Chapter 6
Energy
and
Energy Transfer
1
Introduction to Energy
„
„
„
The concept of energy is one of the
most important topics in science
Every physical process that occurs in
the Universe involves energy and
energy transfers or transformations
Energy is not easily defined
2
Energy Approach to Problems
„
„
The energy approach to describing
motion is particularly useful when the
force is not constant
A global approach to problems involving
energy and energy transfers will be
developed
„
This could be extended to biological
organisms, technological systems and
engineering situations
3
Systems
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A system is a small portion of the
Universe
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We will ignore the details of the rest of the
Universe
This is a simplification model
A critical skill is to identify the system
4
Identifying Systems
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A system may
„
„
„
„
be a single object or particle
be a collection of objects or particles
be a region of space
vary in size and shape
5
Environment
„
There is a system boundary around the
system
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„
„
The boundary is an imaginary surface
It does not necessarily correspond to a
physical boundary
The boundary divides the system from
the environment
„
The environment is the rest of the Universe
6
Work
„
The work, W, done on a system by an
agent exerting a constant force on the
system is the product of the magnitude,
F, of the force, the magnitude Δr of the
displacement of the point of application
of the force, and cos θ, where θ is the
angle between the force and the
displacement vectors
7
Work, cont.
„
W = F Δr cos θ
„
„
„
The displacement is that of the point of
application of the force
A force does no work on the object if the
force does not move through a
displacement
The work done by a force on a moving
object is zero when the force applied is
perpendicular to the displacement of its
point of application
8
Work Example
„
The normal force, n,
and the gravitational
force, m g, do no
work on the object
„
„
cos θ = cos 90° = 0
The force does do
work on the object
9
Units of Work
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Work is a scalar quantity
The unit of work is a joule (J)
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1 joule = 1 newton . 1 meter
J=N·m
10
More About Work
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The system and the environment must be
determined when dealing with work
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The environment does work on the system
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Work by the environment on the system
The sign of the work depends on the direction
of relative to
„
„
Work is positive when projection of onto
is in
the same direction as the displacement
Work is negative when the projection is in the
opposite direction
11
Problem 6.1.
„
A block of mass 2.50 kg is pushed 2.20
m along a frictionless horizontal table by
a constant 16.0-N force directed 25.0°
below the horizontal. Determine the
work done on the block by (a) the
applied force, (b) the normal force
exerted by the table, and (c) the
gravitational force. (d) Determine the
total work done on the block.
12
Scalar Product of Two Vectors
„
The scalar product
of two vectors is
written as
„
It is also called the
dot product
„
„
θ is the angle
between A and B
13
Scalar Product, cont
„
The scalar product is commutative
„
„
The scalar product obeys the distributive
law of multiplication
„
14
Dot Products of Unit Vectors
„
î ⋅ î = ĵ ⋅ ĵ = k̂ ⋅ k̂ = 1
î ⋅ ĵ = î ⋅ k̂ = ĵ ⋅ k̂ = 0
„
Using component form with
and
:
15
Work Done by a Varying
Force
„
„
„
Assume that during a
very small
displacement, Δx, F is
constant
For that displacement,
W1 ≈ F Δx
For all of the intervals,
16
Work Done by a Varying
Force, cont
„
„
Therefore,
„
The work done is
equal to the area
under the curve
17
Work Done By Multiple Forces
„
If more than one force acts on a system
and the system can be modeled as a
particle, the total work done on the
system is the work done by the net
force
18
Work Done by Multiple
Forces, cont.
„
If the system cannot be modeled as a
particle, then the total work is equal to
the algebraic sum of the work done by
the individual forces
19
Hooke’s Law
„
The force exerted by the spring is
Fs = - kx
„
„
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x is the position of the block with respect to the equilibrium
position (x = 0)
k is called the spring constant or force constant and
measures the stiffness of the spring
This is called Hooke’s Law
20
Hooke’s Law, cont.
„
„
„
When x is positive
(spring is stretched),
F is negative
When x is 0 (at the
equilibrium position),
F is 0
When x is negative
(spring is
compressed), F is
positive
21
Hooke’s Law, final
„
„
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The force exerted by the spring is
always directed opposite to the
displacement from equilibrium
F is called the restoring force
If the block is released it will oscillate
back and forth between –xmax and xmax
22
Work Done by a Spring
„
„
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Identify the block as the
system
Calculate the work as
the block moves from xi
= - xmax to xf = 0
The total work done as
the block moves from
–xmax to xmax is zero
23
Spring with an Applied Force
„
„
„
„
Suppose an external
agent, Fapp, stretches
the spring
The applied force is
equal and opposite to
the spring force
Fapp = -Fs = -(-kx) = kx
Work done by Fapp is
equal to -1/2 kx2max
24
A dart is loaded into a spring-loaded toy dart gun by pushing the spring in
by a distance d. For the next loading, the spring is compressed a distance
2d. How much work is required to load the second dart compared to that
required to load the first?
20% 20% 20% 20% 20%
m
m
uc
h
uc
h
10
as
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h
efo
u
on
ha
lf
as
e
th
as
sa
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uc
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e
h
uc
h
m
tim
es
5.
as
4.
tw
o
3.
ur
tim
es
2.
four times as much
two times as much
the same
half as much
one-fourth as much
fo
1.
25
Problem 6.13.
„
When a 4.00-kg object is hung vertically
on a certain light spring that obeys
Hooke’s law, the spring stretches 2.50
cm. If the 4.00-kg object is removed, (a)
how far will the spring stretch if a 1.50kg block is hung on it and (b) how much
work must an external agent do to
stretch the same spring 4.00 cm from its
unstretched position?
26
Kinetic Energy
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Kinetic Energy is the energy of a
particle due to its motion
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K = 1/2 mv2
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K is the kinetic energy
m is the mass of the particle
v is the speed of the particle
A change in kinetic energy is one
possible result of doing work to transfer
energy into a system
27
Kinetic Energy, cont
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Calculating the
work:
28
Work-Kinetic Energy Theorem
„
„
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The Work-Kinetic Energy Theorem states ΣW
= Kf – Ki = ΔK
In the case in which work is done on a system
and the only change in the system is in its
speed, the work done by the net force equals
the change in kinetic energy of the system.
We can also define the kinetic energy
„
K = 1/2 mv2
29
Work-Kinetic Energy Theorem
– Example
„
„
„
The normal and
gravitational forces do
no work since they are
perpendicular to the
direction of the
displacement
W = F Δx
W = ΔK = 1/2 mvf2 - 0
30
Nonisolated System
„
A nonisolated system is one that
interacts with or is influenced by its
environment
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A new analysis model
An isolated system would not interact with
its environment
The Work-Kinetic Energy Theorem can
be applied to nonisolated systems
31
Energy Transfer
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Work is a method of energy transfer
Work has the effect of transferring
energy between the system and the
environment
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If positive work is done on the system,
energy is transferred to the system
Negative work indicates that energy is
transferred from the system to the
environment
32
Internal Energy
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The energy
associated with an
object’s temperature
is called its internal
energy, Eint
The friction does
work and increases
the internal energy
of the surface
33
Ways to Transfer Energy Into
or Out of A System
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Work – transfers by applying a force
and causing a displacement of the point
of application of the force
Mechanical Waves – allow a
disturbance to propagate through a
medium
Heat – is driven by a temperature
difference between two regions in space
34
More Ways to Transfer Energy
Into or Out of A System
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Matter Transfer – matter physically
crosses the boundary of the system,
carrying energy with it
Electrical Transmission – transfer is
by electric current
Electromagnetic Radiation – energy is
transferred by electromagnetic waves
35
Examples of Ways to Transfer
Energy
„
a) Work
„
b) Mechanical
Waves
„
c) Heat
36
Examples of Ways to Transfer
Energy, cont.
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d) Matter transfer
„
e) Electrical
Transmission
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f) Electromagnetic
radiation
37
Conservation of Energy
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Energy is conserved
„
„
This means that energy cannot be created
or destroyed
If the total amount of energy in a system
changes, it can only be due to the fact that
energy has crossed the boundary of the
system by some method of energy transfer
38
Conservation of Energy, cont.
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Mathematically, ΔEsystem = ΣΤ
„
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Esystem is the total energy of the system
T is the energy transferred across the
system boundary
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Established symbols: Twork = W and Theat = Q
Others do not have standard symbols
The Work-Kinetic Energy theorem is a
special case of Conservation of Energy
39
Continuity Equation
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The conservation of energy equation is
an example of an continuity equation
„
„
Specifically, it is the continuity equation for
energy
A continuity equation arises in any
situation in which the change in a
quantity in a system occurs solely
because of transfers across the
boundary
40
Conservation of Energy,
Completed
„
The primary mathematical
representation of the energy analysis of
a nonisolated system is
ΔK + ΔEint = W + TMT + TET + TER
„
„
If any of the terms on the right are zero, the
system is an isolated system
The Work-Kinetic Energy Theorem is a
special case of the more general equation
above
41
Problems Involving Kinetic
Energy
„
„
When kinetic friction is involved in a
problem, you must use a modification of
the work-kinetic energy theorem
ΣW other forces – ƒk d = ΔK
„
„
The term ƒk d is the work associated with
the frictional force
Also, Δeint = ƒk d when friction is the only
force acting in the system
42
Problems Involving Kinetic
Energy, cont
„
„
A friction force transforms the kinetic
energy in a system to internal energy
For a system in which the frictional force
alone acts, the increase in the internal
energy of the system is equal to its
decrease in kinetic energy
43
Power
„
„
The time rate of energy transfer is
called power
The average power is given by
when the method of energy transfer is
work
44
Instantaneous Power
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The instantaneous power is the
limiting value of the average power as
Δt approaches zero
„
This can also be written as
45
Power Generalized
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„
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Power can be related to any type of energy
transfer
In general, power can be expressed as
dE/dt is the rate rate at which energy is
crossing the boundary of the system for a
given transfer mechanism
46
Units of Power
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The SI unit of power is called the watt
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„
A unit of power in the US Customary
system is horsepower
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1 watt = 1 joule / second = 1 kg . m2 / s2
1 hp =550 ft .lb/s = 746 W
Units of power can also be used to
express units of work or energy
„
1 kWh = (1000 W)(3600 s) = 3.6 x106 J
47
Horsepower Ratings of
Automobiles
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The strength of the frictional force exerted on
a car by the roadway is related to the rate at
which energy is transferred to the wheels to
set them into rotation
From Newton’s Second Law, the driving force
is proportional to the acceleration
Therefore, there is a close relationship
between the power rating of a vehicle and its
possible acceleration
48
Horsepower and Acceleration
49