, `0

(a)

(b)

(c)

F(t)

F(t)

x

y

F(t)

problem 5.34:(Filename:summer95f.3)

5.35 A spring and mass system is shown in thefigure.

a) First, as a review, let k1, k2, and k3 equalzero and k4 be nonzero. What is thenatural frequency of this system?

b) Now, let all the springs have non-zerostiffness. What is the stiffness of a sin-gle spring equivalent to the combina-tion of k1, k2, k3, k4? What is the fre-quency of oscillation of mass M?

c) What is the equivalent stiffness, keq , ofall of the springs together. That is, ifyou replace all of the springs with onespring, what would its stiffness have tobe such that the system has the samenatural frequency of vibration?

xk1

k2

k4

k3

M

problem 5.35:(Filename:pfigure.blue.159.3)

5.36 The mass shown in the figure oscillates inthe vertical direction once set in motion by dis-placing it from its static equilibrium position.The position y(t) of the mass is measured fromthe fixed support, taking downwards as posi-tive. The static equilibrium position is ys andthe relaxed length of the spring is `0. At theinstant shown, the position of the mass is y andits velocity y, directed downwards. Draw afree body diagram of the mass at the instant ofinterest and evaluate the left hand side of theenergy balance equation (P = EK).

kys

y

m

problem 5.36:(Filename:pfig2.3.rp1)

5.37 Mass hanging from a spring. A mass mis hanging from a spring with constant k whichhas the length l0 when it is relaxed (i. e., whenno mass is attached). It only moves vertically.

a) Draw a Free Body Diagram of the mass.

b) Write the equation of linear momentumbalance.

c) Reduce this equation to a standard dif-ferential equation in x , the position ofthe mass.

d) Verify that one solution is that x(t) isconstant at x = l0 + mg/k.

e) What is the meaning of that solution?(That is, describe in words what is go-ing on.)

f) Define a new variable x = x (l0 +mg/k). Substitute x = x+(l0+mg/k)into your differential equation and notethat the equation is simpler in terms ofthe variable x .

g) Assume that the mass is released froman an initial position of x = D. Whatis the motion of the mass?

h) What is the period of oscillation of thisoscillating mass?

i) Why might this solution not make phys-ical sense for a long, soft spring ifD > `0 + 2mg/k)?

k

x

l0

m

problem 5.37:(Filename:pg141.1)

The following problem concerns simpleharmonic motion for part of the motion. It in-volves pasting together solutions.

5.38 One of the winners in the egg-drop con-test sponsored by a local chapter of ASME eachspring, was a structure in which rubber bandsheld the egg at the center of it. In this prob-lem, we will consider the simpler case of theegg to be a particle of mass m and the springsto be linear devices of spring constant k. Wewill also consider only a two-dimensional ver-sion of the winning design as shown in the fig-ure. If the frame hits the ground on one of thestraight sections, what will be the frequencyof vibration of the egg after impact? [Assumesmall oscillations and that the springs are ini-tially stretched.]

ground

egg, m

k k

k

problem 5.38:(Filename:pfigure.blue.149.1)

5.39 A person jumps on a trampoline. Thetrampoline is modeled as having an effectivevertical undamped linear spring with stiffnessk = 200 lbf/ ft. The person is modeled as arigid mass m = 150 lbm. g = 32.2 ft/s2.

a) What is the period of motion if the per-sons motion is so small that her feetnever leave the trampoline?

b) What is the maximum amplitude of mo-tion for which her feet never leave thetrampoline?

c) (harder) If she repeatedly jumps so thather feet clear the trampoline by a heighth = 5 ft, what is the period of this mo-tion?

Problems for Chapter 5 21

yyyy

problem 5.39: A person jumps on a tram-poline.

(Filename:pfigure3.trampoline)

5.4 More on vibrations:damping

5.40 If x + cx + kx = 0, x(0) = x0, andx(0) = 0, find x(t).

5.41 A mass moves on a frictionless surface.It is connected to a dashpot with damping coef-ficient b to its right and a spring with constantk and rest length ` to its left. At the instant ofinterest, the mass is moving to the right and thespring is stretched a distance x from its posi-tion where the spring is unstretched. There isgravity.

a) Draw a free body diagram of the massat the instant of interest.

b) Evaluate the left hand side of the equa-tion of linear momentum balance as ex-plicitly as possible.

.x

m

`k b

problem 5.41:(Filename:ch2.11)

5.5 Forced oscillations andresonance

5.42 A 3 kg mass is suspended by a spring(k = 10 N/m) and forced by a 5 N sinusoidallyoscillating force with a period of 1 s. What isthe amplitude of the steady-state oscillations(ignore the homogeneous solution)

5.43 Given that + k2 = sint , (0) = 0,and (0) = 0, find (t) .

5.44 A machine produces a steady-state vi-bration due to a forcing function described byQ(t) = Q0 sint , where Q0 = 5000N . Themachine rests on a circular concrete founda-tion. The foundation rests on an isotropic, elas-tic half-space. The equivalent spring constantof the half-space is k = 2, 000, 000 Nm andhas a damping ratio d = c/cc = 0.125. Themachine operates at a frequency of = 4 Hz.

(a) What is the natural frequency of the sys-tem?

(b) If the system were undamped, whatwould the steady-state displacementbe?

(c) What is the steady-state displacementgiven that d = 0.125?

(d) How much additional thickness of con-crete should be added to the footing toreduce the damped steady-state ampli-tude by 50%? (The diameter must beheld constant.)

5.6 Coupled motions in 1DThe primary emphasis of this section is set-ting up correct differential equations (withoutsign errors) and solving these equations on thecomputer. Experts note: normal modes arecoverred in the vibrations chapter. These firstproblems are just math problems, using someof the skills that are needed for the later prob-lems.

5.45 Write the following set of coupled secondorder ODEs as a system of first order ODEs.

x1 = k2(x2 x1) k1x1x2 = k3x2 k2(x2 x1)

5.46 See also problem 5.47. The solution of aset of a second order differential equations is:

(t) = A sint + B cost + (t) = A cost B sint,

where A and B are constants to be determinedfrom initial conditions. Assume A and B arethe only unknowns and write the equations inmatrix form to solve for A and B in terms of(0) and (0).

5.47 Solve for the constants A and B in Prob-lem 5.46 using the matrix form, if (0) =0, (0) = 0.5, = 0.5 rad/s and = 0.2.5.48 A set of first order linear differential equa-tions is given:

x1 = x2x2 + kx1 + cx2 = 0

Write these equations in the form x = [A]x ,where x =

{x1x2

}.

5.49 Write the following pair of coupled ODEsas a set of first order ODEs.

x1 + x1 = x2 sin tx2 + x2 = x1 cos t

5.50 The following set of differential equationscan not only be written in first order form butin matrix form x = [A]x + c . In generalthings are not so simple, but this linear caseis prevalant in the analytic study of dynamicalsystems.

x1 = x3x2 = x4

x3 + 52x1 42x2 = 22v1x4 42x1 + 52x2 = 2v1

5.51 Write each of the following equations asa system of first order ODEs.

a) + 2 = cos t,b) x + 2px + kx = 0,

22 CONTENTS

c) x + 2cx + k sin x = 0.

5.52 A train is moving at constant absolute ve-locity v. A passenger, idealized as a pointmass, is walking at an absolute absolute veloc-ity u, where u > v. What is the velocity ofthe passenger relative to the train?

5.53 Two equal masses, each denoted by theletter m, are on an air track. One mass is con-nected by a spring to the end of the track. Theother mass is connected by a spring to the firstmass. The two spring constants are equal andrepresented by the letter k. In the rest (springsare relaxed) configuration, the masses are a dis-tance ` apart. Motion of the two masses x1 andx2 is measured relative to this configuration.

a) Draw a free body diagram for eachmass.

b) Write the equation of linear momentumbalance for each mass.

c) Write the equations as a system of firstorder ODEs.

d) Pick parameter values and initial con-ditions of your choice and simulate amotion of this system. Make a plot ofthe motion of, say, one of the masses vstime,

e) Explain how your plot does or doesnot make sense in terms of your under-standing of this system. Is the initialmotion in the right direction? Are thesolutions periodic? Bounded? etc.

k

x1

mk

x2

m

problem 5.53:(Filename:pfigure.s94f1p4)

5.54 Two equal masses, each denoted by theletter m, are on an air track. One mass is con-nected by a spring to the end of the track. Theother mass is connected by a spring to the firstmass. The two spring constants are equal andrepresented by the letter k. In the rest config-uration (springs are relaxed) the masses are adistance ` apart. Motion of the two masses x1and x2 is measured relative to this configura-tion.

a) Write the potential energy of the systemfor arbitrary displacements x1 and x2 atsome time t .

b) Write the kinetic energy of the systemat the same time t in terms of x1, x2, m,and k.

c) Write the total energy of the system.

k

x1

mk

x2

m

problem 5.54:(Filename:pfigure.twomassenergy)

5.55 Normal Modes. Three equal springs (k)hold two equal masses (m) in place. There isno friction. x1 and x2 are the displacements ofthe masses from their equilibrium positions.

a) How many independent normal modesof vibration are there for this system?

b) Assume the system is in a normal modeof vibration and it is observed that x1 =A sin(ct)+ B cos(ct) where A, B, andc are constants. What is x2(t)? (Theanswer is not unique. You may expressyour answer in terms of any of A, B, c,m and k. )

c) Find all of the frequencies of normal-mode-vibration for this system in termsof m and k.

mm

x1

k k k

x2problem 5.55:

(Filename:pfigure.f93f2)

5.56 A two degree of freedom spring-masssystem. A two degree of freedom mass-springsystem, made up of two unequal masses m1 andm2 and three springs with unequal stiffnessesk1, k2 and k3, is shown in the figure. All threesprings are relaxed in the configuration shown.Neglect friction.

a) Derive the equations of motion for thetwo masses.

b) Does each mass undergo simple har-monic motion?

k1 k2 k3x1 x2

m2m1

problem 5.56:(Filename:pfigure.s94h5p1)

5.57 For the three-mass system shown, drawa free body diagram of each mass. Write thespring forces in terms of the displacements x1,x2, and x3.

m

x1 x2 x3

m mkkkk

L L L L

problem 5.57:(Filename:s92f1p1)

5.58 The springs shown are relaxed whenxA = xB = xD = 0. In terms of some or allof m A,m B ,m D, xA, xB , xD, x A, xB , xC , andk1, k2, k3, k4, c1, and F , find the accelerationof block B.

mA

xA xB xD

k1

c1k4

k2 k3mB mD

A B D

problem 5.58:(Filename:pfigure.s95q3)

5.59 A system of three masses, four springs,and one damper are connected as shown. As-sume that all the springs are relaxed whenxA = xB = xD = 0. Given k1, k2, k3, k4,c1, m A , m B , m D , xA , xB , xD , x A , xB , and xD ,find the acceleration of mass B, a B = xB .

mA

xA xB xD

k1c1

k4 k2

k3mB mD

A B D

problem 5.59:(Filename:pfigure.s95f1)

5.60 Equations of motion. Two masses areconnected to fixed supports and each other withthe three springs and dashpot shown. The forceF acts on mass 2. The displacements x1 andx2 are defined so that x1 = x2 = 0 when thesprings are unstretched. The ground is friction-less. The governing equations for the systemshown can be writen in first order form if wedefine v1 x1 and v2 x2.

a) Write the governing equations in a neatfirst order form. Your equations shouldbe in terms of any or all of the constantsm1, m2, k1, k2,k3, C , the constant forceF , and t . Getting the signs right is im-portant.

b) Write computer commands to find andplot v1(t) for 10 units of time. Makeup appropriate initial conditions.

c) For constants and initial conditions ofyour choosing, plot x1 vs t for enoughtime so that decaying erratic oscilla-tions can be observed.

k1 k2 k3

x1

x2F c

m2m1

problem 5.60:(Filename:p.f96.f.3)

5.61 x1(t) and x2(t) are measured positionson two points of a vibrating structure. x1(t) isshown. Some candidates for x2(t) are shown.Which of the x2(t) could possibly be associatedwith a normal mode vibration of the structure?Answer could or could not next to each

Problems for Chapter 5 23

choice. (If a curve looks like it is meant to bea sine/cosine curve, it is.)

a)

b)

c)

d)

e)

X1(t)

X2?

X2?

X2?

X2?

X2?

problem 5.61:(Filename:pfigure.blue.144.1)

5.62 For the three-mass system shown, one ofthe normal modes is described with the eigen-vector (1, 0, -1). Assume x1 = x2 = x3 = 0when all the springs are fully relaxed.

a) What is the angular frequency for thismode? Answer in terms of L ,m, k,and g. (Hint: Note that in this modeof vibration the middle mass does notmove.)

b) Make a neat plot of x2 versus x1 for onecycle of vibration with this mode.

m

x1 x2 x3

m mkkkk

L L L L

problem 5.62:(Filename:pfigure.blue.160.2)

5.63 The three beads of masses m, 2m, and mconnected by massless linear springs of con-stant k slide freely on a straight rod. Let xidenote the displacement of the i th bead fromits equilibrium position at rest.

a) Write expressions for the total kineticand potential energies.

b) Write an expression for the total linearmomentum.

c) Draw free body diagrams for the beadsand use Newtons second law to derivethe equations for motion for the system.

d) Verify that total energy and linear mo-mentum are both conserved.

e) Show that the center of mass must ei-ther remain at rest or move at constantvelocity.

f) What can you say about vibratory (si-nusoidal) motions of the system?

m 2mk k

x1 x2 x3

Displacedconfiguration

Staticequilibrium

configuration m

m m2m

problem 5.63:(Filename:pfigure.blue.161.1)

5.64 The system shown below comprises threeidentical beads of mass m that can slide fric-tionlessly on the rigid, immobile, circular hoop.The beads are connected by three identical lin-ear springs of stiffness k, wound around thehoop as shown and equally spaced when thesprings are unstretched (the strings are un-stretched when 1 = 2 = 3 = 0.)

a) Determine the natural frequencies andassociated mode shapes for the system.(Hint: you should be able to deduce arigid-body mode by inspection.)

b) If your calculations in (a) are correct,then you should have also obtained themode shape (0, 1,1)T . Write downthe most general set of initial conditionsso that the ensuing motion of the systemis simple harmonic in that mode shape.

c) Since (0, 1,1)T is a mode shape,then by symmetry, (1, 0, 1)T and(1,1, 0)T are also mode shapes (drawa picture). Explain how we can havethree mode shapes associated with thesame frequency.

d) Without doing any calculations, com-pare the frequencies of the constrainedsystem to those of the unconstrainedsystem, obtained in (a).

m

m

m k

k

k

g = 0

R = 1

12

3

problem 5.64:(Filename:pfigure.blue.158.1)

5.65 Equations of motion. Two masses areconnected to fixed supports and each otherwith the two springs and dashpot shown. Thedisplacements x1 and x2 are defined so thatx1 = x2 = 0 when both springs are un-stretched.

For the special case that C = 0 and F0 = 0clearly define two different set of initial condi-tions that lead to normal mode vibrations ofthis system.

cM2M1

x1

k1 k2

x2

F0 sin(t)

problem 5.65:(Filename:p.s96.p3.1)

5.66 As in problem 5.59, a system of threemasses, four springs, and one damper are con-nected as shown. Assume that all the springsare relaxed when xA = xB = xD = 0.

a) In the special case when k1 = k2 =k3 = k4 = k, c1 = 0, and m A =m B = m D = m, find a normal modeof vibration. Define it in any clear wayand explain or show why it is a normalmode in any clear way.

b) In the same special case as in (a) above,find another normal mode of vibration.

mA

xA xB xD

k1c1

k4 k2

k3mB mD

A B D

problem 5.66:(Filename:pfigure.s95f1a)

5.67 As in problem 5.143, a system of threemasses, four springs, and one damper are con-nected as shown. In the special case whenc1 = 0, find the normal modes of vibration.

mA

xA xB xD

k1

c1k4

k2 k3mB mD

A B D

problem 5.67:(Filename:pfigure.s95q3a)

5.68 Normal modes. All three masses havem = 1 kg and all 6 springs are k = 1 N/m.The system is at rest when x1 = x2 = x3 = 0.

a) Find as many different initial conditionsas you can for which normal mode vi-brations result. In each case, find theassociated natural frequency. (we willcall two initial conditions [v] and [w]different if there is no constant c so that[v1 v2 v3] = c[w1 w2 w3]. Assumethe initial velocities are zero.)

b) For the initial condition[x0] = [ 0.1 m 0 0 ],

[x0] = [ 0 2 m/s 0 ]what is the initial (immediately after thestart) acceleration of mass 2?

k k

k

k

k

k

x1

m1 m2 m3

x2 x3

problem 5.68:(Filename:pfigure3.f95p2p2)

24 CONTENTS

5.7 Time derivative of a vec-tor: position, velocity andacceleration

5.69 The position vector of a particle in the xy-plane is given as r = 3.0 m + 2.5 m . Find(a) the distance of the particle from the originand (b) a unit vector in the direction of r .

5.70 Given r (t) = A sin(t) + Bt + Ck,find

(a) v (t)

(b) a(t)

(c) r (t) a(t).5.71 A particle of mass =3 kg travels in spacewith its position known as a function of time,r = (sin ts )m+(cos ts )m+te

ts m/sk. At

t = 3 s, find the particlesa) velocity and

b) acceleration.

5.72 A particle of mass m = 2 kg travels in thexy-plane with its position known as a functionof time, r = 3t2 m/s2 + 4t3 m

s3 . At t = 5 s,

find the particles

a) velocity and

b) acceleration.

5.73 If r = (u0 sint) + v0 and r (0) =x0 + y0 , with u0, v0, and , find r (t) =x(t) + y(t) . 5.74 The velocity of a particle of mass mon a frictionless surface is given as v =(0.5 m/s) (1.5 m/s) . If the displacementis given by 1r = v t , find (a) the distancetraveled by the mass in 2 seconds and (b) a unitvector along the displacement.

5.75 For v = vx + vy + vz k and a =2 m/s2 (3 m/s21 m/s3 t) 5 m/s4 t2k,write the vector equation v =

a dt as three

scalar equations.

5.76 Find r (5 s) given thatr = v1 sin(ct)+v2 and r (0) = 2 m+3 mand that v1 is a constant 4 m/s, v2 is a constant5 m/s, and c is a constant 4 s1. Assume and are constant.

5.77 Let r = v0 cos + v0 sin +(v0 tan gt)k, where v0, , , and g areconstants. If r (0) = 0 , find r (t).5.78 On a smooth circular helical path thevelocity of a particle is r = R sin t +R cos t + gtk. If r (0) = R, findr ((/3) s).

5.79 Draw unit vectors along = (4.33 rad/s) + (2.50 rad/s) and r =(0.50 ft) (0.87 ft) and find the angle be-tween the two unit vectors.

5.80 What is the angle between the x-axis and

the vector v = (0.3 2.0 + 2.2k)m/s?

5.81 The position of a particle is given byr (t) = (t2 m/s2 + e ts m ). What are thevelocity and acceleration of the particle?

5.82 A particle travels on a path in the xy-planegiven by y(x) = sin2( xm )m. where x(t) =t3( m

s3). What are the velocity and acceleration

of the particle in cartesian coordinates when

t = () 13 s?

5.83 A particle travels on an elliptical path

given by y2 = b2(1 x2a2) with constant

speed v. Find the velocity of the particle whenx = a/2 and y > 0 in terms of a, b, and v.

5.84 A particle travels on a path in the xy-planegiven by y(x) = (1 e xm )m. Make a plot ofthe path. It is known that the x coordinate ofthe particle is given by x(t) = t2 m/s2. Whatis the rate of change of speed of the particle?What angle does the velocity vector make withthe positive x axis when t = 3 s?

5.85 A particle starts at the origin in the xy-plane, (x0 = 0, y0 = 0) and travels only in thepositive xy quadrant. Its speed and x coordi-

nate are known to be v(t) =

1+ ( 4s2)t2 m/s

and x(t) = t m/s, respectively. What is r (t)in cartesian coordinates? What are the veloc-ity, acceleration, and rate of change of speed ofthe particle as functions of time? What kind ofpath is the particle on? What are the distanceof the particle from the origin and its velocityand acceleration when x = 3 m?

5.8 Spatial dynamics of aparticle

5.86 What symbols do we use for the followingquantities? What are the definitions of thesequantities? Which are vectors and which arescalars? What are the SI and US standard unitsfor the following quantities?

a) linear momentum

b) rate of change of linear momentum

c) angular momentum

d) rate of change of angular momentum

e) kinetic energy

f) rate of change of kinetic energy

g) moment

h) work

i) power

5.87 Does angular momentum depend on refer-ence point? (Assume that all candidate pointsare fixed in the same Newtonian referenceframe.)

5.88 Does kinetic energy depend on refer-ence point? (Assume that all candidate pointsare fixed in the same Newtonian referenceframe.)

5.89 What is the relation between the dynamicsLinear Momentum Balance equation and thestatics Force Balance equation?

5.90 What is the relation between the dynam-ics Angular Momentum Balance equation andthe statics Moment Balance equation?

5.91 A ball of mass m = 0.1 kg is thrown froma height of h = 10 m above the ground withvelocity v = 120 km/h 120 km/h . Whatis the kinetic energy of the ball at its release?

5.92 A ball of mass m = 0.2 kg is thrownfrom a height of h = 20 m above the groundwith velocity v = 120 km/h120 km/h 10 km/hk. What is the kinetic energy of theball at its release?

5.93 How do you calculate P , the power of allexternal forces acting on a particle, from theforces

F i and the velocity

v of the particle?

5.94 A particle A has velocity v A and massm A . A particle B has velocity

v B = 2 v A

and mass equal to the other m B = m A . Whatis the relationship between:

a)LA and

LB,

b)HA/C and

HB/C, and

c) EKA and EKB?

5.95 A bullet of mass 50 g travels with a veloc-ity v = 0.8 km/s + 0.6 km/s . (a) What isthe linear momentum of the bullet? (Answerin consistent units.)

5.96 A particle has position r = 4 m+7 m ,velocity v = 6 m/s 3 m/s , and accelera-tion a = 2 m/s2 + 9 m/s2 . For each po-sition of a point P defined below, find

H P , the

angular momentum of the particle with respectto the point P .

a) r P = 4 m + 7 m ,b) r P = 2 m + 7 m , andc) r P = 0 m + 7 m ,d) r P = 0

5.97 The position vector of a particle of mass1 kg at an instant t is r = 2 m 0.5 m .If the velocity of the particle at this instant isv = 4 m/s + 3 m/s , compute (a) the lin-ear momentum

L = mv and (b) the angular

momentum (HO = r/O (m

v )).

5.98 The position of a particle of mass m =0.5 kg is r (t) = ` sin(t) + h ; where =2 rad/s, h = 2 m, ` = 2 m, and r is measuredfrom the origin.

Problems for Chapter 5 25

a) Find the kinetic energy of the particleat t = 0 s and t = 5 s.

b) Find the rate of change of kinetic energyat t = 0 s and t = 5 s.

5.99 For a particle

EK =1

2m v2.

Why does it follow that EK = m v a? [hint:write v2 as v v and then use the product ruleof differentiation.]

5.100 Consider a projectile of mass m at someinstant in time t during its flight. Let v bethe velocity of the projectile at this instant (seethe figure). In addition to the force of gravity,a drag force acts on the projectile. The dragforce is proportional to the square of the speed(speed |v | = v) and acts in the oppositedirection. Find an expression for the net powerof these forces (P = F v ) on the particle.

v

problem 5.100:(Filename:pfig2.3.rp3)

5.101 A 10 gm wad of paper is tossed in theair (in a strong turbulent wind). The position,velocity, and acceleration of its center of mass

are r = 3 m+3 m+6 mk, v = 9 m/s+24 m/s + 30 m/sk, and a = 10 m/s2 +24 m/s2 + 32 m/s2k, respectively. Find thelinear momentum

L and its rate of change

L

of the the potatos center of mass.

5.102 A 10 gm wad of paper is tossed intothe air. At a particular instant of interest, theposition, velocity, and acceleration of its cen-

ter of mass are r = 3 m + 3 m + 6 mk,v = 9 m/s + 24 m/s + 30 m/sk, anda = 10 m/s2 + 24 m/s2 + 32 m/s2k, re-spectively. What is the translational kinetic en-ergy of the wad at the instant of interest?

5.103 A 2 kg particle moves so that its positionr is given by

r (t) = [5 sin(at) + bt2 + ctk] m

where a = / sec, b = .25/ sec2, c = 2/ sec .a) What is the linear momentum of the par-

ticle at t = 1 sec?b) What is the force acting on the particle

at t = 1 sec?5.104 A particle A has mass mA and velocityvA. A particle B at the same location has massm B = 2 m A and velocity equal to the othervB = vA. Point C is a reference point. Whatis the relationship between:

a)LA and

LB,

b)HA/C and

HB/C, and

c) EKA and EKB?

5.105 A particle of mass m = 3 kg movesin space. Its position, velocity, and acceler-ation at a particular instant in time are r =2 m+3 m+5 mk, v = 3 m/s+8 m/s+10 m/sk, and a = 5 m/s2 + 12 m/s2 +16 m/s2k, respectively. For this particle at theinstant of interest, find its:

a) linear momentumL,

b) rate of change of linear momentumL,

c) angular momentum about the originHO,

d) rate of change of angular momentum

about the originHO,

e) kinetic energy EK, and

f) rate of change of kinetic energy EK.

5.106 A particle has position r = 3 m 2 m+4 mk, velocity v = 2 m/s3 m/s+7 m/sk, and acceleration a = 1 m/s2 8 m/s2 + 3 m/s2k. For each position of apoint P defined below, find the rate of changeof angular momentum of the particle with re-

spect to the point P ,H P .

a) r P = 3 m 2 m + 4 mk,b) r P = 6 m 4 m + 8 mk,c) r P = 9 m + 6 m 12 mk, andd) r P = 0

5.107 A particle of mass m = 5 kg has po-sition, velocity, and acceleration r = 2 m ,v = 3 m/s, and a = 7 m/s2 , respectively,at a particular instant of interest. At the instantof interest find its:

a) linear momentumL,

b) rate of change of linear momentumL,

c) angular momentum about the originHO,

d) rate of change of angular momentum

about the originHO,

e) kinetic energy EK, and

f) rate of change of kinetic energy EK.

g) the net force F on the particle,

h) the net moment on the particle aboutthe origin

MO due to the applied

forces, and

i) rate of change of work W = P done onthe particle by the applied forces.

Particle FBD

F

problem 5.107: FBD of the particle(Filename:pfigure1.1.part.fbd)

5.108 A particle of mass m = 6 kg is moving inspace. Its position, velocity, and accelerationat a particular instant in time are r = 1 m 2 m+4 mk, v = 3 m/s+4 m/s7 m/sk,and a = 5 m/s2 + 11 m/s2 9 m/s2k, re-spectively. For this particle at the instant ofinterest, find its:

a) the net force F on the particle,

b) the net moment on the particle aboutthe origin

MO due to the applied

forces, and

c) the power P of the applied forces.

Particle FBD

F

problem 5.108: FBD of the particle(Filename:pfigure1.1.part.fbdb)

5.109 A particle of mass m = 3 kg is movingin the xz-plane. Its position, velocity, and ac-celeration at a particular instant of interest arer = 4 m+2 mk, v = 3 m/s7 m/sk, anda = 3 m/s2 4 m/s2k, respectively. For thisparticle at the instant of interest, find:

a) the net force F on the particle,

b) the net moment on the particle aboutthe origin

MO due to the applied

forces, and

c) rate of change of work W = P done onthe particle by the applied forces.

Particle FBD

F

problem 5.109: FBD of the particle(Filename:pfigure1.1.part.fbda)

5.110 A particle of mass m = 3 kg is movingin the xy-plane. Its position, velocity, and ac-celeration at a particular instant of interest arer = 2 m + 3 m , v = 3 m/s + 8 m/s ,and a = 5 m/s2 + 12 m/s2 , respectively.For this particle at the instant of interest, findits:

a) linear momentumL,

26 CONTENTS

b) rate of change of linear momentumL,

c) angular momentum about the originHO,

d) rate of change of angular momentum

about the originHO,

e) kinetic energy EK, and

f) rate of change of kinetic energy EK.

5.111 At a particular instant of interest, a par-ticle of mass m1 = 5 kg has position, velocity,and acceleration r 1 = 3 m, v1 = 4 m/s ,and a1 = 6 m/s2 , respectively, and a particleof mass m2 = 5 kg has position, velocity, andacceleration r 2 = 6 m, v2 = 5 m/s , anda2 = 4 m/s2 , respectively. For the systemof particles, find its

a) linear momentumL,

b) rate of change of linear momentumL

c) angular momentum about the originHO,

d) rate of change of angular momentum

about the originHO,

e) kinetic energy EK, and

f) rate of change of kinetic energy EK.

5.112 A particle of mass m = 250 gm is shotstraight up (parallel to the y-axis) from the x-axis at a distance d = 2 m from the origin. Thevelocity of the particle is given by v = vwhere v2 = v20 2ah, v0 = 100 m/s, a =10 m/s2 and h is the height of the particle fromthe x-axis.

a) Find the linear momentum of the parti-cle at the outset of motion (h = 0).

b) Find the angular momentum of the par-ticle about the origin at the outset ofmotion (h = 0).

c) Find the linear momentum of the parti-cle when the particle is 20 m above thex-axis.

d) Find the angular momentum of the par-ticle about the origin when the particleis 20 m above the x-axis.

5.113 For a particle, F = ma . Two forces

F 1 and

F 2 act on a mass P as shown in the

figure. P has mass 2 lbm. The accelerationof the mass is somehow measured to be a =2 ft/s2 + 5 ft/s2 .

a) Write the equation F = ma

in vector form (evaluating each side asmuch as possible).

b) Write the equation in scalar form (useany method you like to get two scalarequations in the two unknowns F1 andF2).

c) Write the equation in matrix form.

d) Find F1 = |F 1| and F2 = |

F 2| by the

following methods:

(a) from the scalar equations usinghand algebra,

(b) from the matrix equation using acomputer, and

(c) from the vector equation using across product.

x

y

30o45o

P

F1 F2

problem 5.113:(Filename:h1.63a)

5.114 Three forces,F 1 = 20 N

5 N ,F 2 = F2x + F2z k, and

F 3 = F3,

where = 12 +

32 , act on a body with

mass 2 kg. The acceleration of the body isa = 0.2 m/s2 + 2.2 m/s2 + 1.7 m/s2k.Write the equation

F = ma as scalar equa-

tions and solve them (most conveniently on acomputer) for F2x , F2z , and F3.

5.115 The rate of change of linear momen-tum of a particle is known in two directions:L x = 20 kg m/s2, L y = 18 kg m/s2 andunknown in the z direction. The forces act-ing on the particle are

F 1 = 25 N+ 32 N +

75 Nk,F 2 = F2x +F2y and

F 3 = F3k.

Using F = L, separate the vector equation

into scalar equations in the x, y, and z direc-tions. Solve these equations (maybe with thehelp of a computer) to find F2x , F2y , and F3.

5.116 A block of mass 100 kg is pulled withtwo strings AC and BC. Given that the tensionsT1 = 1200 N and T2 = 1500 N, neglectinggravity, find the magnitude and direction of theacceleration of the block. [

F = ma ]

y

x

z

1.2m

1.2m

1m1m

AB

C

T1T2

1.2m

1.2m

mg

T1 1

T2 2

y

x

z

1.2m

1.2m

1m1m

AB

C

1.2m

1.2m

FBD

problem 5.116:(Filename:efig1.2.7)

5.117 Neglecting gravity, the only force actingon the mass shown in the figure is from thestring. Find the acceleration of the mass. Usethe dimensions and quantities given. Recallthat lbf is a pound force, lbm is a pound mass,and lbf/ lbm = g. Use g = 32 ft/s2. Notealso that 32 + 42 + 122 = 132.

y

z

x

m

a = ?

m = 2.5 lbm

string pulley

3'

3 lbf

12'

4'

k

problem 5.117:(Filename:pfigure.blue.4.1)

5.118 Three strings are tied to the mass shownwith the directions indicated in the figure. Theyhave unknown tensions T1, T2, and T3. Thereis no gravity. The acceleration of the mass is

given as a = (0.5 + 2.5 + 13 k)m/s2.a) Given the free body diagram in the fig-

ure, write the equations of linear mo-mentum balance for the mass.

b) Find the tension T3.

Problems for Chapter 5 27

y

z

x

k

T2

T1

m = 6 kg

a = (.5 + 2.5 + 13 k) ms2

T33

T3 = ?3 = 113 (3 + 12 + 4k)

problem 5.118:(Filename:pfigure.blue.1.2)

5.119 For part(c) of problem 4.23, assume nowthat the mass at A has non-zero acceleration of(1m/s2) + (2m/s2) + (3m/s2)k. Find thetension in the three ropes at the instant shown.

5.120 A small object (mass= 2 kg) is beingpulled by three strings as shown. The accel-eration of the object at the position shown is

a =(0.6 0.2 + 2.0k

)m/s2.

a) Draw a free body diagram of the mass.

b) Write the equation of linear momentumbalance for the mass. Use s as unitvectors along the strings.

c) Find the three tensions T1, T2, and T3 atthe instant shown. You may find thesetensions by using hand algebra with thescalar equations, using a computer withthe matrix equation, or by using a crossproduct on the vector equation.

y

x

z

Cm

T1

T2T3

1m

4m

1.5m2m

2m

problem 5.120:(Filename:pfigure.s94h2p9)

5.121 Use a computer to draw a square withcorners at (1, 0), (0,1), (1, 0), (0, 1). Thismust be done with scientific software and notwith a purely graphics program.

5.122 Draw a Circle on the Computer. Wewill be interested in keeping track of the mo-tions of systems. A simple example is that ofa particle going in circles at a constant rate.One can draw a circle quite well with a com-pass or with simple drawing programs. But,more complicated motions will be more diffi-cult. Draw a circle on the computer and labelthe drawing (using computer generated letter-ing) with your name and the date.

a) You can program the circular shape anyway that you think is fun (or any otherway if you dont feel like having a goodtime). Your circle should be round.Measure its length and width with aruler, they should be within 10% of eachother (mark the dimensions by hand onyour drawing).

b) A good solution will clearly documentand explain the computer methodology.

5.123 What curve is defined by x = cos(t)2and y = sin(t) cos(t) for 0 t ?Try to figure it out without a computer. Makea computer plot.

5.124 Particle moves on a strange path.Given that a particle moves in the xy plane for1.77 s obeying

r = (5 m) cos2(t2/ s2)+(5 m) sin(t2/ s2) cos(t2/ s2)where x and y are the horizontal distance inmeters and t is measured in seconds.

a) Accurately plot the trajectory of the par-ticle.

b) Mark on your plot where the particle isgoing fast and where it is going slow.Explain how you know these points arethe fast and slow places.

5.125 COMPUTER QUESTION: Whatsthe plot? Whats the mechanics question?Shown are shown some pseudo computer com-mands that are not commented adequately, un-fortunately, and no computer is available at themoment.

a) Draw as accurately as you can, assign-ing numbers etc, the plot that resultsfrom running these commands.

b) See if you can guess a mechanical situ-ation that is described by this program.Sketch the system and define the vari-ables to make the script file agree withthe problem stated.

ODEs = {z1dot = z2 z2dot = 0}ICs = {z1 =1, z2=1}

Solve ODEs with ICs from t=0 to t=5

plot z2 and z1 vs t on the same plotproblem 5.125:

(Filename:pfigure.s94f1p3)

Here begins a series of problems con-cerning the motion of particles over non-infinitesimal time.

5.126 A particle is blown out through the uni-form spiral tube shown, which lies flat on a hor-izontal frictionless table. Draw the particlespath after it is expelled from the tube. Defendyour answer.

problem 5.126:(Filename:pfigure.blue.29.1)

5.127 A ball going to the left with speed v0bounces against a frictionless rigid ramp whichis sloped at an angle from the horizontal. Thecollision is completely elastic (the coefficientof restitution e = 1). Neglect gravity.

a) Find the velocity of the ball after thecollision. You may express your an-swer in terms of any combination of m,

, v0, , , n, and .

n = sin + cos

= cos + sin = sin n+ cos = cos n+ sin

b) For what value of would the verticalcomponent of the speed be maximized?

v0ball

xy

n

problem 5.127:(Filename:ballramp)

5.128 Bungy Jumping. In a new safer bungyjumping system, people jump up from theground while suspended from a rope that runsover a pulley at O and is connected to astretched spring anchored at B. The pulley hasnegligible size, mass, and friction. For the sit-uation shown the spring AB has rest length`0 = 2 m and a stiffness of k = 200 N/m.The inextensible massless rope from A to Phas length `r = 8 m, the person has a mass of100 kg. Take O to be the origin of an xy co-ordinate system aligned with the unit vectors and

a) Assume you are given the position ofthe personr = x +y and the velocityof the personv = x + y . Find her ac-celeration in terms of some or all of herposition, her velocity, and the other pa-rameters given. Use the numbers given,where supplied, in your final answer.

28 CONTENTS

b) Given that bungy jumpers initial posi-tion and velocity are r0 = 1 m5 mandv0 = 0 write MATLAB commandsto find her position at t = /2 s.

c) Find the answer to part (b) with penciland paper (a final numerical answer isdesired).

k

m

10 m

AO

P

B

= 10 m/s2

ground, no contactafter jump off

g

problem 5.128:(Filename:s97p1.3)

5.129 A softball pitcher releases a ball of massm upwards from her hand with speed v0 andangle 0 from the horizontal. The only exter-nal force acting on the ball after its release isgravity.

a) What is the equation of motion for theball after its release?

b) What are the position, velocity, and ac-celeration of the ball?

c) What is its maximum height?

d) At what distance does the ball return tothe elevation of release?

e) What kind of path does the ball followand what is its equation y as a functionof x?

5.130 Find the trajectory of a not-vertically-fired cannon ball assuming the air drag is pro-portional to the speed. Assume the mass is10 kg, g = 10 m/s, the drag proportionalityconstant is C = 5 N/(m/s). The cannon ballis launched at 100 m/s at a 45 degree angle.

Draw a free body diagram of the mass. Write linear momentum balance in vec-

tor form.

Solve the equations on the computerand plot the trajectory.

Solve the equations by hand and thenuse the computer to plot your solution.

5.131 See also problem 5.132. A baseballpitching machine releases a baseball of mass mfrom its barrel with speed v0 and angle 0 fromthe horizontal. The only external forces act-ing on the ball after its release are gravity andair resistance. The speed of the ball is givenby v2 = x2 + y2. Taking into account air re-sistance on the ball proportional to its speedsquared, Fd = bv2et , find the equation ofmotion for the ball, after its release, in cartesiancoordinates.

5.132 The equations of motion from prob-lem 5.131 are nonlinear and cannot be solvedin closed form for the position of the baseball.Instead, solve the equations numerically. Makea computer simulation of the flight of the base-ball, as follows.

a) Convert the equation of motion into asystem of first order differential equa-tions.

b) Pick values for the gravitational con-stant g, the coefficient of resistance b,and initial speed v0, solve for the x andy coordinates of the ball and make aplots its trajectory for various initial an-gles 0.

c) Use Eulers, Runge-Kutta, or other suit-able method to numerically integratethe system of equations.

d) Use your simulation to find the initialangle that maximizes the distance oftravel for ball, with and without air re-sistance.

e) If the air resistance is very high, whatis a qualitative description for the curvedescribed by the path of the ball?

5.133 In the arcade game shown, the object ofthe game is to propel the small ball from theejector device at O in such a way that is passesthrough the small aperture at A and strikes thecontact point at B. The player controls theangle at which the ball is ejected and theinitial velocity vo. The trajectory is confinedto the frictionless xy-plane, which may or maynot be vertical. Find the value of that givessuccess. The coordinates of A and B are (2`,2`) and (3`, `), respectively, where ` is yourfavorite length unit.

x

y

A

B

Oejector device

problem 5.133:(Filename:pfigure.blue.32.1)

5.9 Central force motionand celestial mechanicsExperts note that none of these problems usepolar or other fancy coordinates. Such descrip-tions come later in the text. At this point wewant to lay out the basic equations and the qual-itative features that can be found by numericalintegration of the equations.

5.134 Under what circumstances is the angularmomentum of a system, calculated relative toa point C which is fixed in a Newtonian frame,conserved?

5.135 A satellite is put into an elliptical orbitaround the earth (that is, you can assume theorbit is closed) and has a velocity v P at posi-tion P. Find an expression for the velocity v Aat position A. The radii to A and P are, respec-tively, rA and rP . [Hint: both total energy andangular momentum are conserved.]

A PrA rP

RE

vA

vP

problem 5.135:(Filename:pfigure.blue.64.2)

5.136 The mechanics of nuclear war. A mis-sile, modelled as a point, is launched on a bal-listic trajectory from the surface of the earth.The force on the missile from the earths grav-ity is F = mgR2/r2 and is directed towardsthe center of the earth. When it is launchedfrom the equator it has speed v0 and in the di-rection shown, 45 from horizontal. For thepurposes of this calculation ignore the earthsrotation. That is, you can think of this problemas two-dimensional in the plane shown. If youneed numbers, use the following values:

m = 1000 kg is the mass of the missile,g = 10 m/s2 is earths gravitational con-

stant at the earths surface,R = 6, 400, 000 m is the radius of the

earth, andv0 = 9000 m/sr(t) is the distance of the missile from the

center of the earth.

a) Draw a free body diagram of the mis-sile. Write the linear momentum bal-ance equation. Break this equation intox and y components. Rewrite theseequations as a system of 4 first orderODEs suitable for computer sol