1 Class #7 of 30 Integration of vector quantities CM problems revisited Energy and Conservative...
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Transcript of 1 Class #7 of 30 Integration of vector quantities CM problems revisited Energy and Conservative...
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Class #7 of 30
Integration of vector quantities CM problems revisited
Energy and Conservative forces Stokes Theorem and Gauss’s Theorem Line Integrals Curl Work done by a force over a path
Angular momentum demo
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Test #1 of 4Thurs. 9/26/02 – in classFour problems Bring an index card 3”x5”. Use both sides. Write
anything you want that will help. All calculations to be written out and numbers
plugged in BEFORE solving with a calculator. Full credit requires a units check.
Linear and Angular momentum / ImpulseMoment of Inertia / Center of MassRetarding forces (Stokes / Newton etc.)Conservative forces / Line integrals / curls / energy conservation
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3
Integration in Different Coordinates
ˆ ˆ ˆ
ˆ ˆ ˆcos( ) sin( )
ˆ ˆ ˆcos( )sin( ) sin( )sin( ) cos( )
System Position vector r
Cartesian xx yy zz
Cylindrical r x r y zz
Spherical r x r y r z
:12
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Worked Example L7-1 – Continuous mass
Given hemisphere with uniform mass-density and radius 5 m: Calculate M total Write r in polar coords Write out triple integral, . components
in terms of r and phi Solve integral
Calculate
Given origin O1
CMRAAAAAAAAAAAAAA
( )CM
total total
rdm r r dVR
M M
AAAAAAAAAAAAAA
( )( sin )dV rd r d dr :20
ˆ ˆ ˆ,x y and z
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Worked Example L3-2 – Continuous mass
Given quarter disk with uniform mass-density and radius 2 km: Calculate M total Write r in polar coords Write out double integral, components in terms of r and phi. Solve integral
rO1
2 km
Calculate
Given origin O1
CMRAAAAAAAAAAAAAA
( )CM
total total
rdm r r dAR
M M
AAAAAAAAAAAAAA
( )dA rd dr:30
ˆ ˆx and y
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A force is conservative iff:
1. The force depends only on
2. For any two points P1, P2 the work done by the force is independent of the path taken between P1 and P2.
Conservative Forces
r( )
( , , )
F F r
F F r r t
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, .Dependence on r t not allowed
0
2
1
2
1
0
( )
( 1 2)
( ) ( ) ( )
P
P
P
P
r
r
F dr Const
over all paths
F dr Work P P
U r W r r F r dr
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Line integral and Closed loop integral
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Conservative force
a) P1 and P2 with two possible integration paths. b) and c) P1 and P2 are brought closer together. d) P1 and P2 brought together to an arbitrarily small distance . Geometric
argument that conservative force implies zero closed-loop path integral.
1
2
0
P
PF dr Const
F dr
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L7-2 – Path integrals
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Taylor 4.3 (modified)
ˆ ˆ( )F r yx xy
O
y
xP(1,0)
Q(0,1)
a
c
b
Calculate, along (a)
Calculate, along (b)
Calculate, along (c)
Calculate
Q
PF dr
P
QF dr
P
QF dr
OQPF dr
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Angular Momentum and Central Forces
ˆ( )
ˆ ˆ( ) ( ( ) ) 0 0
:
centralF F r r
dLr F r F r r because r r
dt
L constant Angular momentum conserved by Central forces
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Taylor 3-25
ˆ( )centralF F r r
0 0, ,Given m r and central force
Calculate given new shorter r
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Lecture #7 Wind-up
.
Read Chapter 4First test 9/26
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For conservative forces0
( ) ( )r
rU r F r dr
1
20
P
PF dr C F dr
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Retarding forces summary
.
:72
1gt
vzv v e
vvDFdrag ˆ16
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Linear Drag on a sphere (Stokes)
Quadratic Drag on aSphere (Newton)
xuDFdrag ˆ3
( ) tanh( / )v t v gt v
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Falling raindrops L6-2
A small raindrop falls through a cloud. It has a 1000 m radius. The density of water is 1 g/cc. The viscosity of air is 180 Poise. The density of air is 1.3 g/liter at STP.
a) Draw the free-body diagram.b) What should be the terminal velocity of the
raindrop, using quadratic drag?c) What should be the terminal velocity of the
raindrop, using linear drag?d) Which of the previous of two answers should
we use??e) What is the Reynolds number of this raindrop?
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Stokes and Gauss’s theorem’s
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Gauss – Integrating divergence over a volume is equivalent to integrating function over a surface enclosing that volume.
Stokes – Integrating curl over an area is equivalent to integrating function around a path enclosing that area.
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The curl-o-meter (by Ronco®)
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Conservative force
0 0F dr F
a
c d
b
e fˆ ˆ ˆ
det
x y z
x y z
fx y z
f f f