Universal Gravitation

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Keplers Three Laws of Planetary Motion

Law 1: All planets orbit the Sun in elliptical paths with the
Sun at one focus.

Law 2: An imaginary line joining any planet to the Sun

sweeps out equal areas in equal time intervals.

Formula

T 2 r 3
T 2 kr 3

Law 3: The square of the orbital period of any planet is

proportional to the cube of its average orbital radius.

T r

3
2

Newtons Law of Universal Gravitation

Every particle in the Universe attracts every other particle with a force that is
directly proportional to the product of the masses and that is inversely
proportional to the square of the distance between them.

Two approximations used in deriving the law:

1. Masses are considered to be point masses.
Point mass: infinitely small object (radius = 0) whose mass is m

2. The force between two spherical masses whose separation is large compared to their radii is the same as if
the two spheres were point masses with their masses concentrated at the centers of the spheres.
Sun

Mean Earth-Sun distance = 1.50 x 1011 m

Earth

Mean radius = 6.37 x 106 m

Mean radius = 6.96 x 108 m

Extended spherical body

Point mass

Formula

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m1 m2
r2
G m1 m2
Fg =
r2
Fg

Gravitational Constant :
G = 6.77 x 10-11 N m2/kg2

Re 2Re 3Re 4Re

Newtons Derivation of Keplers Third Law

What provides the centripetal force for orbital motion? gravitation

Application weighing the Sun

Derivation

Fc mac

GMm mv 2

r2
r
GM
v2
r
2 r
but v
T
GM 2 r

r
T
GM 4 2 r 2

r
T2
T 2 4 2

r 3 GM

T 2 4 2

r 3 GM
1 year 365.25days 24hours 3600s
4 2

3
6.67 x1011 M s
1.50 x1011 m
2

M S 2.0 x1030 kg
accepted value=1.99 x1030 kg
2

What is the resultant gravitational force on the Earth from the Sun and Moon, as shown below?
Average Earth-Sun distance = 1.50 x 1011 m

Average Earth-Moon distance = 3.84 x 108 m

Sun

Earth

Moon

Mass = 1.99 x 1030 kg

Mass = 5.98 x 1024 kg

Mass = 7.36 x 1022 kg

Gravitational Field Strength

Gravitational field strength at
a point in a gravitational field:

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the gravitational force exerted per unit mass on a small/point mass

Symbol: g

Deriving formula for gravitational

field strength at any point above the
surface of a planet

Point mass

Formula: g = Fg / m

g = Fg /m
g = (GMm/r2)/m
g = GM/r2

Units: N/kg (m/s2)

Type: vector

Deriving formula for gravitational

field strength at the surface of a planet

1. What is the gravitational field strength of the Earth at its surface?

g = GM/r2
go = G Mp / Rp2

g at the surface of the Earth

2. What is the gravitational field strength at an altitude equal to the radius of Earth?

go = G

ME/Re2

Extended spherical body

go

g ratio
g/go = RE2/r2

Re 2Re 3Re 4Re

Average Earth-Moon distance = 3.84 x 108 m

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Earth

Moon

Mass = 5.98 x 1024 kg

Mass = 7.36 x 1022 kg

3. a) What is the resultant gravitational field strength at a point midway between the Earth and Moon?

b) What is the resultant gravitational force acting on a 1500. kg space probe at this location?

Average Earth-Moon distance = 3.84 x 108 m

Earth

Moon

Mass = 5.98 x 1024 kg

Mass = 7.36 x 1022 kg

4. a) Is there a point where the resultant gravitational field strength of the Earth and Moon is zero? If so, where?

b) What is the resultant gravitational force acting on a 1500. kg space probe at this location?

Gravitational Potential Energy

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Difference in gravitational potential energy between any two points in a gravitational field:

EP = mgh

This difference is path independent.

1. Same EP between any two points no matter what path is
taken between them.
2. Work done in moving a mass between two points in a
gravitational field is independent of the path taken.
3. EP is zero between any two points at the same level no matter
what path is taken.
4. EP is zero for any closed path (a path that begins and ends at
same point).
Old formula for gravitational potential energy:
Ep = mgh

discuss 2 problems with definition

1) g varies above surface
2) arbitrary base level

Base level: infinity

Gravitational PE at infinity: zero

EP = -400 J

Gravitational Potential Energy

of a mass at a point in a
gravitational field:

EP = -100 J

EP = 0

the work done in bringing a small point mass in from

infinity to that point in the gravitational field
Derivation of gravitational potential energy formula

Formula:

Potential energy vs. distance:

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Ep = - GMm/r
EP = -Gm1m2/r
Formula not valid
inside planet

Ep at surface
Ep = -GMpm/Rp

Symbol: V
Units: J
Type: scalar
What is the gravitational potential energy of a 5000 kg satellite:
a) on the surface of the Earth?

b) orbiting the Earth at an altitude of 200 km?

c) How much does the potential energy of the satellite

increase when it is put into this orbit?

Gravitational Potential
Gravitational potential at a
point in a gravitational field:

work done per unit mass to bring a small point mass in from infinity to
that point in the gravitational field

Formulas:

Gravitational Potential
vs. distance

Difference in
gravitational potential:
V = W/m
V = Ep/m
Gravitational
potential at a point:
Symbol: V
V = Ep/m
V = -GM/r

Units: J/kg
Type: scalar

V at surface
Vo = -GMp/Rp
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1. What is the gravitational potential due to the Earths gravitational field:

a) at the surface of the Earth?

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b) At a location three Earth radii from

the center of the Earth?

c) What is the change in potential in moving from the surface to this new location?

d) What is the minimum amount of energy needed to lift a 5000 kg satellite to this location?

2. What is the net gravitational potential at a spot midway between the Earth and the Sun?

Sun

Earth

3. Derive an expression for the gravitational potential at the surface of a planet in terms of the gravitational field strength.

Escape Speed

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Escape Speed: minimum initial speed an object must have at the surface of a planet in order to escape the
gravitational attraction of the planet

Travel to infinity
Just make it means velocity is zero at infinity means EK is zero at infinity as well as EP

Eo = Ep + Ek

Ef = Ep

Planet

Assumptions: planet is isolated ignore air resistance

Derivation:
Note:
1. Direction of travel is irrelevant Ep is path independent

2. independent of mass of rocket

3. More speed (EK) is needed in real life since air friction is

not negligible at lower altitudes

1. What is the escape speed for Earth?

2. If the Earth became a black hole, how large would it be?

Satellite Motion

Natural Satellites
Period of a satellite:

T r
2

Artificial Satellites
Acceleration of a satellite:

Weightlessness

v = 2 r/T

ac = 42 r/T2

3
2

ac = g = GM/r2

Free fall

Orbital speed of a satellite:

Fc mac

ac = v2/r
v = 2 r/T

T 2 kr 3
T r

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Orbital motion

GMm mv 2

r2
r
GM
v2
r
GM
v
r
Deep space

1. Compare the motion of satellites A and B.

A faster, less time (smaller period)
B slower, more time (longer period)
T2/R3 = constant true for all satellites
T = kR3/2

2. What happens to the required orbital speed if:

a) the mass of the satellite increases?
Nothing speed is independent of mass

b) the satellite is boosted into a higher orbit?

Satellite would orbit at a slower
(tangential) speed

3. What would happen to a satellite if it encountered appreciable air resistance?

Slow down, drop to lower orbit, and speed up, encounter even more air molecules (denser), cycle continues spiral to Earth

Energy of Orbiting Satellites

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Compare the energies of the two orbiting satellites.

Gravitational Potential Energy

Kinetic Energy

Energy Derivations

Total Energy

Graphs of the energies of an orbiting satellite

Gravitational Potential Energy

Kinetic Energy

Total Energy

RE

Comparisons:

A 1500 kg satellite is to be put into orbit around the Earth at an altitude of 200 km.
a) How much potential energy will the satellite
have at this altitude?

b) How much kinetic energy will the satellite

need to orbit at this altitude?

d) What is the orbital speed of the satellite?

e) What is the minimum amount of energy needed to lift the satellite

from the surface of the Earth to this altitude?

c) What is the total amount of energy the

satellite has at this altitude?

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Comparisons

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Equipotential surface: a surface on which the potential is the same everywhere

1. The gravitational force does no

work as a mass moves on
along equipotential surface.
2. The work done in moving a
mass between equipotential
surfaces is path independent.
3. The work done in moving a
mass along a closed path is
zero.
one point mass

two point masses

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IB 12
On the diagram at right:
a) Sketch the gravitational field around the point mass.

b) Sketch equipotential surfaces around the point mass.

What is the relationship between the
gravitational field and the equipotential
surfaces?
Perpendicular
Field lines point in direction
of decreasing potential

Gravitational Potential Gradient

gradient: rate of change with respect to something slope or derivative

gravitational potential gradient: the gravitational field is the negative gradient of the gravitational
potential with respect to distance

Formula

-70 J/kg

-80 J/kg

B
derive g = -V/r

What is the average gravitational field strength between

equipotential surfaces A and B if they are 5.0 m apart?

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Practice Questions
1. a) Calculate the gravitational force the Sun exerts
on the Earth.

2. a) Calculate the strength of the gravitational field of the Sun

at a location one million kilometers from the Sun.

3. a) Calculate the strength of the Suns gravitational field at

the surface of the Earth.

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b) Compare this to the gravitational force that the

Earth exerts on the Sun.

b) What is the Suns gravitational force at this

point?

b) Explain why the net gravitational field strength at

the surface of the Earth can be approximated as due
solely to the Earths gravitational field.

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IB 12
4. a) Calculate the resultant gravitational field at a spot
midway between the Earth and Sun.

b) Compare the contributions from the Sun and the

Earth to this resultant field.

c) What is the gravitational force acting on a 5000 kg

space probe at this location?

5. A 5000kg satellite orbits Mars at a distance of 1000 km.

Mass of Mars: 6.42 x 1023 kg

Mean planetary radius: 3.37 x 106 m

a) What is the gravitational potential at the surface of Mars?

Mars

b) How much gravitational potential energy does the satellite have on the surface of Mars?

c) What is the gravitational potential at orbiting altitude?

d) How much gravitational potential energy does

the satellite have at this altitude?

e) What is the minimum energy needed to lift

the satellite to this altitude?

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Universal
Fg m1 m2
Gravitationa
1
l Constant
Fg 2
r
(G):
G = 6.77 x
6. A 5000. kg satellite is placed in a low altitude orbit.
10-11 N
2
2
m /kg
a) If the altitude is sufficiently low, what is the approximate radius of the satellites orbit?

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b) Calculate the satellites orbital speed.

c) Calculate the orbital period of the satellite.

d) Calculate the gravitational potential energy of the satellite.

e) Calculate the kinetic energy of the satellite.

f) Calculate the total energy of the satellite.

g) What is the minimum amount of energy needed to lift the satellite into this orbit?

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