PHY 113 A Fall 2012 -- Lecture 12 19/26/2012
PHY 113 A General Physics I9-9:50 AM MWF Olin 101
Plan for Lecture 12:
Chapter 7 -- The notion of work
1. Kinetic energy and the Work-Kinetic energy theorem
2. Potential energy and work; conservative forces
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Note: Because of the special lecture, the tutorial this evening will be moved to Olin 104 after 6:30 PM.
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Back to work:F
dr
ri rj
rFr
rdW
f
i
fi
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A man must lift a refrigerator of weight mg to a height h to get it to the truck.
iclicker exercise:For which method does the man do more work:
A. Vertically lifting the refrigerator at constant speed to height h?
B. Moving the refrigerator up the ramp of length L at constant speed with h=L sin q.
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FPq
mg
n
fk
xi xf
Assume FP sinq <<mg
Work of gravity? 0
Work of FP? FP cos q (xf-xi)
Work of fk?-mkn (xf-xi)=-mk(mg- FP sin q) (xf-xi)
Multiple forces on block moving from xi to xf
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iclicker exercise:Which of the following statements about friction forces are true.
A. Friction forces always do positive work.B. Friction forces always do negative work.C. Friction forces can do either positive or
negative work.
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Why is work a useful concept?
Consider Newton’s second law:
Ftotal = m a Ftotal · dr= m a · dr
dtdtdmdt
dtd
dtdmd
dtdmdmd
f
i
f
i
f
i
f
i
f
i
total vvrvrvrarFr
r
r
r
r
r
r
r
r
r
Wtotal = ½ m vf2 - ½ m vi
2
Kinetic energy (joules)
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Introduction of the notion of Kinetic energySome more details:
Consider Newton’s second law:
Ftotal = m a Ftotal · dr= m a · dr
dtdtdmdt
dtd
dtdmd
dtdmdmd
f
i
f
i
f
i
f
i
f
i
t
t
t
ttotal vvrvrvrarF
r
r
r
r
r
r
Wtotal = ½ m vf2 - ½ m vi
2
Kinetic energy (joules)
22
21
21
21
if
f
i
t
tmvmvmddmdt
dtdm
f
i
f
i
-
vvvvvv
v
v
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Kinetic energy: K = ½ m v2
units: (kg) (m/s)2 = (kg m/s2) m
N m = joules
Work – kinetic energy relation:
Wtotal = Kf – Ki
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Kinetic Energy-Work theorem
22
21
21
if
f
itotalfi mvmvdW - rF
Example: A ball of mass 10 kg, initially at rest falls a height of 5m. What is its final velocity?
i
f
h
??fv
0iv22
21
21
iffi mvmvmghW -
0
smmghv f /899.9)5)(8.9)(2(2
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ExampleA block, initially at rest at a height h, slides down a frictionless incline. What is its final velocity?
hh=0.5m
22
21
21
iffi mvmvmghW -
0
smmghv f /13.3)5.0)(8.9)(2(2
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Example A mass m initially at rest and attached to a spring compressed a distance x=-|xi|, slides on a frictionless surface. What is the velocity of the mass when x=0 ?
k
) smv
mxmNkkgm
xmkv
mvmvkxW
f
i
if
ififi
/63.02.05.0
5
2.0 /5 5.0For
21
21
21 222
-0
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Special case of “conservative” forces conservative non-dissipative
) ) )if
f
ifi UUdW rrrF
F
--
write topossible isit , forces edissipativ-nonFor
) )
iiff
ifif
f
ifi
mgyUmgyU
mgymgyyymgdW
----
)( and )(
:Earth of surfacenear gravity of Example
rr
rF
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)2
212
21
2212
21
)( and )(
:force spring of Example
iiff
if
x
x
f
ifi
kxUkxU
kxkxdxxkdWf
i
---
rr
rF
k
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iclicker exercise:Why would you want to write the work as the difference between two “potential” energies?
A. Normal people wouldn’t.B. It shows a lack of imagination.C. It shows that the work depends only on the
initial and final displacements, not on the details of the path.
)
dxdUF
dU
x
ref
-
-
that Note
:functionenergy potential Define
rFrr
r
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) ) )
) ) (constant) 21
21
:Or21
21
22
22
EUmvUmv
mvmvUUW
iiff
ififfi
---
rr
rr
Work-Kinetic Energy Theorem for conservative forces:
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Energy diagrams 22
21
21 kxmvE
2max2
1221
2max2
1
max
,0(0)
:0 when :Note ,0
: when :Note
kxmvU
x kxEv
xx
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Example: Model potential energy function U(x) representing the attraction of two atoms
