Download Chapter 6 Work and Energy

Chapter 6
Work and Energy
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0) Energy & conservation laws
• Newton’s laws (with kinematic equations) allow a
complete and continuous description of motion, but
requires
– knowledge of the force at all times
– detailed calculations which may be prohibitive
• Conservation laws allow intermediate details to be ignored
by realizing some quantities are conserved
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Descartes
Conservation of momentum: p = mv - discovered first (consequence of Newton’s laws) - mv was considered an essential quality of motion
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What kind of essential motion remains zero when
bodies are being hurtled through space?
Found mv2 conserved in collisions.
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1) Energy
1807: Associated “Energy” with mv2
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It is important to realize that in physics today, we
have no knowledge of what energy is
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As we cannot give a general definition of energy, the principle of the
conservation of energy simply signifies that there is something which remains
constant. Well, whatever new notions of the world future experiments may give
us, we know beforehand that there will be something which remains constant
and which we will be able to call energy. Henri Poincaré (1854 - 1912)
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• Energy is some quantity, which takes many forms,
associated with a system which we know
intuitively is conserved. Or else we could make a
perpetual motion machine. Feynman writes that
the conservation of energy is the statement that
perpetual motion is not possible.
• State of a system in relation to fundamental forces
• Force is the agent of change; energy is measure of
change
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2) Work
Work produces a change in the energy of a system by the application
of a force acting over a distance.
a) Work along a straight line: F || s
F
m
s
Work done by force F, is
W = Fs
Units: 1 N m = 1 joule = 1 J
If the displacement s is zero, no work is done.
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(b) Force and displacement not parallel
F
m
θ
s
The change in the system is the same as if a parallel force
Fs = F cos θ acted over a distance s, so the work done by F is
W = Fs s = Fs cos θ
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(c) Negative work
If a force acts opposite to the direction of
motion (like friction), it does negative work. It
takes energy out of the system.
f
v
θ
v=0
m
s
W = Fs cos θ = Fs cos180º = −Fs
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Example: A person pulls a block (mass 10 kg) up
an inclined plane at constant speed. The block
moves 1 m along the plane and the tension force
acting on the block is parallel to the plane. The
coefficient of friction is μk = 0.20. Find the work
done on the block by
(a) the tension force
(b) gravity
(c) the normal force
(d) the kinetic friction force (e) the net force
Wnet = Wg + WT + W f + WN
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A cable lifts a 1200-kg elevator at a constant
velocity for a distance of 35 m. What is the work
done by (a) the tension in the cable and (b) the
elevator's weight?
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3) Kinetic Energy
• Work produces a change in energy.
• Work on an otherwise free object produces
motion. • The energy associated with motion is called
kinetic energy, KE. • Work - energy theorem: If as a result of work on an
object (system) the only change is its resulting
motion:
W = ΔKE = KE f − KE i
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v0
v
F
m
s
Using
Define Kinetic Energy:
2nd law: F = ma
KE = mv
1
2
kinematics: v 2 = v02 + 2as
2
ΔKE = 12 mv 2 − 12 mv02
gives for an object starting from rest:
(v 2 − v02 ) 1
Fs = mas = m
= 2 mv 2
2
so that
W = ΔKE
Work = ∆KE
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W = ΔKE
• Work-energy theorem qualifications:
– Force is the net force on the object (system)
– No internal changes to object (system)
e. g. Lifting a rock produces no kinetic energy because there is
no net force on the rock (on average).
e.g. Compressing a spring produces no kinetic energy because
there are internal changes to the system
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A 3.00-kg model airplane has velocity
components of 5.00 m/s due east and 8.00 m/s due
north. What is the plane’s kinetic energy?
a) 134 J
b) 96 J
c) 38 J
d) 254 J
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A 0.075-kg arrow is fired horizontally. The
bowstring exerts an average force of 65 N on the
arrow over a distance of 0.90 m. With what speed
does the arrow leave the bow?
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C&J 6.21An asteroid is moving along a straight line. A force acts
along the displacement of the asteroid and slows it down. The
asteroid has a mass of 4.5 x 104 kg, and the force causes its speed to
change from 7100 to 5500 m/s.
(a) What is the work done by the force?
(b) If the asteroid slows down over a distance of 1.8E6 m,
determine the magnitude of the average force.
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4) Potential Energy
a) Work on a system
• Lifting an object does not
produce kinetic energy.
h
F = mg
m
• Rather, it changes the configuration of
the earth/object system.
• If the object is released, then it gains
kinetic energy.
• The system stores energy by virtue of
its configuration
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• Potential Energy is energy stored in the
configuration of interacting objects.
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(b) Conservative force
• Work done by a conservative force is reversible
• Work done by a conservative force is independent of path
• Work done by a conservative force in a closed loop is zero
– e.g. gravity, spring, electromagnetic
• all fundamental forces
• any force determined by the configuration of the system
e.g. Work done by gravity
m
m
mg
W = −mgh
m
mg
W = mgh
m
mg
mg
W = mg cos 90º = 0
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W = −mgh + 0 + mgh + 0 = 0
• Dissipative forces like friction and air resistance
are non-conservative. The work depends on the
path. • No potential energy can be associated with such
forces.
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C&J conceptual question 12 Air resistance is a nonconservative
force. It always opposes the motion of an object. An airplane flies
from New York to Atlanta and then returns to its point of departure.
The net work done by air resistance during this round trip
___________.
a)
is zero
b)
is positive
c)
is negative
d)
is negative for slow speeds and positive for high speeds.
e)
is positive for slow speeds and negative for high speeds.
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• If, as a result of work (Wext) on a system (against
conservative forces), the only change is its configuration:
Wext (against cons forces) = ΔPE = PE f - PE i
Work done by conservative forces in a system produces a negative
change in the potential energy.
Wc (by cons forces) = −ΔPE
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5) Gravitational potential energy (near earth)
Object lifted by an external force:
The change in the potential energy is
the work done (if there are no other
changes in the system):
h
F = mg
ΔPE = Wext = mgh
m
The position of zero PE is arbitrary;
only changes are interesting. Define PE
= 0 for h = 0. Then,
PE = mgh
Work done by gravity is -mgh, so
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Wg = −ΔPE
ΔPE = mgh f − mgh0
Wg = −ΔPE
Wc = −ΔPE
(quantitative definition)
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7) Conservation of Mechanical Energy
Mechanical energy: E = PE + KE
If all forces in an isolated system are conservative, mechanical energy is
constant
Work-energy: W = ΔKE
If all work is done by conservative forces: W = -ΔPE
Therefore, -ΔPE = ΔKE, or
E = PE + KE = constant
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8) External (or non-conservative) forces
Work-energy: W = ΔKE
Consider work done by conservative and non-conservative (or
external) forces: W = Wc + Wnc
Work done by conservative forces can be accounted for by the change in
potential energy: Wc = -ΔPE
Then
Wnc (or Wext ) = ΔPE + ΔKE
work not accounted for by potential energy changes
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Summary
Wnc (or Wext ) = ΔPE + ΔKE
Δ(mgh)
Fs cosθ
Δ( mv )
1
2
for forces not accounted for by PE
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October 2005 midterm exam
Q 6 A box bounces off a truck while it is at rest on
a bridge and falls into a river that is 160 m below.
As it falls, 40% of its energy is lost due to air
resistance. What is its speed at the moment it
strikes the water?
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Example 8 A Daredevil Motorcyclist
!
A motorcyclist is trying to leap across the canyon by driving
horizontally off a cliff 38.0 m/s. Ignoring air resistance, find
the speed with which the cycle strikes the ground on the other
side.
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A person slides down from rest a large frictionless spherical
surface. At what angle θ does the person leave the surface?
(When the person leaves the surface, the normal force is zero)
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6.84 A 63-kg skier coasts up a snow-covered hill that makes an
angle of 25° with the horizontal. The initial speed of the skier is 6.6
m/s. After coasting 1.9 m up the slope, the skier has a speed of 4.4
m/s.
(a) Find the work done by the kinetic frictional force that acts on
the skis.
(b) What is the magnitude of the kinetic frictional force?
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9) Power
• Time rate of change of work or energy
• Average power:
Wnet ΔE
P=
=
Δt
Δt
Units: 1 watt = 1 W = 1 J/s
(1 hp = 745.7 W = ~ 3/4 kW)
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C&J 6.67
The cheetah is one of the fastest-accelerating animals, because it can go from
rest to 27 m/s (about 60 mi/h) in 4.0 s. If its mass is 110 kg, determine the
average power developed by the cheetah during the acceleration phase of its
motion. Express your answer in watts.
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• Power and speed
W Fs
P=
=
Δt Δt
but
so
for constant force parallel to
motion
s
v=
Δt
P = Fv
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Oct 2005 midterm exam.
Q8 A 50 kg girl runs up a flight of stairs in a time of 12.5 s. The
stairs are 7.0 m long and make an angle of 27° above the horizontal.
What is the average power that she produced?
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10) Variable force
• For constant force parallel to motion
W = Fs
• Represents area of rectangle F by s
Force
F
s
displacement
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Force
Force
F2
F1
Approximate variable force by a
series of constant forces
Total work is sum of rectangular areas, which approximates the area
under the curve. As steps get smaller, the approximations approach
equality:
W = area under F vs s curve
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A basketball of mass 0.60 kg is dropped from rest from a
height of 1.05 m. It rebounds to a height of 0.57 m. (a) How
much mechanical energy was lost during the collision with
the floor? (b) A basketball player dribbles the ball from a
height of 1.05 m by exerting a constant downward force on it
for a distance of 0.080 m. In dribbling, the player
compensates for the mechanical energy lost during each
bounce. If the ball now returns to a height of 1.05 m, what is
the magnitude of the force?
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11) Conservation of Energy
• Energy can be neither created nor destroyed, but
can be converted from one form to another:
– mechanical
– heat
– chemical
– electrical
– nuclear
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