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Gravitation
(Physics Fundamentals 2)

Reynold V. Luna
Physics Instructor, College of Science 1

• Newton’s law of gravitation

• Gravitational Field
• Motion of satellites
• Kepler’s laws of planetary motion
• General Theory of Relativity

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1. When an object is thrown up, the force of gravity _________.

A. is opposite to the direction of motion C. is parallel to the direction of motion
B. becomes zero at the highest point D. increases as it rises up

2. When an object is thrown down, the force of gravity _________.

A. is opposite to the direction of motion C. is parallel to the direction of motion
B. becomes zero at the lowest point D. decreases as it falls down

3. In an Earth-falling apple system, which is accelerating?

A. Earth C. both A and B
B. Apple D. neither A nor B

4. In an Earth-falling apple system, which experienced a greater force of gravity?

A. Earth C. both A and B
B. Apple D. No force of gravity experienced
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Newton’s Law of Gravitation

What causes the APPLE to fall?
What causes YOU to be pulled down?
THE EARTH’S MASS. Anything that has ∝
MASS has a gravitational pull towards it.

The proportionality above says that for

there to be a FORCE DUE TO GRAVITY
on something there must be at least 2
masses involved.

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Newton’s Law of Gravitation

As you move AWAY from the earth, your DISTANCE
increases and your FORCE DUE TO GRAVITY decrease.
This is a special INVERSE relationship called an
Inverse-Square.
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∝
where “r” is the distance between the CENTERS OF
MASS of the two objects.

How did Newton figure this out?

Newton knew that the force on a falling apple (due
to Earth) is in direct proportion to the acceleration
of that apple. He also knew that the force on the
moon is in direct proportion to the acceleration of
the moon, ALSO due to Earth.
∝ ∝
Newton also surmised that the SAME force was
inversely proportional to the distance from the
center of Earth. The problem was that he wasn’t
exactly sure what the exponent was.
1 1
∝ ∝
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How did Newton figure this out?

Since both the acceleration and distance were set up
as proportionalities with the force, he decided to set
up a ratio. 1

∝ → ∝
1

Newton knew the following approximate values:

= 9.8 = 2.72 × 10
= 6.37 × 10 m = 3.85 × 10 m
. . ×
Thus, ∝ → 3600 ∝ 60 → = 2
. × . ×
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Newton’s Law of Gravitation

In 1687, Newton published his work on the law of gravitation in his treatise
Mathematical Principles of Natural Philosophy. It states that:

every particle in the

Universe attracts every
other particle with a force
that is directly proportional =
to the product of their
masses and inversely Where G is universal gravitational
proportional to the square constant and its SI value is
of the distance between
them. = 6.674 × 10 N ∙ m /kg
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Newton’s Law of Gravitation

The gravitational force is expressed mathematically as:

⃗ =− ̂

Properties of gravitational force:

1. It is a conservative force.
2. It is a central force.
3. It is always attractive.
4. It obeys superposition principle.
5. Weakest among the 4 fundamental
forces of nature.
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Gravity Example
How hard do two planets pull on each other if their masses are 1.23  1026 kg
and 5.21  1022 kg and they 230 million kilometers apart?

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Calculating the Gravitational Constant

In 1978, Sir Henry Cavendish used two large
lead spheres, each of mass 12.7 kg and two
smaller spheres, each of mass 9.85 g. The table
below gives the results for the total force on
the fiber with the masses at various distances.

distance (m) 0.05 0.08 0.10 0.12 0.13 0.15

Net force (× 10-10 N) 66.8 26.6 16.6 11.6 9.90 7.40

(a) Plot net force vs. distance. What relationship does this graph suggest?
(b) Determine the slope of the graph using regression analysis.
(c) Divide the slope by the product of the masses to get the value for G. How
does it compare with the accepted value?

Calculating the radius of the Earth

The Greeks approximated Earth’s radius
over 2000 years ago:

= = 6.37 x 106 m

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Calculating the mass of the Earth

ON A NON-ROTATING PLANET
When an object is in (or near) the Earth’s surface:
=

Alternatively:
=
Thus,
= mass of the Earth = 5.97 × 10 kg

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Example
How far from the earth's surface must an astronaut in space be if she is
to feel a gravitational acceleration that is half what she would feel on
the earth's surface?

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Weight, Gravity and Centripetal Force

on a Rotating Planet
Apparent weight = weight – centripetal force

The free-fall acceleration, is:

= −
Where:
= acceleration due to gravitational force
= centripetal acceleration

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Example
A 1.00 kg brick rests on the surface of the Earth at the equator.
Calculate the following:
(a) the gravitational force on the brick;
(b) the centripetal force on the brick;
(c) the weight of the brick;
(d) acceleration due to gravity;
(e) centripetal acceleration;
(f) free-fall acceleration.

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Centripetal and Gravitational Forces

Two particles of equal mass are moving
in a circle of radius under the action of
their mutual gravitation attraction. Find
the speed of each particle.

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Net Gravitational Force due to

Discrete Mass Distribution
Three 0.300-kg billiard balls are placed
on a table at the corners of a right
triangle as shown in the figure. The sides
of the triangle are of lengths a = 0.400
m, b = 0.300 m, and c = 0.500 m.
Calculate the gravitational force vector
on the cue ball (designated m1) resulting
from the other two balls as well as the
magnitude and direction of this force.

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Gravitational Force due to Continuous

Mass Distribution
Find the gravitational force of attraction on the point mass
placed at by a thin rod of mass and length as shown
below.

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Things to consider about the Earth

• You can treat the earth as a point mass with its mass being at the center if
an object is on its surface
• The earth is actually not uniform
• The earth is not a sphere
• The earth is rotating

Assuming that the earth is a uniform sphere.

What would happen to a mass (man) that is
dropped down a hole that goes completely
through the earth?

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Journey to the Center of the Earth

When you jump down and are at a radius “r” from the center, the
portion of Earth that lies OUTSIDE a sphere a radius “r” does NOT
produce a NET gravitational force on you!
The portion that lies INSIDE the sphere does. This implies that as
you fall the “sphere” changes in volume, mass, and density ( due
to different types of rocks)
M 4 3 4r 3
  , Vsphere  3 r  M inside  
V 3
Mm G 4m G 4m
Fg  G  Fg  r k
r r 2
3 3
Fg  kr
This tells us that your “weight” actually DECREASES as you
approach the center of Earth from within the INSIDE of the sphere
and that it behaves like Hooke’s Law. YOU WILL OSCILLATE. 23

Energy Consideration
Recall, the gravitational force is a conservative force and
the work it done is:
= −∆
= − =− −

1 1 1
− =− − = −

Thus, the gravitational potential energy associated with

any pair of particles of masses and separated by a
distance is defined as:
=−
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Gravitational potential
energy depends on
distance
Gravitational potential
energy = − for
The gravitational potential energy of
the earth-astronaut system increases
the Earth-Astronaut
system.
(becomes less negative) as the
astronaut moves away from the
earth, as shown in the figure at the
left.
is
is always
always negative,
negative,
but it becomes less negative
−
with increasing radial distance .
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Example
In Jules Verne’s 1865 story “From the Earth to the Moon”, three men went to
the moon in a shell fired from a giant cannon sunk in the Earth in Florida.
(Neglect air resistance, earth’s rotation, and the gravitational pull of the moon.)
a) Find the minimum muzzle speed Very far from center: Very far from center:
needed to shoot a shell straight up to Very far from
very
very small
center:
small very small

a height above the earth equal to the Farther from center: Very far from than
center:
Speed greater
earth’s radius rE. smaller very small

b) Find the minimum muzzle speed that At surface: Very far from center:
Speed smaller than
very small
would allow a shell to escape from large
the earth completely (the escape
speed, ).

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Types of Celestial Motion

The two main types of periodic motion in
space are:
• elliptical motion — planets about the
Sun and some artificial satellites
• circular motion — moons about their
planets and some artificial satellites.

Satellite means ‘neighbor’ or

‘companion’. To an astrophysicist,
satellites can either be:
• natural (e.g. moons)
• artificial (e.g. communications
satellites). 28

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Artificial Satellites
Artificial satellites are categorized as:
• Geostationary satellites – geosynchronous
• Polar satellites – rotates around and passing the poles

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Circular Orbit
In a circular orbit, a satellite always travels at the
same speed and stays the same distance from Earth.
The earliest measurement of the Moon’s period
shows that it hasn’t changed over the past few
thousand years. It is known to be 27.321 661 days.
The ‘right’ speed for a satellite is such that the
centripetal force needed to keep it in a circular path
exactly equals the force of gravity or its weight. This
velocity is called its critical velocity ( ⃗ )

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Example
A 5000-kg geosynchronous satellite
moves uniformly in a circular path
400 km above the Earth. Calculate
the following:
(a) the gravitational force;
(b) the critical velocity; and
(c) the total mechanical energy.

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1. Planet means ‘wanderer’ but what exactly was

meant by wandering and what is retrograde motion?
2. Galaxy comes from the Greek galas meaning ‘milk’.
What has our galaxy got to do with milk?
3. What is the difference between a pulsar and a
quasar? Name one of each and state their distance
from Earth.
4. The Ptolemaic system used circular motion but still
allowed planets to move in non-circular orbits. Show
how Ptolemy used ‘epicycles’ to contrive his system.
5. Why was Copernicus so reluctant to publish his
theory but finally relented on his deathbed? What
was he scared of?
6. Why was the Church so annoyed with Galileo?
Describe how he was treated. Was he better off than
fellow astronomer Bruno in the hands of the
Inquisition?
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Circular Orbits for Heavenly Bodies

Copernican Model Ptolemaic Model 34

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Elliptical Orbit

It is because Mathematics
provide integrated vision and
Physics gravitate around the right
questions. 35

• The Law of Orbits

• The Law of Areas
• The Law of Periods 36

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Eccentricity Parabola: =1
Hyperbola:
>1

Ellipse:
0< <1
Circle:
Mercury = 0.206 =0
Venus = 0.007
Earth = 0.017
Mars = 0.093
Halley = 0.968
It is the measure of how much the conic
section deviates from being circular.
It is

Kepler’s First Law (1609)

The planets move in elliptical orbits with the sun at one focus.

Distance of the aphelion

from the Sun:
= (1 + )
Distance of the perihelion
from the Sun:
= (1 − )

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Activity

What is the eccentricity of

the ellipse formed inside
the circle?

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Kepler’s Second Law (1609)

The radius vector to a planet sweeps out area at a rate that is independent of
its position in the orbit.

1 1
= =
2 2
1 1
= =
2 2
= ⃗× ⃗ = =

= = constant
2

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Kepler’s Third Law (1619)

The square of the period of revolution about the sun is proportional to the
cube of the semi-major axis of the orbit.
Gravitational force = Centripetal force

= → =
2
= where: = period of revolution
4 4
= → =

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Sample Problems
1. Calculate the mass of the Sun,
noting that the period of the
Earth’s orbit around the Sun is
3.156 × 107 s and its distance
from the Sun is 1.496 × 1011 m.
2. One astronomical unit (AU) is the
distance between Earth and the
sun (about 93 million miles).
Venus is 0.723AU from the sun.
How long is a Venusian year?
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In 1610, Galileo used his telescope to

Galilean Moons discover the four most prominent moons of
Jupiter (the Galilean moons). Their mean
orbital radii and periods are given in the
table. Plot a graph of 3 ( −axis) against 2
( −axis) and comment on what this graph
shows about the relationship between
and .
Name (m) (days)
Io 4.22 × 108 1.77
Europa 6.71 × 108 3.55
Ganymede 10.70 × 108 7.16
Callisto 18.80 × 108 16.70
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Gravitational Fields
Near Earth’s surface the gravitational
field is approximately uniform.

The field lines:

• are radial, rather than parallel,
and point toward center of Earth.
• get farther apart farther from the
surface, meaning the field is
weaker there.
• get closer together closer to the
surface, meaning the field is
stronger there.
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Gravitational Field Intensity due to a

Point Mass

⃗
⃗= =− ̂

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Gravitational Field Intensity due to a

Uniform Circular Ring at a point on its axis

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Gravitational Field Intensity due to a

Uniform Disk at a point on its axis

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Gravitational Field Intensity due to a

Uniform Solid Sphere
Case I: At an external point Case II: At an internal point

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Gravitational Field Intensity due to a

Thin Spherical Shell
Case I: At an external point Case II: At an internal point

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Einstein’s Theory of Gravity is built on the ff:

1. The speed of light is constant.
2. As an object speeds up its clock
runs faster.
3. The effects of gravity cannot be
distinguished from the effects of
acceleration in the absence of
gravity.
4. Everyone is a relative.

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Gravity deforms spacetime

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Tests of GR Gravitational light bending (1922)

Gravity Probe B (2004)

Shapiro delay (1964)

Confirmation of general relativity on large
LIGO scales from weak lensing and galaxy
(2016) velocities (2010)

Black holes are “Gravity Centers of the

Universe” with size ~ 0 and density ~ ∞.
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Application of General Relativity:

Black holes
Anatomy of a Black hole
1. Event horizon - Area around black hole, from
where nothing can be escaped to outside
2. Singularity – center of the black hole
3. Schwarzschild radius, – distance from
singularity to event horizon
2
=
where: = gravitational constant
= 6.67 × 10-11 N·m2/kg2
= mass of the object
= 3 × 10 m/s
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Activity

Schwarzschild
Object Mass (kg)
Radius (m)
Sun 1.989 × 1030
Moon 7.348 × 1022
Earth 5.972 × 1024
You

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STEM Integration
Necessity Beneficiality Practicability

Diwata - 1

HB 3637 and SB 1211

(Philippine Space Act 2016)

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References
1. Young, H., Freedman, R. and Ford, A. (2016) University Physics with
Modern Physics, 14e, Pearson
2. Hewitt, P. (2013) Conceptual Physics, 12e, Addison-Wesley
3. Giancoli, D. (2013) Physics: Principles with Application, Addison-Wesley

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