8. Gravity 1.Toward a Law of Gravity 2. Universal Gravitation 3. Orbital Motion 4. Gravitational Energy 5. The Gravitational Field.

8. Gravity

1. Toward a Law of Gravity

2. Universal Gravitation

3. Orbital Motion

4. Gravitational Energy

5. The Gravitational Field

This TV dish points at a satellite in a fixed position in the sky.

How does the satellite manage to stay at that position?

period = 24 h

Ptolemaic (Geo-Centric) System

epicycle equant 

deferent 

swf

8.1. Toward a Law of Gravity

1543: Copernicus – Helio-centric theory.

1593: Tycho Brahe – Planetary obs.

1592-1610: Galileo – Jupiter’s moons,

sunspots, phases of Venus.

1609-19: Kepler’s Laws

1687: Newton – Universal gravitation.

Phases of Venus:Size would be constant in a geocentric system.

Kepler’s Laws

d Aconst

dt

2 3T a

Explains retrograde motion

Mathematica

8.2. Universal Gravitation

Newton’s law of universal gravitation:

1 212 122

ˆm m

Gr

F rm1 & m2 are 2 point masses.

r12 = position vector from 1 to 2.

F12 = force of 1 on 2.

G = Constant of universal gravitation

= 6.67 1011 N m2 / kg2 .

Law also applies to spherical masses.

m1

m2

r12

F12

Example 8.1. Acceleration of Gravity

Use the law of gravitation to find the acceleration of gravity

(a) at Earth’s surface.

(b) at the 380-km altitude of the International Space Station.

(c) on the surface of Mars.

2Em m

F m g Gr

2Emg Gr

(a)

24

11 2 226

5.97 106.67 10 /

6.37 10

kgg N m kg

m

29.81 /m s

(b)

24

11 2 226 3

5.97 106.67 10 /

6.37 10 380 10

kgg N m kg

m m

28.74 /m s

(c)

24

11 2 226

0.642 106.67 10 /

3.38 10

kgg N m kg

m

23.7 /m s

see App.E

TACTICS 8.1. Understanding “Inverse Square”

Given Moon’s orbital period T & distance R from Earth,

Newton calculated its orbital speed v and hence acceleration a = v2 / R.

He found a ~ g / 3600.

Moon-Earth distance is about 60 times Earth’s radius.

Cavendish Experiment: Weighing the Earth

2E

E

Mg G

R

ME can be calculated if g, G, & RE are known.

Cavendish: G determined using two 5 cm & two 30 cm diameter lead spheres.

Gravity is weakest & long ranged

always attractive

dominates at large range.

EM is strong & long ranged,

can be attractive & repulsive

cancelled out in neutral

objects.

Weak & strong forces: very short-range.

8.3. Orbital Motion

Orbital motion: Motion of object due to gravity from another larger body.

E.g. Sun orbits the center of our galaxy with a period of ~200 million yrs.

Newton’s “thought experiment”

2

2

M m vG m

r r

G Mv

r

Condition for circular orbit

Speed for circular orbit

2 rT

v

Orbital period

3

2r

G M

Kepler’s 3rd law

g = 0

orbit projectiles

Example 8.2. The Space Station

The ISS is in a circular orbit at an altitude of 380 km.

What are its orbital speed & period?

G Mv

r

3

2r

TG M

Orbital speed: 11 2 2 24

6 3

6.67 10 / 5.97 10

6.37 10 380 10

Nm kg kg

m m

7.7 /km s

Orbital period:

36 3

11 2 2 24

6.37 10 380 102 3.1416

6.67 10 / 5.97 10

m m

Nm kg kg

35.5 10 s 90 min

Near-Earth orbit T ~ 90 min.

Moon orbit T ~ 27 d.

Geosynchronous orbit T = 24 h.

Example 8.3. Geosynchronous Orbit

What altitude is required for geosynchronous orbits?

3

2r

TG M

2/3

1/3

2

Tr G M

2/3

1/311 2 2 2424 3600

6.67 10 / 5.97 102 3.1416

sNm kg kg

74.22 10 m

Altitude = r RE7 64.22 10 6.37 10m m 635.80 10 m 35,800 km

Earth circumference = 62 6.37 10 m 40,000 km

Earth not perfect sphere orbital correction required every few weeks.

Elliptical Orbits

Orbits of most known comets, are highly elliptical.

Perihelion: closest point to sun.

Aphelion: furthest point from sun.

Projectile trajectory is parabolic only if curvature of Earth is neglected.

ellipse

Open Orbits

Closed(circle)

Closed(ellipse)

Open(hyperbola)

Borderline(parabola)

Mathematica

8.4. Gravitational Energy

How much energy is required to boost a satellite to geosynchronous orbit?

2

112 dU

r

rF r

2

112 2

r

r

M mU G dr

r

1 2

1 1G M m

r r

U12 depends only on radial positions.U = 0 on this path

… so U12 is the same as if we start here.

Example 8.4. Steps to the Moon

Materials to construct an 11,000-kg lunar observatory are boosted from Earth to geosyn orbit.

There they are assembled & launched to the Moon, 385,000 km from Earth.

Compare the work done against Earth’s gravity on the 2 legs of the trip.

121 2

1 1EW U G M m

r r

1st leg: 11 2 2 246 7

1 16.67 10 / 5.97 10 11,000

6.37 10 4.22 10W Nm kg kg kg

m m

11 2 2 247 8

1 16.67 10 / 5.97 10 11,000

4.22 10 3.85 10W Nm kg kg kg

m m

115.8 10 J

2nd leg:

109.2 10 J

Zero of Potential Energy

G M mU r

r

121 2

1 1U G M m

r r

0U Gravitational potential energy

E > 0, open orbitOpen

ClosedE < 0, closed orbit

Bounded motion Turning point

Example 8.5. Blast Off !

A rocket launched vertically at 3.1 km/s.

How high does it go?

E K U

20

1

2 E

G M mE m v

R Initial state:

G M mE

rFinal state:

Energy conservation:21

2 E

G Mr

G Mv

R

2

1

12 E

vG M R

12

611 2 2 24

3.1 / 1

6.37 102 6.67 10 / 5.97 10

m sr

mNm kg kg

6.90Mm

Altitude = r RE6 66.90 10 6.37 10m m 530 km

Escape Velocity

Body with E 0 can escape to

210

2 E

G M mm v

R

2

escE

G Mv

R

11 2 2 24

6

2 6.67 10 / 5.97 10

6.37 10esc

Nm kg kgv

m

11.2 /km s

Escape velocity

40,300 /km h

Moon trips have v < vesc .

Open

Closed

Energy in Circular Orbits

Circular orbits:2 G Mv a r

r 21

2K m v

2

G M m

r

G M mU

r

2

G M mE K U

r

K 1

2U 0

Higher K or v Lower E & orbit (r) .

0

E

UK

K

Conceptual Example 8.1. Space Maneuvers

Astronauts heading for the International Space Station find themselves in the right circular orbit, but well behind the station.

How should they maneuver to catch up?

1. Fire rocket backward to decrease energy & drop to lower, & faster orbit.

2. Fire to circularize orbit.3. After catching up with the

station, fire to boost to up to its level.

4. Fire to circularize orbit.

Mathematica

energy

0

UG

Altitude

0

E = K+U = U / 2

U

E = K+U < E ( K < K )

h

E = K+U = U / 2

U < U

K K > K

h < h

G M mU

h 1

2 2

G M mK E U

h

GOT IT? 8.3.

Spacecrafts A & B are in circular orbits about Earth, with B at higher altitude.

Which of the statements are true?

(a) B has greater energy.

(b) B is moving faster.

(c) B takes longer to complete an orbit.

(d) B has greater potential energy.

(e) a larger proportion of B’s energy is potential energy.

8.5. The Gravitational Field

Two descriptions of gravity:

1.body attracts another body (action-at-a-distance)

2.Body creates gravitational field.

Field acts on another body.

Near Earth: ˆgg j

2ˆ

G M

rg rLarge scale:

29.8 /g m s

/N kg

Action-at-a-distance instantaneous messages

Field theory finite propagation of information

Only field theory agrees with relativity.

near earth

in spaceGreat advantage of the field approach:No need to know how the field is produced.

Moon’s tidal (differential) force field near Earth

Moon’s tidal (differential) force field at Earth’s surface

E F r f r f r

Mathematica

Application: Tide

Two tidal bulges

Sun + Moon tides with varying strength.

Tidal forces cause internal heating of Jupiter’s moons.

They also contribute to formation of planetary rings.