6 work and kinetic energy work done by a constant force work done by a variable force – straight...
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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.
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
Work Transformation
• Work is a usage of energy, e.g.,
• Burning gasoline produces heat & motion
• battery running a car
• /
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.
Work by Constant Force
xFxFW x cos
Work = Fcosx = (80N)(cos40)(11m) = 674 J
Ex: F = 80N, Angle is 40°, x is 11m,
Total Work on Object
xFxFxFW xnetxxtotal ,21
xxnet maF , x
vva ifx
2
22
xmaW xtotal
2212
21
22
2 ifif
total mvmvxx
vvmW
Kinetic Energy
Energy Kinetic of Definition
221 mvKE
TheoremEnergy -Kinetic -Work
KEWtotal
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
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
Ex. Block pushed 3m with 75N of force while Friction of 50N. Total Work is,
JmN
mNN
xFW xnettotal
75)3)(25(
)3)(5075(,
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
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
Scalar (Dot) Product
zzyyxx BABABABA
ProductScalar - Definition
cosABBA
)ˆˆˆ()ˆˆˆ( kBjBiBkAjAiABA zyxzyx
WorklIncrementa of Definition
)(cos
dFdFdW
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
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)
FFsFW 4)0,3,4)(0,0,(
Power
J/s tt Power wa of Definition
dt
dWP
vFdt
dF
dt
dWP
watt746 lb/sft 550 hp 1
Ex: A car drives at 20m/s and experiences air-drag of 400N.
wattsmNvFP 8000)/20)(400(
hpwatt
hpwatt10
746
1
1
8000
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
Work along Curved Path
2
1
2
1dtvFdFW netnettotal
2
1
2
1
2
1vdvmdtv
dt
vdmdtvam
12
2
1
221 | KEKEmvWtotal
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
• /