Ch. 6: Gravitation & Newton’s Synthesis
This cartoon mixes two legends:
1. The legend of Newton, the
apple & gravity which led to the
Universal Law
of Gravitation.
2. The legend of William Tell &
the apple.
• It was very SIGNIFICANT & PROFOUND in the 1600's when Sir
Isaac Newton first wrote
Newton's Universal Law of Gravitation!
This was done at the young age of about 30. It was this, more than any of his other
achievements, which caused him to be well-known in the world scientific
community of the late 1600's.
• He used this law, along with Newton's 2nd Law
(his 2nd Law!)
plus
Calculus, which he also (co-) invented, to PROVE that the orbits of the
planets around the sun must be ellipses.
– For simplicity, we will assume in Ch. 6 that these orbits are circular.
• Ch. 6 fits THE COURSE THEME OF NEWTON'S LAWS OF
MOTION because Newton used his Gravitation Law & his 2nd Law
in his analysis of planetary motion.
• His prediction that planetary orbits are elliptical is in excellent
agreement with Kepler's analysis of observational data & with Kepler's
empirical laws of planetary motion.
• When Newton first wrote the
Universal Law of Gravitation,
it was the first time, anyone had EVER written a theoretical
expression (physics in math form) & used it to PREDICT something
that is in agreement with observations! For this reason,
Newton's Formulation of his Universal Gravitation Law is considered
THE BEGINNING OF THEORETICAL PHYSICS.
• It also gave Newton his major “claim to fame”. After this, he was
considered to be a “major leader” in science & math among his peers.
• In modern times, this, plus the many other things he did, have led
to the consensus that Sir Isaac Newton was the
GREATEST SCIENTIST
WHO EVER LIVED
Newton’s Grave
& a
Monument
to him are in
Westminster Abbey
in
London, England.
Inscription on Newton’s Gravestone:
“Here is buried Isaac Newton, Knight, who by a
strength of mind almost divine, and mathematical
principles peculiarly his own, explored the course
and figures of the planets, the paths of
comets, the tides of the sea, the
dissimilarities in rays of light, and, what no
other scholar has previously imagined, the
properties of the colors thus produced. Diligent,
sagacious and faithful, in his expositions of
nature, antiquity and the holy Scriptures, he
vindicated by his philosophy the majesty of God
mighty and good, and expressed the simplicity of
the Gospel in his manners. Mortals rejoice that
there has existed such and so great an ornament
of the human race! He was born on 25th
December, 1642, and died on 20th March 1727.
Newton’s Monument in Westminster Abbey.
Sect. 6-1: Newton’s Universal Law of Gravitation
• This is an EXPERIMENTAL LAW describing the
gravitational force of attraction between 2 objects.
• Newton’s reasoning:
the Gravitational force of attraction
between 2 large objects (Earth - Moon, etc.) is the SAME
force as the attraction of objects to the Earth.
• Apple story: This is likely not a true historical account, but
the reasoning discussed there is correct. This story is
probably legend rather than fact.
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 itself!
He further realized that
this same force must be
what keeps the Moon
in its orbit!
The gravitational force on you is half of a Newton’s 3rd Law pair: Earth
exerts a downward force on you, & you exert an upward force on Earth.
When there is such a large difference in the 2 masses, the reaction force
(the force you exert on the Earth) is undetectable, but for 2 objects with masses
closer in size to each other, it can be significant.

This must be true from
Newton’s 3rd Law!
The Force of Attraction between 2 small masses is the same as
the force between Earth & Moon, Earth & Sun, etc.
By observing planetary orbits, Newton also concluded that the gravitational force
decreases as the inverse of the square of the distance r between the masses.
Newton’s Universal Law of Gravitation:
“Every particle in the Universe attracts every other
particle in the Universe with a force that is proportional
to the product of their masses & inversely proportional
to the square of the distance between them:
F12 = -F21  [(m1m2)/r2]

This must be true from
Newton’s 3rd Law!
The direction of this force:
 Along the line joining the 2 masses
Newton’s Universal Gravitation Law
• This force is written as:
G  a constant,
the Universal Gravitational Constant
G is measured & is the same for ALL objects. G must be small!
• The measurement of G in the lab is tedious
& sensitive because it is so small.
– First done by Cavendish in 1789.
• Modern version of Cavendish experiment:
Two small masses are fixed at the ends of a
light horizontal rod. Two larger masses are
placed near the smaller ones.
• The angle of rotation is measured.
• Use Newton’s 2nd Law to get the vector
force between the masses. Relate to angle
of rotation & can extract G.

Cavendish Measurement
Apparatus
• G = the Universal Gravitational Constant
• Measurements Find, in SI Units:
• The force given above is strictly valid only for:
– Very small masses m1 & m2 (point masses)
– Uniform spheres
• For other objects: We need integral calculus!
• The Universal Law of Gravitation is an
example of an Inverse Square Law
– The magnitude of the force varies as the inverse
square of the separation of the particles
• The law can also be expressed in vector form
The negative sign means it’s an attractive force
• Aren’t we glad it’s not repulsive?
Comments
 Force exerted by particle 1
on particle 2
21
 Force exerted by particle 2
on particle 1
F21 = - F12
This tells us that the forces form a
Newton’s 3rd Law
action-reaction pair, as expected.
The negative sign in the above vector equation tells us that
particle 2 is attracted toward particle 1
More Comments
• Gravity is a “field force” that always
exists between two masses, regardless
of the medium between them.
• The gravitational force decreases
rapidly as the distance between the
two masses increases
– This is an obvious consequence of the
inverse square law
Example 6-1: Gravitational Force Between 2 People
A 50-kg person & a 70-kg person are sitting on a
bench close to each other. Estimate the magnitude
of the gravitational force each exerts on the other.
Example 6-2: Spacecraft at 2rE
• Spacecraft at twice the Earth radius
Earth Radius: rE = 6320 km
Earth Mass: ME = 5.98  1024 kg
ME
m
Example 6-2: Spacecraft at 2rE
• Spacecraft at twice the Earth radius
Earth Radius: rE = 6320 km
Earth Mass: ME = 5.98  1024 kg
FG = G(mME/r2)
• At surface (r = rE)
FG = weight
= mg = G[mME/(rE)2]
• At r = 2rE
ME
FG = G[mME/(2rE)2]
= (¼)mg = 4900 N
m
Example 6-3: Force on the Moon
Find the net force on the
Moon due to the gravitational
attraction of both the Earth &
the Sun, assuming they are at
right angles to each other.
ME= 5.99  1024kg
MM=7.35 1022kg
MS = 1.99  1030 kg
rME = 3.85  108 m
rMS = 1.5  1011 m
F = FME + FMS
(vector sum!)
F = FME + FMS
(vector sum!)
FME = G [(MMME)/(rME)2]
= 1.99  1020 N
FMS = G [(MMMS)/(rMS)2]
= 4.34  1020 N
F =[ (FME)2 + (FMS)2]
= 4.77  1020 N
tan(θ) = 1.99/4.34
 θ = 24.6º
Gravitational Force Due to a Mass Distribution
• In can be shown, with integral calculus, that:
The gravitational force exerted by a
SPHERICALLY SYMMETRIC
mass distribution of uniform density on a particle outside the distribution is
the same as if the entire mass of the distribution were
concentrated at the center.
• So, assuming that the Earth is such a sphere, the gravitational force exerted by
the Earth on a particle of mass m on or near the Earth’s surface is
FG = G[(mME)/r2]; ME  Earth Mass, rE  Earth Radius
• Similarly, to treat the gravitational force due to large spherical shaped objects, it
can be shown with calculus, that: 1) If a (point) particle is outside a thin spherical
shell, the gravitational force on the particle is the same as if all the mass of the
sphere were at center of shell. 2) If a (point) particle is inside a thin spherical
shell, the gravitational force on the particle is zero. So, we can model a sphere as
a series of thin shells. For a mass outside any large spherically symmetric mass,
the gravitational force acts as though all the mass of the sphere is at the sphere’s center.
Sect. 6-2: Vector Form of Universal Gravitation Law
In vector form,
The figure gives the directions of
the displacement & force vectors.
If there are many particles, the
total force is the vector sum of
the individual forces:
Example: Billiards (Pool)
• 3 billiard (pool) balls, masses m1 = m2 =
m3 = 0.3 kg on a table as in the figure.
Triangle sides: a = 0.4 m, b = 0.3 m,
c = 0.5 m. Calculate the magnitude &
direction of the total gravitational force
F on m1 due to m2 & m3.
Note: Gravitational force is a vector, so
we have to add the vectors F21 & F31 to
get the vector F (using the vector addition
methods of earlier). F = F21 + F31
Using components:
So,
Fx = F21x + F31x = 0 + 6.67  10-11 N
Fy = F21y + F31y = 3.75  10-11 N + 0
F = [(Fx)2 + (Fy)2]½ = 3.75  10-11 N
tanθ = 0.562, θ = 29.3º