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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

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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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Integration by eyeball

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Conservative force

0F dr

a

c d

b

e f

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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

.

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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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Curl as limit of tiny line-integrals

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0ˆ( ) lim

ia i

F drcurl F n

a

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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

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L8-1 – Area integral of curl

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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,c

Calculate, along a,b

Calculate, inside a,c

Calculate, inside a,b

OQPF dr

OQP

F dA