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Lecture #3

Center of Mass Defined Relation to momentum

Polar, Cylindrical and Spherical CoordinatesWorked problemsDVD Demonstration on momentum cons. and CM motion

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Center of MassCenter of Mass and Center of gravity happen to be equivalentFor a multi-particle discrete mass-distribution

For a continuous mass-distribution

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1

N N

CM Ntotal

m r m rR

Mm

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

( ) ( )

( )CM

total total

rdm rdm r r dV r r dAR

M Mdm r dV

,,,,,,,,,,,,,,

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Worked Example L3-1 – CM Motion

Given m1 to m2

m= m

m = 3m

Calculate Vcm Initial and Final for two cases

1 1

1

N N

CM Ntotal

m v m vV

Mm

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

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0v 0v

Initial Final

0v

Initial

0 / 4v

Final

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Linear Momentum and CM

1

1

1

, 0

N

N

CM CM

total

N

CMtotal total

CM CMtotal external total

CMexternal

m rR M R m r

M

P m r P M R

P F P M R M R

F M R

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,

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5 :60

or

r

Spherical Coordinates and Earth

Spherical coordinates“Phi” or “Fee” – East-west same as longitude

“Theta” – North-south, same as Colatitude

is 0 at north pole, 180 at south pole, 90 at equator

“r” (radius) ˆ( sin cos )r r x

ˆ( sin sin )r yˆ( cos )r z

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Cylindrical and Spherical Coordinates

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Coord. System

AreadA

VolumedV

Cartesian

Spherical

Cylindrical

Polar

dxdydzdxdy

( sin )( )r d rd dr ( sin )( )r d rd

( )rd drdz

( )rd drdz

( )rd dz

( )rd dr

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Worked Example L3-2 – Discrete masses

Given m1 to m10

m= m

m = 3my

x

y

x

O1

O2

1 unit

2 u

nits

Calculate

Given origin O1

For homework given O2

CMR,,,,,,,,,,,,,,

1 1

1

N N

CM Ntotal

m r m rR

Mm

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

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Worked Example L3-3 – 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, in r and phi Solve integral

rO1

2 km

Calculate

Given origin O1

CMR,,,,,,,,,,,,,,

( )CM

total total

rdm r r dAR

M M

,,,,,,,,,,,,,,

( )dA rd dr:60

•REPEAT for Half disk

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Lecture #3 Wind-up

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Office hours today and tomorrow 4-5:30.Homework problems in Taylor, + Supplement.Second homework due in class Thursday 9/4

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CM

total

rdm rdmR

Mdm

,,,,,,,,,,,,,,

CMexternalF M R,,,,,,,,,,,,,,,,,,,,,,,,,,,,