14ap physics c - work and energy
TRANSCRIPT
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Work, Energy, and Power
AP Physics C
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There are many different TYPES of
Energy.
Energy is expressed
in JOULES (J)
4.19 J = 1 calorie
Energy can beexpressed more
specifically by using
the term WORK(W)
Work =The Scalar Dot Product between Force and Displacement.So that means if you apply a force on an object and it covers a
displacement you have supplied ENERGY or done WORK on that
object.
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Scalar Dot Product?
A product is obviously a result of
multiplying 2 numbers. A
scalar is a quantity with NO
DIRECTION. So basically
Work is found by multiplying
the Force times the
displacement and result is
ENERGY, which has no
direction associated with it.
nt vectordisplaceme
cos
r
rFrFW
A dot product is basically a CONSTRAINT
on the formula. In this case it means that
F and x MUST be parallel. To ensure that
they are parallel we add the cosine on the
end.
W = Fx
Area = Base x Height
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WorkThe VERTICAL component of the force DOES NOT
cause the block to move the right. The energy imparted to
the box is evident by its motion to the right. Therefore
ONLY the HORIZONTAL COMPONENT of the force
actually creates energy or WORK.
When the FORCE and DISPLACEMENT are in the SAME
DIRECTION you get a POSITIVE WORK VALUE. The
ANGLE between the force and displacement is ZERO
degrees. What happens when you put this in for the
COSINE?
When the FORCE and DISPLACEMENT are in the
OPPOSITE direction, yet still on the same axis, you get a
NEGATIVE WORK VALUE. This negative doesn't mean
the direction!!!! IT simply means that the force and
displacement oppose each other. The ANGLE between the
force and displacement in this case is 180 degrees. What
happens when you put this in for the COSINE?
When the FORCE and DISPLACEMENT are
PERPENDICULAR, you get NO WORK!!! The ANGLE
between the force and displacement in this case is 90
degrees. What happens when you put this in for the
COSINE?
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ExamplecosrFW
rFW
A box of mass m = 2.0 kg is moving over a
frictional floor ( uk = 0.3) has a force whose
magnitude is F = 25 N applied to it at anangle of 30 degrees, as shown to the left.
The box is observed to move 16 meters in
the horizontal direction before falling off thetable.
a) How much work does F do before takingthe plunge?
JW
NmW
W
rFW
rFW
4.346
4.346
30cos1625
cos
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Example contWhat if we had done this in UNIT VECTOR notation?
JWNmW
W
rFrFW
jiF
yyxx
4.3464.346
)05.12()1665.21(
)()(
5.1265.21
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Example cont
JW
FW
rFW
rFW
N
0
90cos16
cos
Fn
How much work does the
FORCE NORMAL do and
Why?
There is NO WORK since
F and r are perpendicular.
Ff
How much work does the frictional force do?
W
W
rFmgW
rFW
rFW
rFW
N
f
f
180cos16)30cos25)8.9(2(3.0
cos)cos(
cos
cos
-34.08 J
Note:This negative does not
specify a direction in this case
since WORK is a SCALAR. It
simply means that the force is
involved in slowing the object
down.
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What if the FORCE IS NOT CONSTANT?
maFNet
dxdt
dvmdxamW
dxmadxFW
)()(
)(
The function here MUST be a FORCE
function with respect to x or r. Lets lookat a POPULAR force function.
Is this function, with respect to x ? NO!
You can still integrate the function, it simply needs to be modified so that it fits
the model accordingly.
v
vo
dvvmW
dvvmdv
dt
dxmW )(
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Work-Energy Theorem
22
)22()|2|(
)(
22
222
o
ov
v
v
v
mvmvW
vvm
vmW
dvvmW
dvvmdvdtdxmW
o
o
KW
mvK
2
21
EnergyKinetic
Kinetic energy is the ENERGY of MOTION.
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Example W=FrcosA 70 kg base-runner begins to slide into second base when moving
at a speed of 4.0 m/s. The coefficient of kinetic friction betweenhis clothes and the earth is 0.70. He slides so that his speed iszero just as he reaches the base (a) How much energy is lostdue to friction acting on the runner? (b) How far does he slide?
)8.9)(70)(70.0(
mgFF nf
= 480.2 N
f
of
f
W
mvW
KWa
22 )4)(70(2
12
10
)
-560 J
x
r
rFW ff
)180(cos)2.480(560
cos
1.17 m
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Another varying force example..
A ball hangs from a rope attached
to a ceiling as shown. A variableforce F is applied to the ball sothat:
F is always horizontal
Fs magnitude varies so that the ball
moves up the arc at a constant
speed.The balls velocity is very low
Assuming the balls mass is m, how much work does F do as it moves from = 0 to = 1?
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Example Cont
tansin)cos
(
sincos
mgmgF
FTmgT
T
mg
Tcos
Tsin
dr
dyor
dx
dy
drmgdrFW
tan
)tan(
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Example Cont
o
o
y
y
y
y
mgymgyW
yymgymgW
dymgdymgW
drdr
dymgdrmgW
o
o
)(||
)(tan
UWmghmgyU
EnergyPotential
The energy of POSITION or
STORED ENERGY is called
Potential Energy!
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Something is missing.
UW
ymgW
yymgW
FrrFW
gravity
gravity
gravity
gravity
180cos)(
cos
21
Consider a mass m that moves from position 1 ( y1)to position 2 m,(y2), moving with a constant velocity.
How much work does gravity do on the body as itexecutes the motion?
Suppose the mass was thrown UPWARD.
How much work does gravity do on thebody as it executes the motion?
UW
ymgW
yymgW
FrrFW
gravity
gravity
gravity
gravity
0cos)(
cos
12
In both cases, the negative
sign is supplied
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The bottom line..The amount of Work gravity does on a body is
PATH INDEPENDANT. Force fields that actthis way are CONSERVATIVE FORCES
FIELDS. If the above is true, the amount of
work done on a body that moves around a
CLOSED PATH in the field will always be
ZERO
FRICTION is a non conservative force. By NON-CONSERVATIVE
we mean it DEPENDS on the PATH. If a body slides up, and then
back down an incline the total work done by friction is NOT ZERO.
When the direction of motion reverses, so does the force and
friction will do NEGATIVE WORK in BOTH directions.
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Force can be found using the DERIVATIVE
dx
dUF
UFxW
Since work is equal to the NEGATIVEchange in potential energy, the
FORCE of an object is the derivative
of the potential energy with respect
to displacement. Be very careful
handling the negative sign.
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Energy is CONSERVED!
afterbefore
oo
oo
oo
EnergyEnergyUKUK
UUKK
UUKKUKW
)(
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ExampleA 2.0 m pendulum is released from rest when the
support string is at an angle of 25 degrees with thevertical. What is the speed of the bob at the bottomof the string?
L
Lcos
h
h = L Lcos
h = 2-2cos
h = 0.187 m
EB = EA
UO = Kmgho = 1/2mv
2
gho = 1/2v2
1.83 = v2
1.35 m/s = v
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How to we measure energy?One of the things we do everyday is measure how
much energy we use. The rate at which we use itdetermines the amount we pay to our utilitycompany. Since WORK is energy the rate at whichwork is done is referred to as POWER.
The unit is either Joules per second or
commonly called the WATT.
To the left are several various versions of
this formula, including some various
Calculus variations.