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Gravitation

- Newtons Law of Universal Gravitation
- Every mass in the universe attracts every other

mass in the universe according to - F GMm
- r2
- where
- G 6.67 x 10-11 Nm2kg-2

Gravitation

- Newtons Law of Universal Gravitation
- The law deals only with point masses. (A sphere

can be considered a point mass located at the

center of the sphere.) - When dealing with two objects, the gravitational

force acting on each is equal and opposite.

Gravitation

- Newtons Law of Universal Gravitation
- Gravitational forces are always attractive.

(Neither the mass nor the distance between the

objects can be negative.) - Although gravitational force acts between all

objects, the value of G shows the force will be

significant only for very massive objects.

Gravitation

- Newtons Law of Universal Gravitation
- Note The law applies only to two masses at a

time, but the gravitational force vectors may be

added to the force vectors coming from a third

mass in the system. This may be done iteratively

for additional masses. - The resulting mathematical complexity limits

solutions for 3 or more masses to approximate

solutions, (the three body problem).

Gravitation

- Gravitational Field Strength (g)
- g F
- m
- Units are N?kg-1
- Gravitational
- Field Lines ?
- (sphere and point)

Gravitation

- Gravitational Field Strength
- Since F GMm then F GM
- r2 m r2
- Since F mg then F g
- m

Gravitation

- Gravitational Field Strength
- and therefore g GM
- r2
- is the gravitational field at the surface of a

sphere of radius r. Use vectors to get the

resulting field of two objects on a third.

Gravitation

- Gravitational Field Strength
- g GM
- r2
- G 6.67 x 10-11 Nm2kg-2
- Mearth 5.97 1024 kilograms
- rearth 6.4 x 106 kilometers

Gravitation

- LaGrangian Points (Earth Moon)
- L1 - Between the Earth and Moon
- L2 - On the other side of the Moon
- L3 - Opposite the Moon
- L4 - 60 ahead of the Moon
- L5 - 60 behind the Moon

Gravitation

- LaGrangian Points (Earth Moon)
- L1, L2 and L3 are quasistable, station-keeping is

needed to keep an object on location at these

points. - L4 and L5 are stable. Trash will collect at L4

and L5. (Think Jupiter and asteroids.)

Gravitation

- Gravitational Potential Energy
- When a mass moves between two different heights

(h) near the earths surface - difference in energy mgh PE
- works only if (h) is not very large

Gravitation

- Gravitational Potential Energy
- The everyday solution for mgh treats the PE at

the surface of the earth as 0. - The true solution for gravitational PE is 0 for

PE taken at infinity. Therefore, an object at

infinity loses potential energy as it moves

towards a mass M.

Gravitation

- Gravitational Potential Energy
- Therefore the PE is negative for any object at

point P, where P is not infinity. - This means that the work done in moving from

infinity to point P is also negative. - Gravitational PE is a scalar.

Gravitation

- Gravitational Potential Energy
- By math not covered in IB...
- PEG - GMm
- r

Gravitation

- Gravitational Potential Energy
- STOP!
- PROVE
- ?PE mgh

Gravitation

- Gravitational Potential Energy
- Therefore, for r h, the change in PE
- ?PE - GMm (- GMm)
- r r h

Gravitation

- Gravitational Potential Energy
- ?PE - GMm GMm
- r r h
- Since F GMm then Fr GMm
- r2 r

Gravitation

- Gravitational Potential Energy
- and F GMm then F(r h) GMm
- (rh)2 (r h)
- ?PE - Fr F(r h)
- ?PE - Fr Fr Fh

Gravitation

- Gravitational Potential Energy
- ?PE - Fr Fr Fh Fh
- Since F mg
- ?PE Fh mgh

Gravitation

- Gravitational Potential
- A scalar that measures the energy per unit test

mass. - VG W - GM
- m RP
- where W Work m Test Mass RP Planet

Radius

Gravitation

- Escape Speed
- Speed needed for a rocket to go from the planet

surface to infinity - VG - GM since VG W
- RP m
- and the energy difference between the surface and

infinity is

Gravitation

- Escape Speed
- Speed needed for a rocket to go from the planet

surface to infinity - VG (m) 0(m) - (- GM)(m) since W VG (m)
- RP
- W GM(m) and therefore W GMm
- RP RP

Gravitation

- Escape Speed
- By the Work-Kinetic Energy Theorem
- W ?KE ½mv2 GMm
- RP
- therefore v2 2GM and vescape

(2GM/r)½ RP

Gravitation

- Keplers Third Law
- When one object is in orbit around another, the

gravitational and centripetal forces must be

exactly balanced. - FG GMm and FC mv2
- r2 r
- and therefore GMm mv2
- r2 r

Gravitation

- Keplers Third Law
- Therefore the orbital speed is
- vorbital (GM/r)½
- Now rearranging this equation, we note that
- GM v2r

Gravitation

- Keplers Third Law
- We can use geometry to figure the speed
- v 2pr and v2 4p2r2
- T T2
- Substituting into the previous equation
- GM 4p2r3 and therefore GM r3
- T2 4p2 T2

Gravitation

- Keplers Third Law
- Then since G, p, and 4 are all constant, then any

object rotating around M will be subject to the

following constraint, usually known as Keplers

Third Law - r3 constant
- T2

Gravitation

- Total Energy
- Total energy is the sum of potential and kinetic

energies (or TE PE KE) - PEG - GMm and KE ½mv2
- r
- therefore TE - GMm ½mv2
- r

Gravitation

- Total Energy
- Using the equation for orbital velocity
- vorbit (GM/r)½ and vo2 GM/r
- therefore TE - GMm ½mGM
- r r

Gravitation

- Total Energy
- Finally
- TE - ½GMm
- r

Gravitation

- Gravitational Interactions
- In essence, a small object in orbit around a

large object is falling around the large

object. - Satellites, the shuttle and the International

Space Station are examples.

Gravitation

- Gravitational Interactions
- Two large objects rotating around each other

actually rotate around a common center of gravity

between them. - The earth and moon, or the sun and Jupiter, or

double stars are examples.

Gravitation

- Gravitational Interactions
- The gravitational drag of the moon create the

ocean tides on earth. - Tides in high latitudes can be very significant.
- The moon creates 254,000,000,000 watts of tidal

energy daily.

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Gravitation

- Gravitational Interactions
- In order of increasing gravitational attraction

are - Sun
- Blue Giant Star
- Neutron Star
- Black Hole
- (of Sun Mass diameter 1 inch)

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Gravitation

- Satellite Motion
- We usually think of the orbits of satellites and

the planets as circular. - In actuality, most orbits are elliptical, with

the larger body at the focus of the ellipse. - The earth is a satellite of the sun with an

elliptical orbit around the sun.

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Gravitation

- Keplers Three Laws
- 1st Any object bound by a force that varies as

1/r2 travels in an ellipse. - 2nd A line drawn from the sun to any planet

will sweep out equal areas in equal times. - 3rd The ratio r3 /T2 is a constant for any

planet.

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Gravitation

- END OF GRAVITATION

GRAVITATION

- Notes