Kinetics of Particles: Energy and Momentum Methods

VECTOR MECHANICS FOR ENGINEERS: DYNAMICSDYNAMICS

Tenth Tenth EditionEdition

Ferdinand P. BeerFerdinand P. Beer

E. Russell Johnston, Jr.E. Russell Johnston, Jr.

Phillip J. CornwellPhillip J. Cornwell

Lecture Notes:Lecture Notes:

Brian P. SelfBrian P. SelfCalifornia Polytechnic State UniversityCalifornia Polytechnic State University

CHAPTER

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

13Kinetics of Particles: Energy and Momentum Methods


Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics

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Contents

13 - 2

IntroductionWork of a ForcePrinciple of Work & EnergyApplications of the Principle of

Work & EnergyPower and EfficiencySample Problem 13.1Sample Problem 13.2Sample Problem 13.3Sample Problem 13.4Sample Problem 13.5Potential EnergyConservative ForcesConservation of EnergyMotion Under a Conservative

Central Force

Sample Problem 13.6Sample Problem 13.7Sample Problem 13.9Principle of Impulse and MomentumImpulsive MotionSample Problem 13.10Sample Problem 13.11Sample Problem 13.12ImpactDirect Central ImpactOblique Central ImpactProblems Involving Energy and MomentumSample Problem 13.14Sample Problem 13.15Sample Problems 13.16Sample Problem 13.17



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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 ma 2

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

dxkxdxFdUx

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 212

121



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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 :

• 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

dvmvdsFds

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

2212

121

1

1

mvT

v

0lb1151lbft4810002211

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 k = 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

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, k = 0.25

2124900 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 2200kg 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

kf xWU

J377m640.0sm81.9kg60 2

21

J0.112m040.0N3200N2400

N3200m160.0mkN20

N2400m120.0mkN20

21

maxmin21

21

0max

0min

xPPU

xxkP

kxP

e

J112J377212121 kef UUU

0J112J 377-J5.187

:2211

k

TUT

20.0k



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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

kef UUU

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 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:


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


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 k= 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, k= 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

sin 30

(sin 30 cos 30 )

AB

AB k AB

A B AB

k

N mg

F N mg

U mg d F d

mg d

7 m

BC BC

BC k

B C k BC

N mg x

F mg

U mg x

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 1 m/s

2A A AT mv v 21and 2 m/s

2C C CT mv v

2 20

1 1(sin 30 cos30 )

2 2A k k BCmv mg d mg x mv

2 2

2 2

/2 /2

(sin 30 cos30 )

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

sin 30 0.25cos30

C k BC A

k

v g x v gd

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?

k= 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

• At minimum and maximum r, 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

2212

121

T

VVV

kxV

ge

e

Position 2:

22

22

222

12

2

2212

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.

• Apply the principle of conservation of 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

22212

21

1

T

xxkxVVV ge

lbft5.0sft4.64sft2.32

lb5.0

2

1

lbft2ft4lb5.00

222

221

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,

5.1190

9801.0sin

0

0

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 VD0 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

240 ft

W

ND

man

mat

2D

Dv

N W mR

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

vmddtFt

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 whichis small and therefore, may be

neglected – an example of this is the weight of the baseball.

tF



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Sample Problem 13.10

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:

• Apply the principle of impulse and 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:

• Apply the principle of impulse and 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:

• Apply the principle of impulse and 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.

• Apply the principle of impulse and 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

1mv 2mv

+ =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

1mv 2mv

+ =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 Central Impact

• Direct Impact: Impact for which the velocities of the two bodies are directed along the line of impact.

Oblique Central 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 aperiod 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 yieldsB

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 forcesalong the n axis and from external forceexerted 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.

sft0.15 tAtA 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.

sft7.17 nAv 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:

• Apply the principle of conservation of 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.

• Apply the principle of conservation of 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

81.910 333

k

Wx B

• Apply the principle of conservation of energy to determine the maximum deflection of the spring.

2

43

213

4

24

321

34

242

14

4

233212

321

3

2212

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

3213

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, A =0.2, e= 0.8Find (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

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

Datumxd

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)cos 20 2 cos 20 (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]

cos 20 (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

21

(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

1.00nv

a

(1.0) (8.6274 9.81cos 40 )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 ma 2

11 2

t

tmv F dt mv

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Kinetics of Particles: Energy and Momentum Methods

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