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Chapter 12

Gravitation

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Gravitythe force that holds the Moon in its orbit is the same that makes an apple fall

not only does Earth attract an apple and the Moon but every body in the Universe attract every other body

Isaac Newton 1665

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Newton’s law of Gravitation

Isaac Newton – 1665Every particle attracts any other particle with a gravitational force whose magnitude is given by

m1 and m2 are the masses of the particles,r is the distance between them, G is the gravitational constant

Two forces of attraction are equal in magnitude but opposite in direction (Newton’s third law)

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rmmGF =

2211106742.6 /kgmNG ⋅×= −

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ExampleExample

Gravitational force decreases with distance as 1/r2

An increase in distance by a factor of 10 results in a decrease in the force by a factor of 100.

Find the force of gravity between a student (70 kg or 154 lb) and a textbook (2 kg or 4.4 lb) when they are separated by a distance of 0.3 m (or 0.98 ft)

Nm

kgkgkgNmF 72

2211 1004.1)3.0(

)0.2)(0.70()/1067.6( −− ×=×=

the force to keep 1 dime (F=mg) is about

NkgsmFd32 1081.9)001.0()/81.9( −×=×=

i.e. F ≅ 0.00001 Fd

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Questions and problems

1. Why do we care, if the force of gravity between object around us is so small?

2. How to calculate the gravitational force between real objects, i.e. that are not point-like ones?

3. What to do if there are more than two particles?4. Motion and gravity5. What would happen as r->0 ?

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Gravitation and the principle of superposition

The principle of superposition: the net effect is the sum of the individual effectsFor n interacting particles the net force on particle 1 can be written as

nnet FFFFF 1141312,1

rK

rrrr+++=

being practical – see chapter “Vectors”(adding vectors using components)

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Gravitational attraction between real objects1. If the sizes of the objects are small compared to the

distance between them – we may consider the objects like particles (good approximation for Moon-Earth)

REarth = 6,370 km

RMoon = 1,061 km

r = 382,000 km

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Gravitational attraction between real objects2. How about the apple-Earth?

Newton’s shell theorem:

A uniform spherical shell of matter attracts a particle that is outside the shell if all the shell’s mass were concentrated at its center

2rMmGF =

rr

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Gravitation near Earth’s surface

Let us assume that Earth is a uniform sphere of mass M.The magnitude of the gravitational force on a particle of mass m, at a distance r from Earth’s center

2rMmGF =

Newton’s second law

then the free-fall acceleration

mgF =

2rGMg =

rr

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Gravitation near Earth’s surface

The free-fall acceleration is not a constant!

2rGMg =

Besides - the actual free-fall acceleration g is different from the equation above1. Earth is not uniform (density varies)2. Earth is not a sphere3. Earth is rotating

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Gravitation inside Earth or …Journey to the Center of the Earth

More from Newton’s shell theorem

A uniform spherical shell exerts no net gravitational force on a particle located inside it

2rGMg = 3

34 rVM πρρ ==

322 3

4 rrG

rGMg πρ==

rGg ρπ3

4=

for r=0 g=0, or the net gravitational force inside Earth = 0

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The effect of the Earth’s rotation

Uniform circular motion: the period of the rotation – 24 hours

rmvNFg

2

=−

then

gFr

Nr

rvgg

mgr

mvr

MmGr

mvFN

eff

effg

2

2

2

2

−=

=−=−=

Variation of geffCanal zone 9.78243Pittsburgh, PA 9.80118Greenland 9.82534

Linear speed of a point on the Earth’s surface due to the rotation:Norfolk: 827 mphEquator: 1030 mph

Speed around the Sun: 66,000 mph

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Gravitational potential energyPotential energy is energy that can be associated with the configuration of a system of objects that can exerts forces on one another. If the configuration of the system changes, the potential energy of the systems can also change.Gravitational potential energy

rGMmU −=

• U(r) approaches zero as r approaches infinity• for any finite value of r, the value of U(r) is negative• connection between the force and the potential

2rGMm

rGMm

drd

drdUF =⎟

⎠⎞

⎜⎝⎛−−=−=

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Compare two definitions for the Earth

Gravitational potential energy

and r

GMmU −= mghU =

fi

if

if rrrr

GMmr

GMmr

GMmU−

=+−=Δ

if iif rrr <<−

( ) mghrrr

GMmrr

rrGMmU if

ifi

if =−≈−

=Δ 2

The formula we have used in the past, U = mgh, is valid only when .iif rrr <<−

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Gravitational potential energy for many particles

Gravitational potential energy for a system with more than two particles

⎟⎟⎠

⎞⎜⎜⎝

⎛++−=

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rmGm

rmGm

rmGmU

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Projectiles, Satellites and PlanetsHow to explain their motion?

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Part 1: Vertical motion

Conservation of energy

Vertical motion before (potential energy from zero at h=0)

ff

ii mgh

mvmghmv +=+

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Vertical motion now (potential energy from zero at r→∞)

f

f

i

i

rGMmmv

rGMmmv −=−

22

22

ffii UKUK +=+

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Part 1: Vertical motion

Conservation of energy for vertical motion

f

f

i

i

rGMmmv

rGMmmv −=−

22

22

ffii UKUK +=+

irr

ivr

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Vertical motion: Escape speed

Vertical projectile motion: up, up and … up

There is a certain minimum initial speed that will cause a projectile to move upward forever, theoretically coming to rest only at infinity (g is not a constant!).

From conservation of energy

ii R

GMv 2=

for Earth: M=5.98*1024 kg, Ri=6.37*106 m, v=11.2 km/s

022

22

=−=−f

f

i

i

RGMmmv

RGMmmv

escape speed

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Part 2: Orbital motion

It is a matter of initial conditions!

hrvm

hrGMm c

+=

+

2

2)(

if vx=0 then vertical motion

at some vx = vc – circular motion

hrGMvc +

=

For a shuttle: v=7.9 km/s

rr

hr

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Satellites: Orbits and Energy

As a satellite orbits Earth on its elliptical path, both its speed, which fixes its kinetic energy K, and its distance from the center of Earth, which fixes its gravitational potential energy U, fluctuate with fixed periods. However, the mechanical energy E of the satellite remains constant.

For circular orbits (second Newton’s law)

rvm

rGMm 2

2 =

2

2mvK =

22U

rGMmK −==

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Satellites: Orbits and Energy

How to change the orbit?

rGMmUKE

2−=+=

Less kinetic energy = less total energy E (more negative) = less radius r

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Example

A satellite is orbiting the Earth as shown below. At what part of the orbit, if any, are the following quantities largest?

# Kinetic energy # Potential energy # Total energy # Orbital velocity # Gravitational force # Angular momentum

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Checkpoint

An astronaut working outside of the space staion orbiting the Earth releases a wrench. Neglecting air resistance, the wrench will: A) strike Earth under the satellite at the instant of release B) strike Earth under the satellite at the instant of impact C) strike Earth ahead of the satellite at the instant of impact D) strike Earth behind the satellite at the instant of impact E) never strikes Earth

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Planets and satellites: Kepler’s Laws

Three laws from Johannes Keppler (1571-1630)

1. The law of orbits: All planets move in elliptical orbits, with the Sun at one focus

A circle is just a special case of an ellipse.

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Planets and satellites: Kepler’s Laws (cont.)

2. The law of areas: A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal times.

Kepler’s second law is totally equivalent to the law of conservation of angular momentum

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Planets and satellites: Kepler’s Laws (cont.)

3. The law of periods: The square of the period of any planet is proportional to the cube of the semi major axis of its orbit.

Applying Newton’s second law

)( 22

2 rmr

mvr

GMm ω==

2

32 2

⎟⎠⎞

⎜⎝⎛==

TrGM πω

2/332

2 24 rGM

TorrGM

T ππ =⎟⎟⎠

⎞⎜⎜⎝

⎛=The period does not depend on

the mass of the orbiting object.

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For a planet of radius R and density ρ

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382

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34

2

3

eo

e

vR

GMv

RGR

GMv

GRR

GMg

RM

==

==

==

=

ground the above just speed Orbital

speed escape

onaccelerati fall free

planet a of mass

ρπ

πρ

πρ

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Black Holes

Escape from a star

RGR

GMvescape 382 ρπ==

example: escape speed from the surface of the sun is about 2.2 million km/h or 1.5 million mph

For light

is the Schwarzschild radius (nothing, not even light can escape from that body/star)

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cGMR

RGMc s

s

==

sR

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Geosynchronous satellites

Many satellites are moving in a circle in the earth’s equatorial plane. They are at such height that they always remain above the same point.

Find the altitude of such satellites above the earth’s surface

example

mRrh

mGMTr

srGM

T

E

E

E

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73/1

2

2

2/3

1059.3

1023.44

3600*242

⋅=−=

⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛=

==

π

π

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Millennium Simulation:http://www.mpa-garching.mpg.de/galform/press/

The biggest and most detailed supercomputer simulation of the evolution of the Universe from a few hundred thousand years after the Big Bang to the present day.

The Millennium Simulation used 10 billion particles to track the evolution of 20 million galaxies over the history of the universe.

A 3-dimensional visualization of the Millennium Simulation. The movie shows a journey through the simulated universe. During the two minutes of the movie, we travel a distance for which light would need more than 2.4 billion years.

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Physics of Gravitation

search for gravitational waves