Transcript Work - HCC Learning Web

Chapter 5 Energy
Ying Yi PhD
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Forms of Energy
 Mechanical
 Focus for now
 May be kinetic (associated with motion) or potential
(associated with position)
 Chemical
 Electromagnetic
 Nuclear
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Energy Conservation Assumption
 The total amount of energy in the Universe never
changes. Energy can be transformed from one
form to another.
 Essential to the study of physics, chemistry, biology,
geology, astronomy
 Can be used in place of Newton’s laws to solve
certain problems more simply
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Outline
 Work
 Work-Energy Theorem
 Potential Energy
 Gravitational Potential Energy
 Spring Potential Energy
 Power
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Work (Link between force and energy)
 Definition: The work, W, done by a constant force on
an object is defined as the product of the component
of the force along the direction of displacement and
the magnitude of the displacement
 Units: Newton • meter = Joule
 N•m=J
 J = kg • m2 / s2
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Work (Same direction)
 W = F x
 This equation applies
when the force is in the
same direction as the
displacement
 F and x are in the
same direction
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Work (Different Direction)
 W = (F cos )x
 F is the magnitude of
the force
 Δ x is the magnitude
of the object’s
displacement
 q is the angle between
F and x
Question: Does work have anything to do with velocity and
acceleration? Is work a scalar or vector?
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Work (Multiple forces)
If there are multiple forces acting on an object, the total
work done is the algebraic sum of the amount of work
done by each force
Wnet  Fnet cos x  W1  W2  ......
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False or True Questions
 When the force is perpendicular to the displacement
cos 90° = 0, the work done by a force is zero.
True
 Work can be positive and negative.
True
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When Work is Zero?
 Displacement is
horizontal
 Force is vertical
 cos 90° = 0
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Work Can Be Positive or Negative
 Work is positive when
lifting the box
 Work would be
negative if lowering
the box
 The force would still be
upward, but the
displacement would be
downward
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Work and Dissipative Forces
 Work can be done by friction
 The energy lost to friction by an object goes into
heating both the object and its environment
 Some energy may be converted into sound
 For now, the phrase “Work done by friction” will
denote the effect of the friction processes on
mechanical energy alone
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Example 5.1 Sledding
An Eskimo returning from a successful fishing trip pulls a sled
loaded with salmon. The total mass of the sled and salmon is
50.0 kg, and the Eskimo exerts a force of magnitude
1.20×102N on the sled by pulling on the rope.
(a) How much work does he do on the sled if the rope is
horizontal to the ground (Ɵ=0° in Fig. 5.6) and he pulls the
sled 5.00 m?
(b) How much work does he do on the sled if Ɵ=30.0° and he
pulls the sled the same distance? (Treat the sled as a point
particle, so details such as the point of attachment of the rope
make no difference.)
(c) At a coordinate position of 12.4 m, the Eskimo lets up on
the applied force. A friction force of 45.0 N between the ice and
the
sled
PHYSbrings
I @ HCC the sled to rest at a coordinate position of 18.2
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m. How much work does friction do on the sled?
Work (Varying Force)
 The work done by a
variable force acting
on an object that
undergoes a
displacement is equal
to the area under the
graph of Fx versus x
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Spring Example
 Spring is slowly
stretched from 0 to
xmax
 Fapplied = -Fs = kx
 W = ½kx²
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Example 5.15 Work required to stretch
a Spring
One end of a horizontal spring (k=80.0 N/m) is held
fixed while an external force is applied to the free end,
stretching it slowly from xA=0 to xB=4.00 cm. (a) find
the work done by the applied force on the spring. (b)
Find the additional work done in stretching the spring
from xB=4.00 cm to xC=7.00 cm.
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Work & Kinetic Energy
Wnet  Fnet x  (ma)x
v  v0
a
2x
2
Recall motion equation:
Wnet
v 2  v0 2
1 2 1
 (ma)x  m
x  mv  mv0 2
2x
2
2
Work Energy Theory
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Kinetic Energy
 Energy associated with the motion of an object

KE 
1
mv 2
2
 Scalar quantity with the same units as work
 Work is related to kinetic energy
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Work-Kinetic Energy Theorem
 When work is done by a net force on an object
and the only change in the object is its speed, the
work done is equal to the change in the object’s
kinetic energy
 Wnet  KEf  KEi  KE
 Speed will increase if work is positive
 Speed will decrease if work is negative
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Example 5.3 Collision Analysis
The driver of a 1.00×103 kg car traveling on the
interstate at 35.0 m/s (nearly 80.0 mph) slams on his
brakes to avoid hitting a second vehicle in front of him,
which had come to rest because of congestion ahead
(Fig. 5.9). After the brakes are applied, a constant
kinetic friction force of magnitude 8.00×103N acts on
the car. Ignore air resistance. (a) At what minimum
distance should the brakes be applied to avoid a collision
with the other vehicle? (b) If the distance between the
vehicles is initially only 30.0 m, at what speed would the
collision occurs?
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Types of Forces
 There are two general kinds of forces
 Conservative
 Work and energy associated with the force can be recovered
 Nonconservative
 The forces are generally dissipative and work done against it
cannot easily be recovered
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Conservative Forces
 A force is conservative if the work it does on an object
moving between two points is independent of the path
the objects take between the points
 Examples of conservative forces include:
 Gravity
 Spring force
 Electromagnetic forces
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Nonconservative Forces
 A force is nonconservative if the work it does on an object
depends on the path taken by the object between its final
and starting points.
 Examples of nonconservative forces
 Kinetic friction, air drag, propulsive forces
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Nonconservative Force (Friction)
 The blue path is
shorter than the red
path
 The work required is
less on the blue path
than on the red path
 Friction depends on
the path and so is a
non-conservative force
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Work done by conservative force
Wg  F y  mg ( y f  yi )
Wnet  Wnc  Wg  KE
Wnet  Wnc  mg ( y f  yi )  KE
Wnc  KE  mgy f  mgyi
Gravitational potential energy
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Potential Energy
 Potential energy is associated with the position of the
object within some system
 Potential energy is a property of the system, not the
object
 A system is a collection of objects interacting via forces
or processes that are internal to the system
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Work and Potential Energy
 For every conservative force a potential energy
function can be found
 Evaluating the difference of the function at any two
points in an object’s path gives the negative of the
work done by the force between those two points
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Gravitational Potential Energy
 Gravitational Potential Energy is the energy
associated with the relative position of an object in
space near the Earth’s surface
 Objects interact with the earth through the
gravitational force
 Actually the potential energy is for the earth-object
system
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Work and Gravitational Potential
Energy
 PE = mgy
 Wgravity  PEi  PE f
 Units of Potential
Energy are the same
as those of Work and
Kinetic Energy
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Work and Gravitational Potential
Energy
 PE = mgy
 Wgrav ity  PEi  PEf
 Units of Potential
Energy are the same
as those of Work and
Kinetic Energy
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Work-Energy Theorem, Extended
 The work-energy theorem can be extended to
include potential energy:
Wnc  (KEf  KEi )  (PEf  PEi )
 If other conservative forces are present, potential
energy functions can be developed for them and
their change in that potential energy added to the
right side of the equation
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Reference Levels for Gravitational Potential Energy
 Often the Earth’s surface
 May be some other point suggested by the problem
 Once the position is chosen, it must remain fixed for
the entire problem
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Reference Levels, cont
 At location A, the desk
may be the convenient
reference level
 At location B, the floor
could be used
 At location C, the ground
would be the most logical
reference level
 The choice is arbitrary,
though
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Example 5.4 Skis
A 60.0 kg skier is at the top of a slope, as shown in
Figure 5.14, At the initial point A, she is 10.0 m
vertically above point B. (a) Setting the zero level for
gravitational potential energy at B, find the
gravitational potential energy of this system when the
skier is at A and then at B. Finally, find the change in
potential energy of the skier-Earth system at the skier
goes from point A to point B. (b) Repeat this problem
with the zero level at point A. (c) Repeat again, with the
zero level 2.00 m higher than point B.
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Conservation of Mechanical Energy
 Conservation in general
 To say a physical quantity is conserved is to say that the
numerical value of the quantity remains constant
throughout any physical process although the quantities
may change form
 In Conservation of Energy, the total mechanical
energy remains constant
 In any isolated system of objects interacting only through
conservative forces, the total mechanical energy of the system
remains constant.
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Conservation of Energy, cont.
 Total mechanical energy is the sum of the kinetic and
potential energies in the system
Ei  E f
KEi  PEi  KE f  PE f
 Other types of potential energy functions can be added
to modify this equation
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Problem Solving with Conservation of Energy
 Define the system
 Verify that only conservative forces are present
 Select the location of zero gravitational potential energy,
where y = 0
 Do not change this location while solving the problem
 Identify two points the object of interest moves between
 One point should be where information is given
 The other point should be where you want to find out something
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Problem Solving, cont
 Apply the conservation of energy equation to the
system
 Immediately substitute zero values, then do the algebra
before substituting the other values
 Solve for the unknown
 Typically a speed or a position
 Substitute known values
 Calculate result
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Work-Energy With Nonconservative Forces
 If nonconservative forces are present, then the full
Work-Energy Theorem must be used instead of the
equation for Conservation of Energy
 Often techniques from previous chapters will need to
be employed
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Group Problem
Waterslides are nearly frictionless, hence can provide
bored students with high-speed thrills (Fig. 5.18). One
such slide, Der Stuka, named for the terrifying German
dive bombers of World War II, is 72.0 feet high (21.9
m), found at Six Flags in Dallas, Texas. (a) Determine
the speed of a 60.0 kg woman at the bottom of such a
slide, assuming no friction is present. (b) If the woman
is clocked at 18.0 m/s at the bottom of the slide, find
the work done on the woman by friction.
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Potential Energy Stored in a Spring
 Involves the spring constant, k
 Hooke’s Law gives the force
F=-kx
 F is the restoring force
 F is in the opposite direction of x
 k depends on how the spring was formed, the material it is made
from, thickness of the wire, etc.
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Potential Energy in a Spring
 Elastic Potential Energy
 Related to the work required to compress a spring from
its equilibrium position to some final, arbitrary, position
x

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PE s 
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1 2
kx
2
Spring Potential Energy, Example
 A) The spring is in
equilibrium, neither
stretched or compressed
 B) The spring is
compressed, storing
potential energy
 C) The block is released
and the potential energy is
transformed to kinetic
energy of the block
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Work-Energy Theorem Including a Spring
 Wnc = (KEf – KEi) + (PEgf – PEgi) + (PEsf – PEsi)
 PEg is the gravitational potential energy
 PEs is the elastic potential energy associated with a
spring
 PE will now be used to denote the total potential energy
of the system
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Conservation of Energy Including a Spring
 The PE of the spring is added to both sides of the
conservation of energy equation

(KE  PE g  PE s )i  (KE  PE g  PE s )f
 The same problem-solving strategies apply
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Example5.9 Spring
A block with mass of 5.00 kg is attached to a horizontal
spring with spring constant k=4.00×102N/m, as in
Figure 5.21. The surface the block rests upon is
frictionless. If the block is pulled out to xi=0.0500 m
and released , (a) find the speed of the block when it
first reaches the equilibrium point, (b) find the speed
when x=0.0250 m, and (c) repeat part (a) if friction acts
on the block, with coefficient µk=0.150.
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Nonconservative Forces with Energy
Considerations
 When nonconservative forces are present, the
total mechanical energy of the system is not
constant
 The work done by all nonconservative forces
acting on parts of a system equals the change in
the mechanical energy of the system

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Wnc  Energy
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Nonconservative Forces and Energy
 In equation form:
Wnc   KEf  KEi   (PEi  PEf ) or
Wnc  (KEf  PEf )  (KEi  PEi )
 The energy can either cross a boundary or the
energy is transformed into a form of nonmechanical energy such as thermal energy
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Transferring Energy
 By Work
 By applying a force
 Produces a displacement of the system
 Heat
 The process of transferring heat by collisions between
atoms or molecules
 For example, when a spoon rests in a cup of coffee, the
spoon becomes hot because some of the KE of the
molecules in the coffee is transferred to the molecules of
the spoon as internal energy
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Transferring Energy
 Mechanical Waves
 A disturbance propagates through a medium
 Examples include sound, water, seismic
 Electrical transmission
 Transfer by means of electrical current
 This is how energy enters any electrical device
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Transferring Energy
 Electromagnetic radiation
 Any form of electromagnetic waves
 Light, microwaves, radio waves
 For example
 Cooking something in your microwave oven
 Light energy traveling from the Sun to the Earth
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Notes About Conservation of Energy
 We can neither create nor destroy energy
 Another way of saying energy is conserved
 If the total energy of the system does not remain
constant, the energy must have crossed the boundary by
some mechanism
 Applies to areas other than physics
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Power
 Often also interested in the rate at which the
energy transfer takes place
 Power is defined as this rate of energy transfer

W

 Fv
t
 SI units are Watts (W)

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J kg m2
W  
s
s2
Power, cont.
 US Customary units are generally hp
 Need a conversion factor
ft lb
1 hp  550
 746 W
s
 Can define units of work or energy in terms of units of
power:
 kilowatt hours (kWh) are often used in electric bills
 This is a unit of energy, not power
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Example 5.12 Power
A 1.00×103 kg elevator car carries a maximum load of
8.00×102 kg. A constant frictional force of 4.00×103 N
retards its motion upward, as in Figure 5.26. What
minimum power, in kiloatts and in horsepower, must the
motor deliver to lift the fully loaded elevator car at a
constant speed of 3.00 m/s?
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