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afs p53f09 L13
Chapter 6Work and Energy
afs p53f09 L13
Goals for Chapter 6
• Study work as defined in physics • Relate work to kinetic energy• Consider work done by a variable force• Study potential energy• Understand energy conservation• Include time and relationship of work to power
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Introduction• In previous chapters we studied motion
– Sometimes force and motion are not enough to solve a problem
– We introduce work and energy as the next step
• One of the most important concepts in physics
– Alternative approach to mechanics
• Many applications beyond mechanics
• Very useful tools
– You will learn new (sometimes much easier) ways to solve problems
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Work, Energy, Power
Chapter 6 in a nutshell
Work is Force times ‘Distance’.The change in Kinetic Energy is equal to the work.
Power is Work per unit time.
TWP
KW
dFW
net
/
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sFW J joule 1 mN 1
6.1 Work done by a constant force
Force distance
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Work is force times distance…but!
Only the force component in the direction of motion counts!
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scosFW
1180cos
090cos
10cos
6.1 Work done by a constant force
SI Unit of work: Joule
Component of the force along the displacement is used in defining work
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Typical Values of Work
10–20Breaking a bond in DNA
10–7Hop of a flea
10–3Turning page of a book
0.5Heartbeat
6000Lighting a 100-W bulb for 1 minute
107
104
Human food intake/day
Melting an ice cube
108Burning one gallon of gas
1018Mt. St. Helens eruption
8 x 1019Annual U.S. energy use
Equivalent work (J)Activity
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Find the work done if the force is 45.0-N, the angle is 50.0 degrees, and the displacement is 75.0 m.
J 2170
Example: Pulling a suitcase-on-wheels
scosFW m 0.750.50cosN 0.45
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Fss0cosFW o
Example: Bench-Pressing
An athlete is bench-pressing a barbell whose weight is 710 N. First, he raises the barbell a distance of 0.65 m above his chest, and then he lowers it the same distance. The weight is raised and lowered at constant velocity. Determine the work done on the barbell.
Lifting phase
m65.00cosN710 o J460
Fss180cosFW o Lowering phase
m65.0180cosN710 o J460Work is negative as force is opposite to displacment
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The truck is accelerating at a rate of +1.50 m/s2. The mass of the crate is 120-kg and it does not slip. The magnitude of the displacement is 65 m.
What is the total work done on the crate by all of the forces acting on it?
Example: Accelerating a crate
The angle between the displacementand the normal force is 90 degrees.
0s90cosFW
The angle between the displacement and the weight is also 90 degrees.
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The angle between the displacementand the friction force is 0 degrees.
m 650cosN180W
mafs 2sm5.1kg 120
J102.1 4
total work is done by frictional force here
N180
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Steve pushes his stalled car 19 m to clear the intersection. If he pushes with a constant force with magnitude 210 N (about 47 lb), how much work does he do on the car (a) if he pushes in the direction the car is heading, (b) if he pushes at 30o to that direction ?
Example: Pushing a stalled car
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Consider a constant net external force acting on an object.
The work is simply
s
F
smasFW
6.2 The Work-Energy Theorem and Kinetic Energy
The object is displaced a distance s, in the same direction as the net force.
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asmW
as2vv 2o
2f 2
o2f2
1 vvas
DEFINITION OF KINETIC ENERGY
221 mvKE
The Work-Energy Theorem and Kinetic Energy
The kinetic energy KE of and object with mass mand speed v is given by
2o
2f2
1 vvmW 2o2
12f2
1 mvmv
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• Kinetic: Energy of motion.
– A car on the highway has kinetic energy.
– We have to remove this energy to stop it.
– The breaks of a car get HOT!
– This is an example of turning one form of energy into another (thermal energy).
2vm2
1K Kinetic Energy
Kinetic energy
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THE WORK-ENERGY THEOREM
2o2
12f2
1of mvmvKEKEW
The Work-Energy Theorem and Kinetic Energy
When a net external force does work on and object, the kinetic energy of the object changes according to
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The mass of the space probe is 474-kg and its initial velocity is 275 m/s. If the 56.0-mN force acts on the probe through a displacement of 2.42×109m, what is its final speed?
Example: Deep Space 1
2o2
12f2
1 mvmvW
scosFW
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m1042.20cosN105.60 9-2
2o2
12f2
1 mvmvscosF
sm805vf
2212
f21 sm275kg 474vkg 474
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The previous problem can be solved using kinematic equations.
First using Newton’s second law, F=ma to find acceleration.
a = F/m= 0.056/474 = 1.18x10-4 m/s2
Vo = 275 m/s, and d = 2.42x109 m.
So
smV
V
daVV
f
f
of
/804
1042.21018.12)275(
2
942
22
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Summary
Kinetic Energy
Work
Work - (Kinetic) Energy Theorem
KKKW iftotal afs p53f09 L13
• Provides a link between force and energy
• 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
x)cosF(W
Work done by a constant force
(F cos ) is the component of the force in the direction of the displacement
x is the displacement
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FTot
rF1
F2
Suppose FTot = F1 + F2
and the displacement is r.
W1 = (F1 cos1 ) r
W2 = (F2 cos2 ) r
WTOT = (FTot cos) r
WTot = W1 + W2
Work done by multiple forces
It’s the total force that matters!!
The work done by each force is: FTot
rF1
F2
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– Speed will increase if work is positive
– Speed will decrease if work is negative
KEKEKEW iftotal
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
2i
2ftotal mv
2
1mv
2
1W
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Example: Downhill SkiingA 50-kg skier is coasting down a 25o slope. Near the top of the slope, her speed is 3.6 m/s. She accelerates down the slope because of the gravitational force, even though a kinetic friction force of magnitude 71 N opposes her motion. Ignoring air resistance, determine the speed at a point that is displaced 57 m downhill.
iftotal KEKEW
2i2
12f2
1 mvWmv
2vm2
1KE
- solve for vf
- need W
scosFW afs p53f09 L13
In this case the net force is
kf25sinmgF
scosFW - need F
N7125sins
m80.9kg50F
2 N170
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2i2
12f2
1 mvWmv putting it all together
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• Gravitational Potential Energyis the energy associated with the relative position of an object in space near the Earth’s surface
6.3 Gravitational Potential Energy
– Objects interact with the earth through the gravitational force
– The potential energy of the earth-object system
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• Consider block of mass m at initial height yi
• Work done by the gravitational force
PEPEPEW figravity
s)cosmg(scosFWgrav
This quantity is called potential energy:
mgyPE
Important: work is related to the difference in PE’s!
Work and Gravitational Potential Energy
,1cos,yys:but fi figrav yymgW:Thus
fi mgymgy
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The gymnast leaves the trampoline at an initial height of 1.20 mand reaches a maximum height of 4.80 m before falling back down. What was the initial speed of the gymnast?
Example: A Gymnast on a trampoline
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2o2
12f2
1 mvmvW fogravity hhmgW
2o2
1fo mvhhmg
foo hhg2v
m 80.4m 20.1sm80.92v 2o sm40.8
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We can solve this problem using kinematics
)(2
)(2
)()(20
)(2
2
2
22
foo
foo
ofo
ofof
hhgV
hhgV
hhgV
hhaVV
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• 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 or
particles interacting via forces or processes that are internal to the system
• Units of Potential Energy are the same as those of Work and Kinetic Energy [J]
Potential Energy
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fogravity mghmghW
DEFINITION OF GRAVITATIONAL POTENTIAL ENERGY
The gravitational potential energy PE is the energy that anobject of mass m has by virtue of its position relative to thesurface of the earth. That position is measured by the height h of the object relative to an arbitrary zero level:
mghPE
J joule 1 mN 1
Gravitational Potential Energy
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Version 1 A force is conservative when the work it doeson a moving object is independent of the path between the object’s initial and final positions.
fogravity hhmgW
6.4 Conservative versus Nonconservative Forces
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Conservative Forces - Independence from Path
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Version 2 A force is conservative when it does no work on an object moving around a closed path, starting andfinishing at the same point.
fo hh fogravity hhmgW
Conservative versus Nonconservative Forces
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Conservative versus Nonconservative Forces
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An example of a nonconservative force is the kinetic frictional force fk.
sfs180cosfscosFW kk
The work done by the kinetic frictional force is always negative.Thus, it is impossible for the work it does on an object that moves around a closed path to be zero.
The concept of potential energy is not defined for a nonconservative force.
Conservative versus Nonconservative Forces
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• 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 nonconservative force
Friction depends on the path - nonconservative force
• The friction force transforms kinetic energy of the object into a type of energy associated with temperature (heat)
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In normal situations both conservative and non-conservative forces act simultaneously on an object, so the work done by the net external force can be written as
ncc WWW
KEKEKEW of
PEPEPEmghmghWW fofogravityc
Conservative versus Nonconservative Forces
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ncc WWW
ncWPE
THE WORK-ENERGY THEOREM
PEKEWnc
Conservative versus Nonconservative Forces
KE
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PEKEWnc
ooff PEKEPEKE ncW
ofnc EEW If the net work on an object by nonconservative forcesis zero, then its energy does not change:
of EE
6.5 The conservation of mechanical energy
ofof PEPEKEKE
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THE PRINCIPLE OF CONSERVATION OF MECHANICAL ENERGY
The total mechanical energy (E = KE + PE) of an object remains constant as the object moves, provided that the net work done by external non-conservative forces is zero.
The conservation of mechanical energy
of EE
ooff EPEKPEKE
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The conservation of mechanical energy
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A motorcyclist is trying to leap across the canyon by driving horizontally off a cliff with v=38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side.
Example: A daredevil motorcyclist
‘projectile motion’, but use conservation of mechanical energy
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of EE 2o2
1o
2f2
1f mvmghmvmgh
2o2
1o
2f2
1f vghvgh
2ofof vhhg2v
22f sm0.38m0.35sm8.92v sm2.46
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Example 10: The Kingda Ka roller coaster
The Kingda Ka is a giant roller coaster. The ride includes a vertical drop of 127 m. Suppose the coaster has a speed of 6.0 m/s at the top of the drop. Neglect friction and air resistance and find the speed of the riders at the bottom.
Conservation of mechanical energy
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ncc WWW
ncWPE THE WORK-ENERGY THEOREM
PEKEWnc
KE
KEKEKEW if
iiff EPEKPEKE If Wnc = 0 CONSERVATION OF MECHANICAL ENERGY
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• Define the system
• Determine whether or not nonconservative forces are present
• If only conservative forces are present, apply conservation of energy and solve for the unknown
• Select the location of zero gravitational potential energy
– Do not change this location while solving the problem
Problem Solving with Work-Energy theorem
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Fireworks: m = 0.20 kg. The nonconservative force generated by the burning propellant does 425 J of work, the rocket is 29 m above its starting point. What is the final speed of the rocket. Ignore air resistance.
2f2
1fnc mvmghW
Example 12: Fireworks -- Work-energy theorem
WORK-ENERGY THEOREM
ofnc EEW
2o2
1o mvmgh
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2o2
12f2
1ofnc mvmvmghmghW
2
f21
2
vkg 20.0
m 0.29sm80.9kg 20.0
2f2
1ofnc mvhhmgW
sm61vf
J 425
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A location where the gravitational potential energy is zero must be chosen for each problem
Gravitational Potential Energy
figravity PEPEW
Gravitational potential energy:mgyPE Important: work done by gravity is related to the difference in gravitational PE’s!
The choice is arbitrary since the change in the potential energy gives the work done
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V1
F = - mg
V2
y = -h
PEyFW KW
0v1
Example: Falling Raindrops
Mass M dropped from height H. What is speed just before hitting the ground? (Neglect friction of air)
ghv 22
222
1mvmgh
)()( hmg
mgh21
22 2
1
2
1mvmv
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ghv 22
Initial Final
Independence from Path (Conservative Forces)
Find speed at bottom of the slide
mgh0
ghv 2
02
1 2 mv
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Rough spot, coefficient of friction .
MV0
V1
ML
A block of mass M is moving initially with speed V0. It passes over a rough patch of table of length L. Given the mass M, the initial speed V0, and the coefficient of friction, find the final speed of the block.
1. Draw the picture. 2. What is given? 3. What are the relationships?
Non-conservative work: MgLLFWNC Work-energy: KWNC
Example: Slowing down with friction force
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Rough spot, coefficient of friction .
MV0
V1
ML
MgLLFWNC
KWNC
Example: Slowing down with friction force
gL2VV 201
20
21 V
2
MV
2
MK
20
21 V
2
MV
2
MMgL
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• Often also interested in the rate at which the energy transfer (work done) takes place
• Power is defined as this rate of energy transfer (work done)
2
2
s
mkg
s
JoulesWatts
– SI units are Watts
Average Power
horse power 1 hp = 746 W = 0.746 kW
Power
t
W
Time
WorkP
vFP
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Power
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Example: A marathon stair climb
A marathon runner with mass 50.0 kg runs up the stairs to the top of the Sears Tower (443 m) in 15.0 minutes. What is her average power output in Watts ?
∆ ∆
. .
.
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Example: Power in a Jet Engine
A jet airplane engine develops a thrust (a forward force on the plane) of 15,000 N. When the plane is flying at 300 m / s (roughly 600 mi / h), what horsepower does the engine develop ?
P = F v
. . W
.
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Energy can neither be created nor destroyed, but can only be converted from one form to another.
PRINCIPLE OF CONSERVATION OF ENERGY
• True for any isolated system.
– i.e. when we put on the brakes, the kinetic energy of the car is turned into heat using friction in the brakes. The total energy of the “car-breaks-road-atmosphere” system is the same.
– The energy of the car “alone” is not conserved...
• It is reduced by the braking.
• Doing “work” on an isolated system will change its “energy”...
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• Energy is conserved• Kinetic energy describes motion and relates to the
mass of an object and its energy squared• Energy on earth originates from the sun (nuclear
fusion)• Chemical energy is released by metabolism• Energy can be stored as potential energy in an
objects height and mass and also through elastic deformation
• Energy can be dissipated as heat and noise
Elastic potential energy stored in a stretched rubber band
Overview of Energy
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Transformation of gravitational potential energy to kinetic energy
Energy and its transformation
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Energy can be dissipated as heat (molecular motions)
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ii x)cosF(W
Split total displacement (xf-xi)into many small displacements x
For each small displacement:
Thus, total work is:
i
ixi
itot xFWW
Graphical Representation of Work (variable force)
which is total area under the F(x) curve!
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Summary Conservationof mech. Energy
Work - Energy Theorem
Gravitational Potential Energy
KKKW iftotal
vFt
WP
Average Power
Conservative Forces:
PEU
UUW fininic
ffii UKUK
PEKEWnc
mgyUPE
