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6 Work and Kinetic Energy
• Work Done by a Constant Force
• Work Done by a Variable Force – Straight Line Motion
• The Scalar Product
• Work-Kinetic Energy Theorem – Curved Paths
• Hk: 27, 41, 49, 51.
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Energy: The work that a physical system is capable of doing in changing from its actual state to a specified reference state … (American Heritage Dictionary)
Energy: The capacity to do work. (Physics)
What is Work?
Some Definitions
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Work Transformation
• Work is a usage of energy, e.g.,
• Burning gasoline produces heat & motion
• battery running a car
• /
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Work
• Work is force x distance (N·m = joule), force parallel to motion (no work done by perpendicular component)
• It takes energy to do work.
• Less stored energy is available after productive work is done.
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Work by Constant Force
xFxFW x cos
Work = Fcosx = (80N)(cos40)(11m) = 674 J
Ex: F = 80N, Angle is 40°, x is 11m,
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Total Work on Object
xFxFxFW xnetxxtotal ,21
xxnet maF , x
vva ifx
2
22
xmaW xtotal
2212
21
22
2 ifif
total mvmvxx
vvmW
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Kinetic Energy
Energy Kinetic of Definition
221 mvKE
TheoremEnergy -Kinetic -Work
KEWtotal
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Example: 20kg moving at 5m/s. 250J of work (total) are done on it. What is its final speed?
2212
21
iftotal mvmvW 2
212
21 5)20()20(250 fv
251010250 2 fv25010250 2 fv
210500 fv 502 fv
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Negative Work (object slows down)
negative becan cos cos xFW
Ex. A car moves 10 meters while a braking force of 500 newtons acts.
Energy Kinetic of 500Jlost car
50010)180(cos500 JmNW
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Ex. Block pushed 3m with 75N of force while Friction of 50N. Total Work is,
JmN
mNN
xFW xnettotal
75)3)(25(
)3)(5075(,
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Work by a Variable Force, Straight Line Motion
with xchanges force where
2
1x
x xdxFW
1
0 212
212
21
1
0
221 01
xalong 1 to0 moving Ex.
|xxdxW
xFx
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Hooke’s Law
• Elastic restoring force proportional to deformation
• F = -kx k = elastic constant (N/m)
• Ex. Lab springs, k = 8N/m, 0.1kg mass:
• mg = kx
• (0.1kg)(9.8N/kg) = 8N/m(x)
• x = 0.98N/(8N/m) = 0.1225 m
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Scalar (Dot) Product
zzyyxx BABABABA
ProductScalar - Definition
cosABBA
)ˆˆˆ()ˆˆˆ( kBjBiBkAjAiABA zyxzyx
WorklIncrementa of Definition
)(cos
dFdFdW
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5)0)(0()0)(1()5)(1( BA
Ex: A = (1, 1, 0), B = (5, 0, 0)
552
45cos52
cos
22
ABBA
2011 222 A5005 222 B
45
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5)0)(1()0)(1()5)(1( BA
cosABBA 3111 222222 zyx AAAA
5005 222222 zyx BBBB
cos535
7.54
3
1
53
5cos
Example: Find the angle between A = (1, 1, 1) and B = (5, 0, 0)
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FFsFW 4)0,3,4)(0,0,(
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Power
J/s tt Power wa of Definition
dt
dWP
vFdt
dF
dt
dWP
watt746 lb/sft 550 hp 1
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Ex: A car drives at 20m/s and experiences air-drag of 400N.
wattsmNvFP 8000)/20)(400(
hpwatt
hpwatt10
746
1
1
8000
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What size motor needed when Operating Speed is 10cm/s?
Cube of bricks ~ 1 ton
1 ton = 2000 lbs ~ 9000 N
Minimum Power:
P = Fv = (9000N)(0.1m/s)
P = 900 W = 1.2 hp
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Work along Curved Path
2
1
2
1dtvFdFW netnettotal
2
1
2
1
2
1vdvmdtv
dt
vdmdtvam
12
2
1
221 | KEKEmvWtotal
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Summary
• Work is force parallel to path x distance (force constant)
• Negative total work (object slows down)
• Work is integral of force·distance (Scalar Product)
• Power is rate work is done
• Total work = change in KE
• /