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GRAVITATION

Newton’s Law of Universal Gravitation

If the force of gravity is being exerted on

objects on Earth, what is the origin of that

force?

Newton’s realization was

that the force must come

from the Earth.

He further realized that

this force must be what

keeps the Moon in its

orbit.

A force of attraction between objects that is due to their masses.

Because gravity is

less on the moon

than on Earth,

walking on the

moon’s surface

was a very bouncy

experience for the

Apollo astronauts.

Compared with “all” the objects around you, Earth has a enormous mass.

Any two bodies with masses can attract each other. This universal effect is known as gravitation

The force with which one body attracts the other due to their masses is known as gravitational force

Newton’s Law of Universal Gravitation states that the gravitational force FG between any two bodies of mass m1 and m2, separated by a distance r, is described by,

It is a center-to center attraction between all forms of matter.

2

21

r

mmFG

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Newton’s Law of Universal Gravitation

The gravitational force on you is one-half of a third law

pair: the Earth exerts a downward force on you, and

you exert an upward force on the Earth.

When there is such a disparity in masses, the reaction

force is undetectable, but for bodies more equal in

mass it can be significant.

>The moon is actually falling toward Earth but

has great enough tangential velocity to

avoid hitting Earth.

>If the moon did not fall, it would follow a

straight-line path.

INVERSE SQUARE LAW

• SMALL ‘d’

• LARGE ‘F’ • LARGE ‘d’ • SMALL ‘F’

This applies to any case where the effect from a localized source spreads out evenly OTHER EXAMPLES WHERE THE INVERSE SQUARE LAW IS APPLIED: Light, Radiation, Sound NOTE: The force between any two objects NEVER reaches zero, it just gets very small (Asymptotically approaches zero).

8.1.1 NEWTON’S CONFIRMATION OF 1/r2

Inverse-square law:

relates the intensity of an effect to the inverse- square of the distance from the cause.

in equation form: intensity = 1/distance2.

for increases in distance, there are decreases in force.

even at great distances, force approaches but never reaches zero.

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The gravitational force on either one of the two object is proportional to both m1 and m2,

2

21

r

mmFG

Mass

m1 m2

force varies directly with masses

Fg α m1 m2

decrease mass Decreased force

Gravitational force increases as mass increases.

◦ Imagine an Elephant and a Cat

◦ Or imagine the Earth and the Moon

Gravitational force decreases as distance increases. ◦ Gravity between you and the Earth

◦ Gravity between you and the Sun

Gravity is the weakest of four known fundamental forces

With the gravitational constant G, we have the equation

Universal gravitational constant:

G = 6.67 10-11 Nm2/kg2 Once the value was known, the mass of

Earth was calculated as 6 1024 kg

2

21

r

mmGFG

NxKgKgx

d

mGmFg

11

2

11

2

21 1075.34

)3)(3)(1067.6(

What is the force of gravity between two 3 kg blocks that are placed 4

meters apart?

How much is the force of gravity from the Earth acting on a 90 Kg man?

(Mass of the Earth = 6.0 x 1024 Kg ; radius of the Earth = 6400 km)

NKgxKgx

d

mGmFg 879

)6400000(

)100.6)(90)(1067.6(2

2411

2

21

Newton’s Law of Universal Gravitation

Example 6-2: Spacecraft at 2rE.

What is the force of gravity acting on a

2000-kg spacecraft when it orbits two

Earth radii from the Earth’s center (that

is, a distance rE = 6380 km above the

Earth’s surface)? The mass of the Earth

is mE = 5.98 x 1024 kg.

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Newton’s Law of Universal Gravitation

Example 6-3: Force on the Moon.

Find the net force on the Moon

(mM = 7.35 x 1022 kg) due to the

gravitational attraction of both the

Earth (mE = 5.98 x 1024 kg) and the

Sun (mS = 1.99 x 1030 kg),

assuming they are at right angles

to each other.

Gravitational acceleration on the moon and nine planets:

PLANET GRAVITATIONAL ACCELERATION (m/s2)

Mercury 3.7

Venus 8.9

Earth 9.8

Moon 1.6

Mars 3.7

Jupiter 26

Saturn 12

Uranus 11

Neptune 12

Pluto 2

Joe Average's mass is same everywhere in the universe, but his weight at various

places is not the same For a long time, most scientists thought all

satellites travel in perfectly circular orbits ◦ NOT TRUE

Using the circular orbit theory… they could not make accurate predicts of their motion

Planets, moons, etc. were not where they were supposed to be! ◦ Planets did not follow these predicted paths

◦ So something must be wrong

◦ Then…… along came Johannes Kepler

First Law

1) The paths of the planets are ellipses, with the sun at one focus (the other focus is just a point in space)

An Imaginary line from the sun to a planet sweeps out equal areas in equal time intervals.

This means planets move faster when they are closer to the sun and slower when they are further away

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The square of the ratio of the periods of any two planets revolving about the sun is equal to the cube of the ratio of their average distances from the sun. Thus, if Ta and Tb are the planets periods, and ra and rb are their average distances from the sun we get

(Ta/Tb)2 =(ra/rb)

3

(1) V2 = G m2

r (2) V2 r= G m2

(3) V = ωr

(4) ω= 2πf

(5) T = 1/f

(6) ω = 2π / T

(7) (ωr)2r = Gm2

(8) ω2 r3 = Gm2 (9) ( 2π / T)2 r3 = Gm2

(10) 4π2 r3 / T2 = Gm2

(11) T2 / r3 = 4π2/ Gm2 = a constant T2 / r3 = a constant

Kepler’s Third Law

Velocity increases (perigee)

Velocity decreases (apogee)

Equal Areas In

Equal Times

A satellite does not fall because it is moving, being given a tangential velocity by the rocket that launched it. It does not travel off in a straight line because Earth’s gravity pulls it toward the Earth.

The tangential speed of an object in a circular orbit is given by:

If the period of the orbit is known, the velocity may be determined using:

The period of a satellite can be determined by:

r

MGv E

T

r2v

v

r2T

Fg = G = Mem _________

r² ____

r

mv2

Speed of a Satellite:

r

Gmv e

Speed of a Satellite:

Velocity Vectors

Acceleration Vectors

Force Vectors

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Earth

A satellite is a projectile shot from a very high elevation and is in free fall about the Earth.

Inertial position

Centripetal force

Centripetal force

Centripetal force

Centripetal force

Gravity supplies centripetal force inward towards

the center of the circular path

Planet

Force of gravity Fg

Centripetal force Fc

Fg = Fc

Fg = G m1 m2 r2

Fc = m v2

r

G m1 m2 = m1 v2

r2 r Canceling m1 & r on both sides

V2 = G m2

r

5000 km

Re r = re + h

me = 6 x 10 24 kg Re = 6.4 x 10 6 m

smv

v

r

Gmv

r

Gmv

a

e

/5900

100.5104.6(

)100.6)(1067.6(

,

.

66

2411

22

hrsT

T

v

rT

T

rv

b

4.3102.1

5900

)104.11(2

22

.

4

6

At what speed must a spacecraft be injected into orbit if it is to circle the Earth at treetop height?

Given: rearth= 6.4×106 m, mearth= 6.0×1024 kg

13

6

242211

109.7

104.6

)100.6)(/1067.6(

msv

m

kgkgNmv

r

Gmv

What is gravity in outer space? Where space shuttle orbits…g = 8.7m/s2

How come astronauts are “floating” then?

g = F/m

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Weight:

force an object exerts against a supporting surface Examples:

• standing on a scale in an elevator accelerating downward, less compression in scale springs; weight is less

• standing on a scale in an elevator accelerating upward, more compression in scale springs; weight is greater

• at constant speed in an elevator, no change in weight

Weightlessness:

no support force, as in free fall

Example: Astronauts in

orbit are without support forces and are in a continual state of weightlessness.

How you feel weight, is different than your actual weight. As long as you are near the surface of the Earth you will always have the same weight but you may “feel” like you have a different weight This can happen if you are accelerating up or down Imagine an elevator……..