Tenth Edition · 2015. 12. 14. · Vector Mechanics for Engineers: Dynamics dition Contents 13 - 2 Introduction Work of a Force Principle of Work & Energy Applications of the Principle

VECTOR MECHANICS FOR ENGINEERS:

DYNAMICS

Tenth Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

Phillip J. Cornwell

Lecture Notes:

Brian P. Self California Polytechnic State University

CHAPTER

© 2013 The McGraw-Hill Companies, Inc. All rights reserved.

13 Kinetics of Particles:

Energy and Momentum

Methods


Vector Mechanics for Engineers: Dynamics

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Contents

13 - 2

Introduction

Work of a Force

Principle of Work & Energy

Applications of the Principle of

Work & Energy

Power and Efficiency

Sample Problem 13.1

Sample Problem 13.2

Sample Problem 13.3

Sample Problem 13.4

Sample Problem 13.5

Potential Energy

Conservative Forces

Conservation of Energy

Motion Under a Conservative

Central Force

Sample Problem 13.6

Sample Problem 13.7

Sample Problem 13.9

Principle of Impulse and Momentum

Impulsive Motion

Sample Problem 13.10



Impact

Direct Central Impact

Oblique Central Impact

Problems Involving Energy and

Momentum



Sample Problems 13.16




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Energy and Momentum Methods

2 - 3

The pogo stick allows the boy

to change between kinetic

energy, potential energy from

gravity, and potential energy

in the spring.

Accidents are often analyzed

by using momentum methods.



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Introduction

13 - 4

• Previously, problems dealing with the motion of

particles were solved through the fundamental

equation of motion,

• The current chapter introduces two additional

methods of analysis.

.F ma

• Method of work and energy: directly relates force,

mass, velocity and displacement.

• Method of impulse and momentum: directly

relates force, mass, velocity, and time.



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Introduction

2 - 5

Forces and

Accelerations

Velocities and

Displacements

Velocities and

Time

Approaches to Kinetics Problems

Newton’s Second

Law (last chapter)

Work-Energy Impulse-

Momentum

2211 TUT GF ma2

11 2

t

tmv F dt mv



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Work of a Force

13 - 6

• Differential vector is the particle displacement. rd

• Work of the force is

dzFdyFdxF

dsF

rdFdU

zyx

cos

• Work is a scalar quantity, i.e., it has magnitude and

sign but not direction.

force. length • Dimensions of work are Units are

J 1.356lb1ftm 1N 1 J 1 joule



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Work of a Force

13 - 7

• Work of a force during a finite displacement,

2

1

2

1

2

1

2

1

cos

21

A

Azyx

s

st

s

s

A

A

dzFdyFdxF

dsFdsF

rdFU

• Work is represented by the area under the

curve of Ft plotted against s.

• Ft is the force in the direction of the

displacement ds



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Work of a Force

13 - 8

What is the work of a constant force in

rectilinear motion?

1 2 cosU F x

1 2 sinU F x

1 2U F x

1 2 0U

a)

b)

c)

d)



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Work of a Force

13 - 9

• Work of the force of gravity,

yWyyW

dyWU

dyW

dzFdyFdxFdU

y

y

zyx

12

21

2

1

• Work of the weight is equal to product of

weight W and vertical displacement y.

• In the figure above, when is the work done by the weight positive?

a) Moving from y1 to y2 b) Moving from y2 to y1 c) Never



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Work of a Force

13 - 10

• Magnitude of the force exerted by a spring is

proportional to deflection,

lb/in.or N/mconstant spring

k

kxF

• Work of the force exerted by spring,

222

1212

121

2

1

kxkxdxkxU

dxkxdxFdU

x

x

• Work of the force exerted by spring is positive

when x2 < x1, i.e., when the spring is returning to

its undeformed position.

• Work of the force exerted by the spring is equal to

negative of area under curve of F plotted against x,

xFFU 2121

21



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Work of a Force

13 - 11

As the block moves from A0 to A1, is

the work positive or negative?

Positive Negative

As the block moves from A2 to Ao, is

the work positive or negative?

Positive Negative

Displacement is

in the opposite

direction of the

force



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Work of a Force

13 - 12

Work of a gravitational force (assume particle M

occupies fixed position O while particle m follows path

shown),

12221

2

2

1r

MmG

r

MmGdr

r

MmGU

drr

MmGFdrdU

r

r



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2 - 13

Does the normal force do work as the

block slides from B to A?

YES NO

Does the weight do work as

the block slides from B to A?

YES NO

Positive or

Negative work?



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Work of a Force

13 - 14

Forces which do not do work (ds = 0 or cos 0:

• Weight of a body when its center of gravity

moves horizontally.

• Reaction at a roller moving along its track, and

• Reaction at frictionless surface when body

in contact moves along surface,

• Reaction at frictionless pin supporting rotating body,



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Particle Kinetic Energy: Principle of Work & Energy

13 - 15

dvmvdsF

ds

dvmv

dt

ds

ds

dvm

dt

dvmmaF

t

tt

• Consider a particle of mass m acted upon by force F

• Integrating from A1 to A2 ,

energykineticmvTTTU

mvmvdvvmdsFv

v

s

st

221

1221

212

1222

12

1

2

1

• The work of the force is equal to the change in

kinetic energy of the particle.

F

• Units of work and kinetic energy are the same:

JmNms

mkg

s

mkg

2

22

21

mvT



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Applications of the Principle of Work and Energy

13 - 16

• The bob is released

from rest at position A1.

Determine the velocity

of the pendulum bob at

A2 using work & kinetic

energy.

• Force acts normal to path and does no

work.

P

glv

vg

WWl

TUT

2

2

10

2

22

2211

• Velocity is found without determining

expression for acceleration and integrating.

• All quantities are scalars and can be added

directly.

• Forces which do no work are eliminated from

the problem.



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Applications of the Principle of Work and Energy

13 - 17

• Principle of work and energy cannot be

applied to directly determine the acceleration

of the pendulum bob.

• Calculating the tension in the cord requires

supplementing the method of work and energy

with an application of Newton’s second law.

• As the bob passes through A2 ,

Wl

gl

g

WWP

l

v

g

WWP

amF nn

32

22

glv 22

If you designed the rope to hold twice the weight of the bob, what would happen?



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Power and Efficiency

13 - 18

• rate at which work is done.

vF

dt

rdF

dt

dU

Power

• Dimensions of power are work/time or force*velocity.

Units for power are

W746s

lbft550 hp 1or

s

mN 1

s

J1 (watt) W 1

•

inputpower

outputpower

input work

koutput wor

efficiency



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Sample Problem 13.1

13 - 19

An automobile weighing 4000 lb is

driven down a 5o incline at a speed of

60 mi/h when the brakes are applied

causing a constant total breaking force

of 1500 lb.

Determine the distance traveled by the

automobile as it comes to a stop.

SOLUTION:

• Evaluate the change in kinetic energy.

• Determine the distance required for the

work to equal the kinetic energy change.



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Sample Problem 13.1

13 - 20

SOLUTION:

• Evaluate the change in kinetic energy.

lbft481000882.324000

sft88s 3600

h

mi

ft 5280

h

mi60

2

212

121

1

1

mvT

v

0lb1151lbft481000

2211

x

TUT

ft 418x

• Determine the distance required for the work to

equal the kinetic energy change.

x

xxU

lb1151

5sinlb4000lb150021

00 22 Tv



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Sample Problem 13.2

13 - 21

Two blocks are joined by an inextensible

cable as shown. If the system is released

from rest, determine the velocity of block

A after it has moved 2 m. Assume that the

coefficient of friction between block A

and the plane is mk = 0.25 and that the

pulley is weightless and frictionless.

SOLUTION:

• Apply the principle of work and

energy separately to blocks A and B.

• When the two relations are combined,

the work of the cable forces cancel.

Solve for the velocity.



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Sample Problem 13.2

13 - 22

SOLUTION:

• Apply the principle of work and energy separately

to blocks A and B.

221

221

2211

2

kg200m2N490m2

m2m20

:

N490N196225.0

N1962sm81.9kg200

vF

vmFF

TUT

WNF

W

C

AAC

AkAkA

A

mm

221

221

2211

2

kg300m2N2940m2

m2m20

:

N2940sm81.9kg300

vF

vmWF

TUT

W

c

BBc

B



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Sample Problem 13.2

13 - 23

• When the two relations are combined, the work of the

cable forces cancel. Solve for the velocity.

221 kg200m2N490m2 vFC

221 kg300m2N2940m2 vFc

221

221

kg500J 4900

kg300kg200m2N490m2N2940

v

v

sm 43.4v



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13.2 – Alternate Solution, Group Problem Solving

2 - 24

Could you apply work-energy to the combined

system of blocks?

Given: v1= 0, distance = 2 m, mk = 0.25

212

4900 J 500kg v

Note that vA = vB

1 2 0.25 200 9.81 2m 300 9.81 2m 4900 JU

What is T1 of the system?

1 0T

What is the total work done between points 1 and 2?

2 m

2 m

2 2 2 21 1 1 12 2 2 2 2

200kg 300kgA BT m v m v v v

What is T2 of the system?

Solve for v

sm 43.4v

1

1

2

2



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Sample Problem 13.3

13 - 25

A spring is used to stop a 60 kg package

which is sliding on a horizontal surface.

The spring has a constant k = 20 kN/m

and is held by cables so that it is initially

compressed 120 mm. The package has a

velocity of 2.5 m/s in the position shown

and the maximum deflection of the spring

is 40 mm.

Determine (a) the coefficient of kinetic

friction between the package and surface

and (b) the velocity of the package as it

passes again through the position shown.

SOLUTION:

• Apply the principle of work and energy

between the initial position and the

point at which the spring is fully

compressed and the velocity is zero.

The only unknown in the relation is the

friction coefficient.

• Apply the principle of work and energy

for the rebound of the package. The

only unknown in the relation is the

velocity at the final position.



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Sample Problem 13.3

13 - 26

SOLUTION:

• Apply principle of work and energy between initial

position and the point at which spring is fully compressed.

0J5.187sm5.2kg60 22

212

121

1 TmvT

kk

kfxWU

mm

m

J377m640.0sm81.9kg60 2

21

J0.112m040.0N3200N2400

N3200m160.0mkN20

N2400m120.0mkN20

21

maxmin21

21

0max

0min

xPPU

xxkP

kxP

e

J112J377212121 kefUUU m

0J112J 377-J5.187

:2211

k

TUT

m 20.0km



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Sample Problem 13.3

13 - 27

• Apply the principle of work and energy for the rebound

of the package.

232

1232

132 kg600 vmvTT

J36.5

J112J377323232

kefUUU m

232

1

3322

kg60J5.360

:

v

TUT

sm103.13 v



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Sample Problem 13.4

13 - 28

A 2000 lb car starts from rest at point 1

and moves without friction down the

track shown.

Determine:

a) the force exerted by the track on

the car at point 2, and

b) the minimum safe value of the

radius of curvature at point 3.

SOLUTION:

• Apply principle of work and energy to

determine velocity at point 2.

• Apply Newton’s second law to find

normal force by the track at point 2.

• Apply principle of work and energy to

determine velocity at point 3.

• Apply Newton’s second law to find

minimum radius of curvature at point 3

such that a positive normal force is

exerted by the track.



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Sample Problem 13.4

13 - 29

SOLUTION:

• Apply principle of work and energy to determine

velocity at point 2.

sft8.50ft2.32ft402ft402

2

1ft400:

ft40

2

10

222

2

222211

21

22

222

121

vsgv

vg

WWTUT

WU

vg

WmvTT

• Apply Newton’s second law to find normal force by

the track at point 2.

:nn amF

WN

g

g

Wv

g

WamNW n

5

ft20

ft402

2

22

lb10000N



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Sample Problem 13.4

13 - 30

• Apply principle of work and energy to determine

velocity at point 3.

sft1.40sft2.32ft252ft252

2

1ft250

323

233311

vgv

vg

WWTUT

• Apply Newton’s second law to find minimum radius of

curvature at point 3 such that a positive normal force is

exerted by the track.

:nn amF

33

23 ft252

g

g

Wv

g

W

amW n

ft503



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Sample Problem 13.5

13 - 31

The dumbwaiter D and its load have a

combined weight of 600 lb, while the

counterweight C weighs 800 lb.

Determine the power delivered by the

electric motor M when the dumbwaiter

(a) is moving up at a constant speed of

8 ft/s and (b) has an instantaneous

velocity of 8 ft/s and an acceleration of

2.5 ft/s2, both directed upwards.

SOLUTION:

Force exerted by the motor

cable has same direction as

the dumbwaiter velocity.

Power delivered by motor is

equal to FvD, vD = 8 ft/s.

• In the first case, bodies are in uniform

motion. Determine force exerted by

motor cable from conditions for static

equilibrium.

• In the second case, both bodies are

accelerating. Apply Newton’s

second law to each body to

determine the required motor cable

force.



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Sample Problem 13.5

13 - 32

• In the first case, bodies are in uniform motion.

Determine force exerted by motor cable from

conditions for static equilibrium.

slbft1600

sft8lb 200

DFvPower

hp 91.2slbft550

hp 1slbft1600

Power

Free-body C:

:0 yF lb 4000lb8002 TT

Free-body D:

:0 yF

lb 200lb 400lb 600lb 600

0lb 600

TF

TF



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Sample Problem 13.5

13 - 33

• In the second case, both bodies are accelerating. Apply

Newton’s second law to each body to determine the required

motor cable force.

2212 sft25.1sft5.2 DCD aaa

Free-body C:

:CCy amF lb5.38425.12.32

8002800 TT

Free-body D:

:DDy amF

lb 1.2626.466005.384

5.22.32

600600

FF

TF

slbft2097sft8lb 1.262 DFvPower

hp 81.3slbft550

hp 1slbft2097

Power



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Group Problem Solving

2 - 34

Packages are thrown down an

incline at A with a velocity of 1

m/s. The packages slide along

the surface ABC to a conveyor

belt which moves with a

velocity of 2 m/s. Knowing that

mk= 0.25 between the packages

and the surface ABC, determine

the distance d if the packages

are to arrive at C with a velocity

of 2 m/s.

SOLUTION:

The problem deals with a change in

position and different velocities, so use

work-energy.

• Find the kinetic energy at points A

and C.

• Determine the work done between

points A and C as a function of d.

• Use the work-energy relationship

and solve for d.

• Draw FBD of the box to help us

determine the forces that do work.



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2 - 35

SOLUTION:

Given: vA= 1 m/s, vC= 2 m/s, mk= 0.25

Find: distance d

Will use:

Draw the FBD of the

block at points A and C

A A B B C CT U U T

Determine work done A → B

cos30

0.25 cos 30

sin30

(sin30 cos 30 )

AB

AB k AB

A B AB

k

N mg

F N mg

U mg d F d

mg d

m

m

7 m

BC BC

BC k

B C k BC

N mg x

F mg

U mg x

m

m

Determine work done B → C



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2 - 36

A A B B C CT U U T

Determine kinetic energy at A and at C

Substitute values into

21and 1m/s

2A A AT mv v 21

and 2 m/s2

C C CT mv v

2 20

1 1(sin30 cos30 )

2 2A k k BCmv mg d mg x mvm m

2 2

2 2

/2 /2

(sin30 cos30 )

(2) /(2)(9.81) (0.25)(7) (1) /(2)(9.81)

sin30 0.25cos30

C k BC A

k

v g x v gd

m

m

6.71 md

Divide by m and solve for d



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2 - 37

If you wanted to bring the package to a complete stop at the

bottom of the ramp, would it work to place a spring as shown?

Would the package ever come to a stop?

mk= 0.25

No, because the potential energy of the spring would turn

into kinetic energy and push the block back up the ramp

Yes, eventually enough energy would be dissipated

through the friction between the package and ramp.



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2 - 38

The potential energy stored at

the top of the roller coaster is

transferred to kinetic energy

as the cars descend.

The elastic potential energy

stored in the trampoline is

transferred to kinetic energy

and gravitational potential

energy as the girl flies upwards.



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Potential Energy

2 - 39

If the work of a force only depends on differences in

position, we can express this work as potential energy.

Can the work done by the following forces be expressed as

potential energy?

Weight

Friction

Spring force

Normal force

Yes

No

Yes

No

No

Yes

Yes

No



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Potential Energy

13 - 40

2121 yWyWU

• Work of the force of gravity , W

• Work is independent of path followed; depends

only on the initial and final values of Wy.

WyVg

potential energy of the body with respect

to force of gravity.

2121 gg VVU

• Units of work and potential energy are the same:

JmN WyVg

• Choice of datum from which the elevation y is

measured is arbitrary.



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Potential Energy

13 - 41

• Previous expression for potential energy of a body

with respect to gravity is only valid when the

weight of the body can be assumed constant.

• For a space vehicle, the variation of the force of

gravity with distance from the center of the earth

should be considered.

• Work of a gravitational force,

1221

r

GMm

r

GMmU

• Potential energy Vg when the variation in the

force of gravity can not be neglected,

r

WR

r

GMmVg

2



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Potential Energy

13 - 42

• Work of the force exerted by a spring depends

only on the initial and final deflections of the

spring,

222

1212

121 kxkxU

• The potential energy of the body with respect

to the elastic force,

2121

221

ee

e

VVU

kxV

• Note that the preceding expression for Ve is

valid only if the deflection of the spring is

measured from its undeformed position.



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Conservative Forces

13 - 43

• Concept of potential energy can be applied if the

work of the force is independent of the path

followed by its point of application.

22211121 ,,,, zyxVzyxVU

Such forces are described as conservative forces.

• For any conservative force applied on a closed path,

0 rdF

• Elementary work corresponding to displacement

between two neighboring points,

zyxdV

dzzdyydxxVzyxVdU

,,

,,,,

Vz

V

y

V

x

VF

dzz

Vdy

y

Vdx

x

VdzFdyFdxF zyx

grad



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Conservation of Energy

13 - 44

• Work of a conservative force,

2121 VVU

• Concept of work and energy,

1221 TTU

• Follows that

constant

2211

VTE

VTVT

• When a particle moves under the action of

conservative forces, the total mechanical

energy is constant.

WVT

WVT

11

11 0

WVT

VWgg

WmvT

22

2222

12 02

2

1• Friction forces are not conservative. Total

mechanical energy of a system involving

friction decreases.

• Mechanical energy is dissipated by friction

into thermal energy. Total energy is constant.



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Motion Under a Conservative Central Force

13 - 45

• When a particle moves under a conservative central

force, both the principle of conservation of angular

momentum

and the principle of conservation of energy

may be applied.

sinsin 000 rmvmvr

r

GMmmv

r

GMmmv

VTVT

221

0

202

1

00

• Given r, the equations may be solved for v and j.

• At minimum and maximum r, j 90o. Given the

launch conditions, the equations may be solved for

rmin, rmax, vmin, and vmax.



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Sample Problem 13.6

13 - 46

A 20 lb collar slides without friction

along a vertical rod as shown. The

spring attached to the collar has an

undeflected length of 4 in. and a

constant of 3 lb/in.

If the collar is released from rest at

position 1, determine its velocity after

it has moved 6 in. to position 2.

SOLUTION:

• Apply the principle of conservation of

energy between positions 1 and 2.

• The elastic and gravitational potential

energies at 1 and 2 are evaluated from

the given information. The initial kinetic

energy is zero.

• Solve for the kinetic energy and velocity

at 2.



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Sample Problem 13.6

13 - 47

SOLUTION:

• Apply the principle of conservation of energy between

positions 1 and 2.

Position 1:

0

lbft20lbin.24

lbin.24in. 4in. 8in.lb3

1

1

2

212

121

T

VVV

kxV

ge

e

Position 2:

22

22

222

12

2

2

212

221

311.02.32

20

2

1

lbft 5.5lbin. 6612054

lbin. 120in. 6lb 20

lbin.54in. 4in. 01in.lb3

vvmvT

VVV

WyV

kxV

ge

g

e

Conservation of Energy:

lbft 5.50.311lbft 20 22

2211

v

VTVT

sft91.42v



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Sample Problem 13.7

13 - 48

The 0.5 lb pellet is pushed against the

spring and released from rest at A.

Neglecting friction, determine the

smallest deflection of the spring for

which the pellet will travel around the

loop and remain in contact with the

loop at all times.

SOLUTION:

• Since the pellet must remain in contact

with the loop, the force exerted on the

pellet must be greater than or equal to

zero. Setting the force exerted by the

loop to zero, solve for the minimum

velocity at D.


energy between points A and D. Solve

for the spring deflection required to

produce the required velocity and

kinetic energy at D.



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Sample Problem 13.7

13 - 49

SOLUTION:

• Setting the force exerted by the loop to zero, solve for the

minimum velocity at D.

:nn maF

222

2

sft4.64sft32.2ft 2

rgv

rvmmgmaW

D

Dn

• Apply the principle of conservation of energy between

points A and D.

0

18ftlb360

1

22

212

21

1

T

xxkxVVV ge

lbft5.0sft4.64sft2.32

lb5.0

2

1

lbft2ft4lb5.00

22

2

2

21

2

2

D

ge

mvT

WyVVV

25.0180 2

2211

x

VTVT

in. 47.4ft 3727.0 x



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Sample Problem 13.9

13 - 50

A satellite is launched in a direction

parallel to the surface of the earth

with a velocity of 36900 km/h from

an altitude of 500 km.

Determine (a) the maximum altitude

reached by the satellite, and (b) the

maximum allowable error in the

direction of launching if the satellite

is to come no closer than 200 km to

the surface of the earth

SOLUTION:

• For motion under a conservative central

force, the principles of conservation of

energy and conservation of angular

momentum may be applied simultaneously.

• Apply the principles to the points of

minimum and maximum altitude to

determine the maximum altitude.

• Apply the principles to the orbit insertion

point and the point of minimum altitude to

determine maximum allowable orbit

insertion angle error.



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Sample Problem 13.9

13 - 51

• Apply the principles of conservation of energy and

conservation of angular momentum to the points of minimum

and maximum altitude to determine the maximum altitude.

Conservation of energy:

1

212

1

0

202

1

r

GMmmv

r

GMmmvVTVT AAAA

Conservation of angular momentum:

1

0011100

r

rvvmvrmvr

Combining,

2001

0

1

0

02

1

202

021 2

111vr

GM

r

r

r

r

r

GM

r

rv

23122622

60

0

sm10398m1037.6sm81.9

sm1025.10hkm36900

km6870km500km6370

gRGM

v

r

km 60400m104.60 61 r



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Sample Problem 13.9

13 - 52

• Apply the principles to the orbit insertion point and the point

of minimum altitude to determine maximum allowable orbit

insertion angle error.

Conservation of energy:

min

2max2

1

0

202

100

r

GMmmv

r

GMmmvVTVT AA

Conservation of angular momentum:

min

000maxmaxmin000 sinsin

r

rvvmvrmvr

Combining and solving for sin j0,

5.1190

9801.0sin

0

0

j

5.11error allowable



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2 - 53

A section of track for a roller coaster

consists of two circular arcs AB and CD

joined by a straight portion BC. The radius

of CD is 240 ft. The car and its occupants,

of total weight 560 lb, reach Point A with

practically no velocity and then drop freely

along the track. Determine the normal

force exerted by the track on the car at

point D. Neglect air resistance and rolling

resistance.

SOLUTION:

• This is two part problem – you

will need to find the velocity of

the car using work-energy, and

then use Newton’s second law

to find the normal force.

• Draw a diagram with the car

at points A and D, and

define your datum. Use

conservation of energy to

solve for vD

• Draw FBD and KD of the car

at point D, and determine the

normal force using Newton’s

second law.



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2 - 54

SOLUTION:

Given: vA= 0 ft/s, rCD= 240 ft, W=560 lbs

Find: ND

Define your datum, sketch the

situation at points of interest

Datum

Use conservation of energy to find vD A A D DT V T V

0 0A Av T

(560 lb)(90 60)=84,000 ft lbsA AV Wy

2 2 21 1 5608.6957

2 2 32.2D D D DT mv v v

Find TA

Find VA

Find TD

Find VD 0 0D Dy V

Solve for vD

28.6957 84000Dv

98.285 ft/sDv



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2 - 55

Draw FBD and KD at point D

Use Newton’s second law in the normal direction

n nF ma

W

ND

man

mat

2D

D

vN W m

R

2560 98.285560

32.2 240DN

1260 lbsDN

en

et



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2 - 56

What happens to the normal

force at D if….

a) ND gets larger

b) ND gets smaller

c) ND stays the same

…we move point A higher? …the radius is smaller?

…we include friction?

a) ND gets larger

b) ND gets smaller


a) ND gets larger

b) ND gets smaller




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Impulsive Motion

2 - 57

The impulse applied to the railcar

by the wall brings its momentum

to zero. Crash tests are often

performed to help improve safety

in different vehicles.

The thrust of a rocket acts

over a specific time period

to give the rocket linear

momentum.



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Principle of Impulse and Momentum

13 - 58

• From Newton’s second law,

vmvmdt

dF

linear momentum

2211

21 force theof impulse 2

1

vmvm

FdtFt

t

Imp

Imp

• The final momentum of the particle can be

obtained by adding vectorially its initial

momentum and the impulse of the force during

the time interval.

12

2

1

vmvmdtF

vmddtF

t

t

• Dimensions of the impulse of

a force are

force*time.

• Units for the impulse of a

force are

smkgssmkgsN 2



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Impulsive Motion

13 - 59

• Force acting on a particle during a very short

time interval that is large enough to cause a

significant change in momentum is called an

impulsive force.

• When impulsive forces act on a particle,

21 vmtFvm

• When a baseball is struck by a bat, contact

occurs over a short time interval but force is

large enough to change sense of ball motion.

• Nonimpulsive forces are forces for which

is small and therefore, may be

neglected – an example of this is the weight

of the baseball.

tF



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13 - 60

An automobile weighing 4000 lb is

driven down a 5o incline at a speed of

60 mi/h when the brakes are applied,

causing a constant total braking force of

1500 lb.

Determine the time required for the

automobile to come to a stop.

SOLUTION:

• Apply the principle of impulse and

momentum. The impulse is equal to the

product of the constant forces and the

time interval.



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13 - 61

SOLUTION:


momentum.

2211 vmvm

Imp

Taking components parallel to the

incline,

015005sin4000sft882.32

4000

05sin1

tt

FttWmv

s49.9t



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13 - 62

A 4 oz baseball is pitched with a

velocity of 80 ft/s. After the ball is hit

by the bat, it has a velocity of 120 ft/s

in the direction shown. If the bat and

ball are in contact for 0.015 s,

determine the average impulsive force

exerted on the ball during the impact.

SOLUTION:


momentum in terms of horizontal and

vertical component equations.



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13 - 63

SOLUTION:

• Apply the principle of impulse and momentum in

terms of horizontal and vertical component equations.

2211 vmvm

Imp

x

y

x component equation:

lb89

40cos1202.32

16415.080

2.32

164

40cos21

x

x

x

F

F

mvtFmv

y component equation:

lb9.39

40cos1202.32

16415.0

40sin0 2

y

y

y

F

F

mvtF

lb5.97,lb9.39lb89 FjiF



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13 - 64

A 10 kg package drops from a chute

into a 24 kg cart with a velocity of 3

m/s. Knowing that the cart is initially at

rest and can roll freely, determine (a)

the final velocity of the cart, (b) the

impulse exerted by the cart on the

package, and (c) the fraction of the

initial energy lost in the impact.

SOLUTION:


momentum to the package-cart system

to determine the final velocity.

• Apply the same principle to the package

alone to determine the impulse exerted

on it from the change in its momentum.



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13 - 65

SOLUTION:

• Apply the principle of impulse and momentum to the package-cart

system to determine the final velocity.

2211 vmmvm cpp

Imp

x

y

x components: 2

21

kg 25kg 1030cosm/s 3kg 10

030cos

v

vmmvm cpp

m/s 742.02 v



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13 - 66

• Apply the same principle to the package alone to determine the impulse

exerted on it from the change in its momentum.

x

y

2211 vmvm pp

Imp

x components:

2

21

kg 1030cosm/s 3kg 10

30cos

vtF

vmtFvm

x

pxp

sN56.18 tFx

y components:

030sinm/s 3kg 10

030sin1

tF

tFvm

y

yp

sN15 tFy

sN 9.23sN 51sN 56.1821 tFjitF

Imp



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13 - 67

To determine the fraction of energy lost,

221 11 12 2

221 12 22 2

10 kg 3m s 45 J

10 kg 25 kg 0.742m s 9.63 J

p

p c

T m v

T m m v

786.0J 45

J 9.63J 45

1

21

T

TT



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2 - 68

The jumper approaches the takeoff

line from the left with a horizontal

velocity of 10 m/s, remains in contact

with the ground for 0.18 s, and takes

off at a 50o angle with a velocity of 12

m/s. Determine the average impulsive

force exerted by the ground on his

foot. Give your answer in terms of the

weight W of the athlete.

SOLUTION:

• Draw impulse and momentum diagrams

of the jumper.


momentum to the jumper to determine

the force exerted on the foot.



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2 - 69

Draw impulse and momentum diagrams of the jumper

1mv2mv

+ = 50º

Use the impulse momentum equation in y to find Favg

W t

Given: v1 = 10 m/s, v2= 12 m/s at 50º,

Δt= 0.18 s

Find: Favg in terms of W

avgF t

1 2( ) 0.18 sm t m t v P W v

x

y



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2 - 70

1mv2mv

+ = 50º

Use the impulse momentum equation in x and y to find Favg

W tavgF t

1 2( ) 0.18 savgm t m t v F W v

0 ( )(0.18) (12)(sin 50 )

(12)(sin 50 )

(9.81)(0.18)

avg y

avg y

WF W

g

F W W

(10) ( )(0.18) (12)(cos 50 )

10 (12)(cos 50 )

(9.81)(0.18)

avg x

avg x

W WF

g g

F W

1.295 6.21avg W W F i j

x

y

Favg-x is positive, which means we

guessed correctly (acts to the left)



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2 - 71

Car A and B crash into

one another. Looking

only at the impact, which

of the following

statements are true?

The total mechanical energy is the

same before and after the impact

If car A weighs twice as much as car B,

the force A exerts on car B is bigger

than the force B exerts on car A.

The total linear momentum is the same

immediately before and after the impact

True False

True False

True False



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2 - 72

The coefficient of restitution is used

to characterize the “bounciness” of

different sports equipment. The U.S.

Golf Association limits the COR of

golf balls at 0.83

Civil engineers use the

coefficient of restitution to

model rocks falling from

hillsides



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Impact

13 - 73

• Impact: Collision between two bodies which

occurs during a small time interval and during

which the bodies exert large forces on each other.

• Line of Impact: Common normal to the surfaces

in contact during impact.

• Central Impact: Impact for which the mass

centers of the two bodies lie on the line of impact;

otherwise, it is an eccentric impact..


• Direct Impact: Impact for which the velocities of

the two bodies are directed along the line of

impact.


• Oblique Impact: Impact for which one or both of

the bodies move along a line other than the line of

impact.



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13 - 74

• Bodies moving in the same straight line,

vA > vB .

• Upon impact the bodies undergo a

period of deformation, at the end of which,

they are in contact and moving at a

common velocity.

• A period of restitution follows during

which the bodies either regain their

original shape or remain permanently

deformed.

• Wish to determine the final velocities of the

two bodies. The total momentum of the

two body system is preserved,

BBBBBBAA vmvmvmvm

• A second relation between the final

velocities is required.



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13 - 75

• Period of deformation: umPdtvm AAA

• Period of restitution: AAA vmRdtum

10

e

uv

vu

Pdt

Rdt

nrestitutio of tcoefficien e

A

A

• A similar analysis of particle B yields B

B

vu

uv e

• Combining the relations leads to the desired

second relation between the final velocities.

BAAB vvevv

• Perfectly plastic impact, e = 0: vvv AB vmmvmvm BABBAA

• Perfectly elastic impact, e = 1:

Total energy and total momentum conserved. BAAB vvvv



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13 - 76

• Final velocities are

unknown in magnitude

and direction. Four

equations are required.

• No tangential impulse component;

tangential component of momentum

for each particle is conserved.

tBtBtAtA vvvv

• Normal component of total

momentum of the two particles is

conserved.

nBBnAAnBBnAA vmvmvmvm

• Normal components of relative

velocities before and after impact

are related by the coefficient of

restitution.

nBnAnAnB vvevv



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13 - 77

• Block constrained to move along horizontal

surface.

• Impulses from internal forces

along the n axis and from external force

exerted by horizontal surface and directed

along the vertical to the surface.

FF

and

extF

• Final velocity of ball unknown in direction and

magnitude and unknown final block velocity

magnitude. Three equations required.



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13 - 78

• Tangential momentum of ball is

conserved.

tBtB vv

• Total horizontal momentum of block

and ball is conserved.

xBBAAxBBAA vmvmvmvm

• Normal component of relative

velocities of block and ball are related

by coefficient of restitution.

nBnAnAnB vvevv

• Note: Validity of last expression does not follow from previous relation for

the coefficient of restitution. A similar but separate derivation is required.



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Problems Involving Energy and Momentum

13 - 79

• Three methods for the analysis of kinetics problems:

- Direct application of Newton’s second law

- Method of work and energy

- Method of impulse and momentum

• Select the method best suited for the problem or part of a problem

under consideration.



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13 - 80

A ball is thrown against a frictionless,

vertical wall. Immediately before the

ball strikes the wall, its velocity has a

magnitude v and forms angle of 30o

with the horizontal. Knowing that

e = 0.90, determine the magnitude and

direction of the velocity of the ball as

it rebounds from the wall.

SOLUTION:

• Resolve ball velocity into components

normal and tangential to wall.

• Impulse exerted by the wall is normal

to the wall. Component of ball

momentum tangential to wall is

conserved.

• Assume that the wall has infinite mass

so that wall velocity before and after

impact is zero. Apply coefficient of

restitution relation to find change in

normal relative velocity between wall

and ball, i.e., the normal ball velocity.



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13 - 81

• Component of ball momentum tangential to wall is conserved.

vvv tt 500.0

• Apply coefficient of restitution relation with zero wall

velocity.

vvv

vev

n

nn

779.0866.09.0

00

SOLUTION:

• Resolve ball velocity into components parallel and

perpendicular to wall.

vvvvvv tn 500.030sin866.030cos

n

t

7.32500.0

779.0tan926.0

500.0779.0

1vv

vvv tn



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13 - 82

The magnitude and direction of the

velocities of two identical

frictionless balls before they strike

each other are as shown. Assuming

e = 0.9, determine the magnitude

and direction of the velocity of each

ball after the impact.

SOLUTION:

• Resolve the ball velocities into components

normal and tangential to the contact plane.

• Tangential component of momentum for

each ball is conserved.

• Total normal component of the momentum

of the two ball system is conserved.

• The normal relative velocities of the

balls are related by the coefficient of

restitution.

• Solve the last two equations simultaneously

for the normal velocities of the balls after

the impact.



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13 - 83

SOLUTION:

• Resolve the ball velocities into components normal and

tangential to the contact plane.

sft0.2630cos AnA vv sft0.1530sin AtA vv

sft0.2060cos BnB vv sft6.3460sin BtB vv

• Tangential component of momentum for each ball is

conserved.

sft0.15tAtA vv sft6.34

tBtB vv

• Total normal component of the momentum of the two

ball system is conserved.

0.6

0.200.26

nBnA

nBnA

nBBnAAnBBnAA

vv

vmvmmm

vmvmvmvm



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13 - 84

6.557.23

6.34tansft9.41

6.347.23

3.407.17

0.15tansft2.23

0.157.17

1

1

B

ntB

A

ntA

v

v

v

v

t

n

• The normal relative velocities of the balls are related by the

coefficient of restitution.

4.410.200.2690.0

nBnAnBnA vvevv

• Solve the last two equations simultaneously for the normal

velocities of the balls after the impact.

sft7.17nAv sft7.23

nBv



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13 - 85

Ball B is hanging from an inextensible

cord. An identical ball A is released

from rest when it is just touching the

cord and acquires a velocity v0 before

striking ball B. Assuming perfectly

elastic impact (e = 1) and no friction,

determine the velocity of each ball

immediately after impact.

SOLUTION:

• Determine orientation of impact line of

action.

• The momentum component of ball A

tangential to the contact plane is

conserved.

• The total horizontal momentum of the

two ball system is conserved.

• The relative velocities along the line of

action before and after the impact are

related by the coefficient of restitution.

• Solve the last two expressions for the

velocity of ball A along the line of action

and the velocity of ball B which is

horizontal.



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13 - 86

SOLUTION:

• Determine orientation of impact line of action.

30

5.02

sin

r

r

• The momentum component of ball A

tangential to the contact plane is

conserved.

0

0

5.0

030sin

vv

vmmv

vmtFvm

tA

tA

AA

• The total horizontal (x component)

momentum of the two ball system is

conserved.

0

0

433.05.0

30sin30cos5.00

30sin30cos0

vvv

vvv

vmvmvm

vmvmtTvm

BnA

BnA

BnAtA

BAA



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13 - 87

• The relative velocities along the line of action before

and after the impact are related by the coefficient of

restitution.

0

0

866.05.0

030cos30sin

vvv

vvv

vvevv

nAB

nAB

nBnAnAnB

• Solve the last two expressions for the velocity of ball

A along the line of action and the velocity of ball B

which is horizontal.

00 693.0520.0 vvvv BnA

0

10

00

693.0

1.16301.46

1.465.0

52.0tan721.0

520.05.0

vv

vv

vvv

B

A

ntA



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13 - 88

A 30 kg block is dropped from a height

of 2 m onto the the 10 kg pan of a

spring scale. Assuming the impact to be

perfectly plastic, determine the

maximum deflection of the pan. The

constant of the spring is k = 20 kN/m.

SOLUTION:


energy to determine the velocity of the

block at the instant of impact.

• Since the impact is perfectly plastic, the

block and pan move together at the same

velocity after impact. Determine that

velocity from the requirement that the

total momentum of the block and pan is

conserved.


energy to determine the maximum

deflection of the spring.



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13 - 89

SOLUTION:

• Apply principle of conservation of energy to

determine velocity of the block at instant of impact.

sm26.6030 J 5880

030

J 588281.9300

2222

1

2211

2222

1222

12

11

AA

AAA

A

vv

VTVT

VvvmT

yWVT

• Determine velocity after impact from requirement that

total momentum of the block and pan is conserved.

sm70.41030026.630 33

322

vv

vmmvmvm BABBAA



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13 - 90

Initial spring deflection due to

pan weight:

m1091.4

1020

81.910 3

33

k

Wx B

• Apply the principle of conservation of energy to

determine the maximum deflection of the spring.

2

43

213

4

24

3

21

34

242

14

4

233

212

321

3

2

212

321

3

10201091.4392

1020392

0

J 241.01091.410200

J 4427.41030

xx

xxx

kxhWWVVV

T

kx

VVV

vmmT

BAeg

eg

BA

m 230.0

10201091.43920241.0442

4

24

3

213

4

4433

x

xx

VTVT

m 1091.4m 230.0 334

xxh m 225.0h



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2 - 91

A 2-kg block A is pushed up against a spring

compressing it a distance x= 0.1 m. The block is

then released from rest and slides down the 20º

incline until it strikes a 1-kg sphere B, which is

suspended from a 1 m inextensible rope. The

spring constant k=800 N/m, the coefficient of

friction between A and the ground is 0.2, the

distance A slides from the unstretched length of

the spring d=1.5 m, and the coefficient of

restitution between A and B is 0.8. When =40o,

find (a) the speed of B (b) the tension in the rope.

SOLUTION:

• This is a multiple step problem.

Formulate your overall approach.

• Use work-energy to find the

velocity of the block just

before impact

• Use conservation of

momentum to determine

the speed of ball B after

the impact

• Use work energy to find

the velocity at

• Use Newton’s 2nd Law to

find tension in the rope



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2 - 92

Given: mA= 2-kg mB= 1-kg,

k= 800 N/m, mA =0.2, e= 0.8

Find (a) vB (b) Trope

• Use work-energy to find the velocity of

the block just before impact

Determine the friction force acting on the block A

cos 0AN m g

cos

(2)(9.81)cos 20

18.4368 N

(0.2)(18.4368)

3.6874 N

A

f k

N m g

F N

m

0:yF

Sum forces in the y-direction

Solve for N



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2 - 93

Set your datum, use work-energy to determine vA at impact.

1

2

Datum x d

1 1 1 1 2 2 2 2( ) ( ) ( ) ( )e g e gT V V U T V V

Determine values for each term.

2 21 1 1

1 10, ( ) (800)(0.1) 4.00 J

2 2eT V k x

1 1( ) ( )sin (2)(9.81)(1.6)sin 20 10.7367 Jg A AV m gh m g x d

1 2 ( ) (3.6874)(1.6) 5.8998 JfU F x d

2 2 22 2

1 1(1)( ) 1.000 0

2 2A A A AT m v v v V

21 1 1 2 2 2: 0 4.00 10.7367 5.8998 1.000 0AT V U T V v

2 2 28.8369 m /sAv

2.9727 m/sA v

Substitute into the Work-Energy equation and solve for vA



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2 - 94

• Use conservation of momentum to determine

the speed of ball B after the impact

• Draw the impulse diagram

Note that the ball is constrained to move only

horizontally immediately after the impact.

Apply conservation of

momentum in the x direction

(2)(2.9727)cos20 2 cos20 (1.00)A Bv v

cos 0 cosA A A A B Bm v m v m v ( ) ( ) [( ) ( ) ]B n A n B n A nv v e v v

cos [ 0]

cos20 (0.8)(2.9727)

B A A

B A

v v e v

v v

Use the relative velocity/coefficient

of restitution equation

(1) (2)

Solve (1) and (2) simultaneously

1.0382 m/s 3.6356 m/sA Bv v



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2 - 95

• Use work energy to find the velocity at

Set datum, use Work-Energy to determine vB at = 40o

1

2

Datum

1 1 1 1 2 2 2 2( ) ( ) ( ) ( )e g e gT V V U T V V

Determine values for each term.

21 1

22 2 2 2

1( ) 0

2

1(1 cos )

2

B B

B B B

T m v V

T m v V m gh m gl

2 21 1 2 2 2

2 22

2

2 2

1 1: ( ) 0 (1 cos )

2 2

( ) 2 (1 cos )

(3.6356) (2)(9.81)(1 cos 40 )

8.6274 m /s

B B B B

B

T V T V m v m v m g

v v gl

2 2.94 m/sv

Substitute into the Work-Energy equation and solve for vA



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2 - 96

• Use Newton’s 2nd Law to find tension in the rope

• Draw your free-body and

kinetic diagrams

:n B nF m a

cos

( cos )

B B n

B n

T m g m a

T m a g

et

en

222

1.00 m

8.62748.6274 m/s

1.00n

va

(1.0)(8.6274 9.81cos40 )T

• Sum forces in the normal direction • Determine normal acceleration

16.14 NT

• Substitute and solve



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Concept Question

2 - 97

If the coefficient of restitution is

smaller than the 0.8 in the

problem, the tension T will be…

Smaller Bigger

If the rope length is smaller than

the 1 m in the problem, the

tension T will be…

Smaller Bigger

If the coefficient of friction is

smaller than 0.2 given in the

problem, the tension T will be…

Smaller Bigger

If the mass of A is smaller

than the 2 kg given in the

problem, the tension T will

be…

Smaller Bigger

Compare the following statement to the problem you just solved.



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Summary

2 - 98

Forces and

Accelerations

Velocities and

Displacements

Velocities and

Time

Approaches to Kinetics Problems

Newton’s Second

Law (last chapter)

Work-Energy Impulse-

Momentum

2211 TUT GF ma2

11 2

t

tmv F dt mv

Documents

Tenth Edition · 2015. 12. 14. · Vector Mechanics for Engineers: Dynamics dition Contents 13 - 2 Introduction Work of a Force Principle of Work & Energy Applications of the Principle