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Chapter 7
Work and Energy
Copyright © 2009 Pearson Education, Inc.
Copyright © 2009 Pearson Education, Inc.
7-1 Work Done by a Constant Force
The work done by a constant force is defined as
the distance moved multiplied by the component
of the force in the direction of displacement:
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7-1 Work Done by a Constant Force
In the SI system, the units of work are joules:
As long as this person does
not lift or lower the bag of
groceries, he is doing no
work on it. The force he
exerts has no component in
the direction of motion.
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7-1 Work Done by a Constant Force
Example 7-1: Work done on a crate.
A person pulls a 50-kg crate 40 m along a horizontal floor by a
constant force FP = 100 N, which acts at a 37° angle as shown. The
floor is smooth and exerts no friction force. Determine
(a) the work done by each force acting on the crate
crate, and
(b) the net work done on the crate.
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Copyright © 2009 Pearson Education, Inc.
7-1 Work Done by a Constant Force
Solving work problems:
1. Draw a free-body
y diagram.
g
2. Choose a coordinate system.
3. Apply Newton’s laws to determine any
unknown forces.
4. Find the work done by a specific force.
5. To find the net work, either
a) find the net force and then find the work it
does, or
b) find the work done by each force and add.
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7-1 Work Done by a Constant
Force
Example 7
7-2:
2: Work on a backpack.
backpack
(a) Determine the work a hiker must do
on a 15.0
15.0-kg
kg backpack to carry it up a
hill of height h = 10.0 m, as shown.
Determine also
(b) the work done by gravity on the
backpack, and
(c) the net work done on the backpack.
backpack
For simplicity, assume the motion is
smooth and at constant velocity (i.e.,
acceleration
l
i is
i zero).
)
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Copyright © 2009 Pearson Education, Inc.
7-1 Work Done by a Constant Force
Conceptual Example 7-3: Does the Earth do
work on the Moon?
The Moon revolves around the Earth
in a nearly circular orbit, with
approximately constant tangential
speed, kept there by the
gravitational force exerted by the
Earth Does gravity do
Earth.
(a) positive work,
(b) negative work,
work or
(c) no work at all on the Moon?
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Copyright © 2009 Pearson Education, Inc.
7-2 Scalar Product of Two Vectors
Definition of the scalar, or dot, product:
Therefore we can write:
Therefore,
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7-2 Scalar Product of Two Vectors
Example 7-4: Using the dot product.
The force shown has magnitude FP = 20 N and
makes an angle of 30° to the ground. Calculate
the work done by this force,
force using the dot
product, when the wagon is dragged 100 m
along the ground
ground.
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Copyright © 2009 Pearson Education, Inc.
7-3 Work Done by a Varying Force
Particle racted on by a varying force.
Clearly F·dd is not constant!
Clearly,
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7-3 Work Done by a Varying Force
For a force that varies, the work can be
approximated
pp
by
y dividing
g the distance up
p into
small pieces, finding the work done during
each, and adding
g them up.
p
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7-3 Work Done by a Varying Force
In the limit that the pieces become
infinitesimally
y narrow, the work is the area
under the curve:
Or:
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7-3 Work Done by a Varying Force
Work done by a spring force:
The force exerted
b a spring
by
i is
i
given by:
.
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7-3 Work Done by a Varying Force
Plot of F vs. x. Work
d
done
is
i equall to
t the
th
shaded area.
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7-3 Work Done by a Varying Force
Example 7
7-5:
5: Work done on a spring.
spring
(a) A person pulls on a spring, stretching it 3.0 cm, which
requires a maximum force of 75 N. How much work
does the person do?
(b) If, instead, the person compresses the spring 3.0 cm,
how much work does the person do?
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Copyright © 2009 Pearson Education, Inc.
7-3 Work Done by a Varying Force
Example 7-6: Force as a function of x.
A robot arm that controls the position of a video
camera in
i an automated
t
t d surveillance
ill
system
t
is
i
manipulated by a motor that exerts a force on the
arm The force is given by
arm.
where
e e F0 = 2.0
0 N,, x0 = 0
0.0070
00 0 m,, a
and
d x is
s tthe
e
position of the end of the arm. If the arm moves
from x1 = 0.010 m to x2 = 0.050 m, how much work
did the motor do?
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Copyright © 2009 Pearson Education, Inc.
7-4 Kinetic Energy and the Work-Energy
Principle
Energy was traditionally defined as the ability to
do work.
work We now know that not all forces are
able to do work; however, we are dealing in these
chapters with mechanical energy,
energy which does
follow this definition.
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7-4 Kinetic Energy and the Work-Energy
Principle
If we write the acceleration in terms of the
velocity and the distance,
distance we find that the
work done here is
We define the kinetic energy as:
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7-4 Kinetic Energy and the Work-Energy
Principle
This means that the work done is equal to the
change in the kinetic energy:
• If the net work is positive, the kinetic energy
increases.
• If the net work is negative, the kinetic energy
decreases.
decreases
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7-4 Kinetic Energy and the Work-Energy
Principle
Because work and kinetic energy can be
equated, they must have the same units:
kinetic energy is measured in joules. Energy
can be considered as the ability to do work:
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7-4 Kinetic Energy and the Work-Energy
Principle
Example 7-7: Kinetic energy and work
done on a baseball.
A 145-g baseball is thrown so that it
acquires
i
a speed
d off 25 m/s.
/ (a)
( ) What
Wh t is
i
its kinetic energy? (b) What was the net
work
kd
done on the
th ball
b ll to
t make
k it reach
h
this speed, if it started from rest?
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Copyright © 2009 Pearson Education, Inc.
7-4 Kinetic Energy and the Work-Energy
Principle
Example
p 7-8: Work on a car,, to increase its
kinetic energy.
How much net work is required to
accelerate a 1000-kg car from 20 m/s to 30
m/s?
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Copyright © 2009 Pearson Education, Inc.
7-4 Kinetic Energy and the Work-Energy
Principle
Example 7-9: Work to stop a car.
A car traveling 60 km/h can brake to a stop
within a distance d of 20 m. If the car is
going twice as fast, 120 km/h, what is its
stopping
pp g distance? Assume the maximum
braking force is approximately independent
of speed.
p
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Copyright © 2009 Pearson Education, Inc.
7-4 Kinetic Energy and the Work-Energy
Principle
Example 7-10: A compressed spring.
A horizontal spring has spring constant k = 360 N/m.
N/m
(a) How much work is required to compress it from its uncompressed length
(x = 0) to x = 11.0 cm?
(b) If a 1.85-kg block is placed against the spring and the spring is released,
what will be the speed of the block when it separates from the spring
at x = 0? Ignore friction.
(c) (c) Repeat part (b) but assume that the block is moving on a table and
that some kind of constant drag force FD = 7.0 N is acting to slow it down,
such as friction (or perhaps your finger).
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Copyright © 2009 Pearson Education, Inc.
Copyright © 2009 Pearson Education, Inc.
Summary of Chapter 7
• Work:
• Work done by a variable force:
• Kinetic energy
gy is energy
gy of motion:
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Summary of Chapter 7
• Work-energy principle: The net work done
on an object
bj t equals
l the
th change
h
in
i its
it
kinetic energy.
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Copyright © 2009 Pearson Education, Inc.
8-1 Conservative and Nonconservative
Forces
A force is conservative if:
th workk done
the
d
by
b the
th force
f
on an object
bj t
moving from one point to another depends
only
l on the
th initial
i iti l andd final
fi l positions
iti
off the
th
object, and is independent of the particular
path taken
taken.
Example: gravity.
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8-1 Conservative and Nonconservative
Forces
Another definition of a conservative force:
a force is conservative if the net work done by the force
on an object moving around any closed path is zero.
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8-1 Conservative and Nonconservative
Forces
If friction is present, the work done depends not
only on the starting and ending points, but also
on the path taken. Friction is called a
nonconservative force.
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8-1 Conservative and Nonconservative
Forces
Potential
P
t ti l energy can
only be defined for
conservative
ti forces.
f
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8-2 Potential Energy
An object can have potential energy by virtue of
its surroundings.
Familiar examples of potential energy:
• A wound
wound-up
up spring
• A stretched elastic band
• An object at some height above the ground
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8-2 Potential Energy
In raising a mass m to a height
h, the work done by the
external force is
.
We therefore define the
gravitational potential
energy at a height y above
some reference point:
.
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8-2 Potential Energy
This potential energy can become kinetic energy
if the object is dropped.
Potential energy is a property of a system as a
whole,
h l nott just
j t off the
th object
bj t (because
(b
it depends
d
d
on external forces).
If Ugrav = mgy, where do we measure y from?
It turns
t
outt nott to
t matter,
tt as long
l
as we are
consistent about where we choose y = 0. Only
changes
h
i potential
in
t ti l energy can be
b measured.
d
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8-2 Potential Energy
Example 8-1: Potential energy changes for a roller coaster.
A 1000-kg roller-coaster car moves from point 1 to point 2 and then to
point 3.
(a) What is the gravitational potential energy at points 2 and 3 relative to
point 1? That is, take y = 0 at point 1.
(b) What is the change in potential energy when the car goes from point
2 to point 3?
(c) (c) Repeat parts (a) and (b)
(b), but take the reference point (y = 0) to be
at point 3.
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8-2 Potential Energy
General definition of gravitational
potential energy:
For any conservative force:
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8-2 Potential Energy
A spring has potential
energy,
gy called elastic
potential energy, when it
is compressed.
p
The force
required to compress or
stretch a spring
p g is:
where k is called the
g constant, and
spring
needs to be measured for
each spring.
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8-2 Potential Energy
Then the potential energy is:
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8-2 Potential Energy
I one dimension,
In
di
i
We can invert this equation
q
to find U(x)
( )
if we know F(x):
I three
In
th
dimensions:
di
i
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8-3 Mechanical Energy and Its
Conservation
If there are no nonconservative forces, the sum
of the changes in the kinetic energy and in the
potential energy is zero—the kinetic and
potential energy changes are equal but opposite
in sign.
This allows us to define the total mechanical
energy:
gy
And its conservation:
.
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8-3 Mechanical Energy and Its
Conservation
The principle of conservation of mechanical
energy:
If only conservative forces are doing work,
th ttotal
the
t l mechanical
h i l energy off a system
t
neither increases nor decreases in any
process. It stays constant—it is conserved.
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8-4 Problem Solving Using Conservation of
Mechanical Energy
In the image on the left,
the total mechanical
energy at any point is:
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8-4 Problem Solving Using Conservation of
Mechanical Energy
Example
p 8-3: Falling
g rock.
If the original height of the
rock is y1 = h = 3.0 m,
calculate the rock’s
rock s speed
when it has fallen to 1.0 m
above the ground.
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Copyright © 2009 Pearson Education, Inc.
8-4 Problem Solving Using Conservation of
Mechanical Energy
Example 8-4: Roller-coaster car speed using
energy conservation.
Assuming the height of the hill is 40 m
m, and
the roller-coaster car starts from rest at the
top calculate (a) the speed of the rollertop,
coaster car at the bottom of the hill, and (b)
at what height it will have half this speed.
speed
Take y = 0 at the bottom of the hill.
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Copyright © 2009 Pearson Education, Inc.
8-4 Problem Solving Using Conservation of
Mechanical Energy
Conceptual Example 8-5: Speeds on two
water slides
slides.
Two water slides at a pool
are shaped differently
differently, but
start at the same height h.
Two riders, Paul and
Kathleen, start from rest at
the same time on different
slides.
lid
(a)
( ) Which
Whi h rider,
id Paul
P l
or Kathleen, is traveling
faster at the bottom? (b)
Which rider makes it to the
bottom first? Ignore friction
and assume both slides have
the same path length.
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8-4 Problem Solving Using Conservation of
Mechanical Energy
Which to use for solving problems?
Newton s laws: best when forces are
Newton’s
constant
Work and energy: good when forces are
constant; also may succeed when forces
are not constant
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8-4 Problem Solving Using Conservation of
Mechanical Energy
Example 8-6: Pole vault.
Estimate the kinetic
energy and the speed
required for a 70-kg
pole vaulter to just pass
over a bar 5.0 m high.
Assume the vaulter
vaulter’s
s
center of mass is
initially 0
0.90
90 m off the
ground and reaches its
maximum height at the
level of the bar itself.
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8-4 Problem Solving Using Conservation of
Mechanical Energy
For an elastic force, conservation of energy tells
us:
Example 8-7:
8 7: Toy dart gun
gun.
A dart of mass 0.100 kg is
pressed against the spring of a
toy dart gun. The spring (with
spring stiffness constant k = 250
N/ and
N/m
d iignorable
bl mass)) is
i
compressed 6.0 cm and released.
If the dart detaches from the
spring when the spring reaches
its natural length (x = 0), what
speed
d does
d
the
th dart
d t acquire?
i ?
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For an elastic force, conservation of energy :
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8-4 Problem Solving Using Conservation of
Mechanical Energy
Example
p 8-8: Two kinds of p
potential energy.
gy
A ball of mass m = 2.60 kg, starting
from rest, falls a vertical distance
h = 55.0
55 0 cm b
before
f
striking
t iki a vertical
ti l
coiled spring, which it compresses an
amount Y = 15.0 cm. Determine the
spring stiffness constant of the spring
spring.
Assume the spring has negligible
mass, and ignore air resistance.
Measure all distances from the point
where the ball first touches the
uncompressed spring (y = 0 at this
point).
point)
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Using Conservation of Mechanical Energy
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8-4 Problem Solving Using Conservation of
Mechanical Energy
Example 8-9: A swinging pendulum.
This simple pendulum consists of a small bob of
mass m suspended by a massless cord of length l.
p
) at t = 0, where
The bob is released ((without a push)
the cord makes an angle θ = θ0 to the vertical.
(a) Describe the motion of the
bob in terms of kinetic energy
and potential energy. Then
determine the speed of the bob
(b) as a function of position θ as
itt s
swings
gs bac
back a
and
d forth,
o t , and
a d (c)
at the lowest point of the swing.
(d)
r Find the tension in the cord,
FT. Ignore friction and air
resistance.
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Copyright © 2009 Pearson Education, Inc.
Copyright © 2009 Pearson Education, Inc.
8-5 The Law of Conservation of Energy
Nonconservative, or dissipative, forces:
Friction
Heat
Electrical energy
Chemical energy
and more
do not conserve mechanical energy. However,
when these forces are taken into account, the
total energy is still conserved:
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8-5 The Law of Conservation of Energy
The law of conservation of energy is one of
the most important principles in physics.
The total energy
gy is neither increased nor
decreased in any process. Energy can be
transformed from one form to another,
another and
transferred from one object to another, but
th ttotal
the
t l amountt remains
i constant.
t t
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8-6 Energy Conservation with Dissipative
Forces: Solving Problems
Problem Solving:
1. Draw a picture.
2 D
2.
Determine
t
i th
the system
t
f which
for
hi h energy will
ill
be conserved.
3. Figure out what you are looking for, and
decide on the initial and final positions.
4. Choose a logical reference frame.
5. Apply conservation of energy.
6 Solve.
6.
S l
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8-6 Energy Conservation with Dissipative
Forces: Solving Problems
Example 8-11: Friction with a spring.
A block of mass m sliding
along
g a rough
g horizontal
surface is traveling at a
speed v0 when it strikes a
massless
l
spring
i head-on
h d
and compresses the spring a
maximum distance X.
X If the
spring has stiffness constant
k,, determine the coefficient
of kinetic friction between
block and surface.
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Copyright © 2009 Pearson Education, Inc.
8-7 Gravitational Potential Energy and
Escape Velocity
Far from the surface of the Earth, the force
of gravity is not constant:
The work done on an object
moving in the Earth’s
gravitational field is given by:
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8-7 Gravitational Potential Energy and
Escape Velocity
Solving the integral gives:
Because the value of the integral depends
only on the end points,
points the gravitational
force is conservative and we can define
gravitational potential energy:
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8-7 Gravitational Potential Energy and
Escape Velocity
Example
p 8-12: Package
g dropped
pp from highg
speed rocket.
A box of empty film canisters is allowed to
fall from a rocket traveling outward from
Earth at a speed of 1800 m/s when 1600 km
above the Earth’s surface. The package
eventually falls to the Earth.
Earth Estimate its
speed just before impact. Ignore air
resistance.
resistance
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8-7 Gravitational Potential Energy and
Escape Velocity
If an object’s
bj t’ initial
i iti l kinetic
ki ti energy is
i equall to
t
the potential energy at the Earth’s surface, its
t t l energy will
total
ill be
b zero. The
Th velocity
l it att which
hi h
this is true is called the escape velocity; for
E th
Earth:
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8-7 Gravitational Potential Energy and
Escape Velocity
Example
p 8-13: Escaping
p g the Earth or
the Moon.
(a) Compare the escape velocities of a
rocket from the Earth and from the
Moon.
Moon
((b)) Compare
p
the energies
g
required
q
to
launch the rockets. For the Moon,
MM = 7.35 x 1022 kg
g and rM = 1.74 x
106 m, and for Earth, ME = 5.98 x 1024
kg
g and rE = 6.38 x 106 m.
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8-8 Power
Power is the rate at which work is done
done.
Average power:
Instantaneous power:
In the SI system, the units of power
are watts:
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8-8 Power
Power can also be described as the rate at
which energy is transformed:
In the British system, the basic unit for
power is the foot-pound per second,
second but
more often horsepower is used:
1 hp = 550 ft·lb/s = 746 W.
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8-8 Power
Example 8-14: Stair-climbing power.
A 60-kg jogger runs up a
long
o g flight
g t of
o stairs
sta s in 4.0
0s
s.
The vertical height of the
stairs is 4.5 m. ((a)) Estimate
the jogger’s power output
in watts and horsepower.
p
(b) How much energy did
this require?
q
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8-8 Power
Power is also needed for
acceleration and for moving against
the force of friction.
The power can be written in terms of
the net force and the velocity:
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8-8 Power
Example 8-15:
8 15: Power needs of a car.
car
Calculate the power required of a 1400-kg car
under the following circumstances: (a) the car
climbs a 10° hill (a fairly steep hill) at a steady 80
km/h; and (b) the car accelerates along a level
road from 90 to 110 km/h in 6.0 s to pass another
car. Assume that the average retarding force on
the car is FR = 700 N throughout.
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