Transcript Inverse Square Law

Gravity
Sucks!
If I have seen further it is by
standing on ye sholders of
Giants.
This is the statement written in a letter to
Robert Hooke. It also symbolizes that
the discoveries of Newton including
gravity.
 So we’re going to start with a brief
history are in the bronze age.

Why is it so difficult to find out about the
state of astronomical knowledge of
bronze-age civilizations?
1.
2.
3.
4.
5.
Written documents from that time are in a language
that we don’t understand.
There are no written documents documents from that
time.
Different written documents about their astronomical
knowledge often contradict each other.
Due to the Earth’s precession, they had a completely
different view of the sky than we have today.
They didn’t have any astronomical knowledge at all.
Ancient Greek Astronomers

Models were based on unproven “first principles”,
believed to be “obvious” and were not questioned:
1. Geocentric “Universe”: The Earth is
at the Center of the “Universe”.
2. “Perfect Heavens”: The motions of all
celestial bodies can be described by
motions involving objects of “perfect” shape,
i.e., spheres or circles.
Ptolemy
He was GrecoRoman writer who
lived in Alexandria.
 Lived between 90 168 AD
 One of the few
surviving text about
ancient astronomy

• Ptolemy: Geocentric model, including epicycles
Central guiding principles:
1. Imperfect, changeable Earth,
2. Perfect Heavens (described by spheres)
What were the epicycles in Ptolemy’s
model supposed to explain?
1.
2.
3.
4.
5.
The fact that planets are moving against the
background of the stars.
The fact that the sun is moving against the background
of the stars.
The fact that planets are moving eastward for a short
amount of time, while they are usually moving
westward.
The fact that planets are moving westward for a short
amount of time, while they are usually moving
eastward.
The fact that planets seem to remain stationary for
substantial amounts of time.
Epicycles
Introduced to explain retrograde
(westward) motion of planets
The ptolemaic system was considered
the “standard model” of the Universe
until the Copernican Revolution.
The Copernican Revolution
Nicolaus Copernicus (1473 – 1543):
Heliocentric Universe (Sun in the Center)
Johannes Kepler (1571 – 1630)
• Used the precise observational
tables of Tycho Brahe (1546 –
1601) to study planetary motion
mathematically.
• Found a consistent description
by abandoning both
1. Circular motion and
2. Uniform motion.
• Planets move around the sun on elliptical paths,
with non-uniform velocities.
New (and correct) explanation for
retrograde motion of the planets:
Retrograde
(westward)
motion of a
planet occurs
when the Earth
passes the
planet.
This made
Ptolemy’s
epicycles
unnecessary.
Described in Copernicus’ famous book “De Revolutionibus
Orbium Coelestium” (“About the revolutions of celestial objects”)
Galileo Galilei (1564-1642 A.D.)
- Founder of Modern Mechanics and Astronomical use of the
Telescope
Proved Aristotle Wrong
1. Many more stars too faint to be seen with eye
2. Moon has mountains and craters like Earth….
Earth and Space made of same material
3. Discovered imperfection in Sun (SUNSPOTS)…Sun is not
perfect
Provided more evidence for a Heliocentric Solar System (Venus
exhibits a full cycle of phases which is only possible in a Heliocentric
system + Jupiter appears as a mini solar system which means that
Kepler’s Laws apply for all planets)
Galileo’s Sketch of the Moon Using a Telescope
Isaac Newton (1643 - 1727)
• Adding physics interpretations to
the mathematical descriptions of
astronomy by Copernicus,
Galileo and Kepler
Major achievements:
1. Invented Calculus as a necessary tool to solve
mathematical problems related to motion
2. Discovered the three laws of motion
3. Discovered the universal law of mutual gravitation
The “Discovery” of Gravity
We’ve all heard the story…
an apple fell on Newton’s
head and he discovered
gravity.
 Most scholars believe that
Newton did see an apple fall
and it got him wondering
about the rules for falling
objects.
 He wondered if the force that
pulled the apple down also
affected the Moon.

Remember Newton’s 1st Law
He had already explained that straight-line
motion was perfectly natural and moving in a
circle required a force
 Johannes Kepler had shown that planets and
moons moved in an ellipse.
 Newton wanted to understand what made
them move that way.
 His breakthrough was to explain how the
same rules apply to little things like apples
and big things like the moon.

Making sense of one force and two very
different results
We already learned that the apple will accelerate at
about 9.8 m/s2 downward
 Newton also knew that the moon accelerates
toward the Earth at 0.00272 m/s2

Distance is only part of the story
Distance has a large effect on the force of gravity
(we’ll explore this more in a minute)
 Remember, though F = m • a

 The force of gravity is also affected by the mass
 or more correctly, both masses
Gravitational Force





Gravitational Force is the mutual force of
attraction between particles of matter
This force always exists between two masses,
regardless of the medium that separates them
It is not just between large masses, like the
sun and the Earth.
The chair you are sitting on is attracted to the
person next to you.
However, the force of friction between the
chair and the carpet is so great that you don’t
move.
m1m2
Fg  G 2
r



Newton’s Law of Universal Gravitation is an example of an
inverse-square law
This is because the force decreases the further the two
objects get from each other.
The distance is measured from the center of each mass.

Remember, the Force of gravity (Fgrav) that acts
on an object is the same as that object’s weight
(in Newtons)
Inverse Square Law
4
1/4
9
1/9
16
1/16
Inverse Square Law
At 4d
3d
2d
5d apple weighs
1/9
1/25
1/4
1/16NN
Connecting the two formulas
Example #1
Find the force of gravity between a 30 kg girl and her
10 kg cat if they are 2 meters apart.
m1 = 30 kg
m1m m
2 kg
=
10
2
Fg  G
2
G = 6.67
r x 10-11 N·m2/kg2
r = 2m
Fg  6.67 x10
11
30kg  10kg
2
(2m )
9
Fg  5.0025 x10 N
Example #2
Find the distance between a 0.300 kg billiard ball
and a 0.400 kg billiard ball if the magnitude of the
gravitational force is 8.92 x 10-11 N.
m1  m2 m1 = 0.3 kg
Fg  G
m2 = 0.4 kg
r2
-11 N
=
8.92
x
10
r 2  Fg  G Fm

m
g1
2
r
=
?
m

m
2
r2 G 1
Fg
m1  m2
r G

Fg
6.67 x10 11  0.300  0.400
8.92 x10 11
r  0.2996 m about 30cm
Determine the magnitude of the
gravitational force between a baseball
player with a mass of 100 kg and Earth
(5.98 X 1024 kg), if they are separated by a
distance of 5.38 X 106 m.
[Option 1]
B. [Option 2]
C. [Option 3]
D. [Option 4]
A.
If a large meteor hits the moon,
causing it to get closer to the earth. If
the moon’s orbits the earth at half of
its original radius, would its force be?
Double the original force
B. Half the original force
C. Four times the original force
D. One fourth the original force
A.
The acceleration due to gravity on the
International space station is 8.7 m/s2. If
an 50 kg astronaut stood on a scale what
would it read?
435 N
B. 5.7 N
C. .17 N
D. 0 N
A.
Tides
Newton also used the inverse square
law to explain the tides.
 People had known for centuries that the
moon affects the tides.
 No one until Newton knew how it did
this.

d-R
d
d+R
Which of the two forces:
moon on left mass (m) or
moon on right mass (m)
is stronger and why? Fd-R
Tidal Bulges
Ocean tides are the alternate rising and
falling of the surface of the ocean that
usually occurs in two intervals everyday,
between the hours of 7a.m. to 7p.m.
 It is caused by the gravitational attraction
of the moon occurring unequally on
different parts of the earth.

KEPLER:
the laws of planetary motion
KEPLER’S
FIRST
LAW
KEPLER’S
SECOND
LAW
KEPLER’S
THIRD
LAW
INTERESTING
APPLETS
Johannes Kepler
 Born on December 27, 1571
in Germany
 Studied the planetary motion of
Mars
 Used
HOME
observational data of Brahe
Instruments
 Tyco Brahe
 only compass and sextant
 No telescope – naked eye
HOME
Kepler’s FIRST Law
HOME
 “The orbit of each planet is an
ellipse and the Sun is at one focus”
 Kepler proved Copernicus wrong –
planets didn’t move in circles
Focus
HOME
 Focus – one of two special points on
the major axis of an ellipse
 Foci – plural of focus
 A+B is always
the same on
any point on
the ellipse
KEPLER’S
FIRST
LAW
Kepler’s SECOND Law
HOME
 “The line joining the planet to
the sun sweeps out equal areas
in equal intervals of time”
In Another Words…
HOME
 The area from one time to another
time is equal to another area with
the same time interval
 All of the areas (in yellow and peach)
have equal intervals
of time
KEPLER’S
SECOND
LAW
Acceleration of Planets
HOME
 Planet moves faster when closer to the
sun
 Force
acting on the planet increases as
distance decreases and planet accelerates in
its orbit
 Planet moves slower
when farther
from the sun
KEPLER’S
SECOND
LAW
Kepler’s THIRD Law
HOME
 “The square of the period of any
planet is proportional to the cube
of the semi-major of its axis”
 Also referred to as the Harmonic Law
T²  a³
HOME
 T = orbital period in years
 a = semi-major axis in astronomical
unit (AU)
 Can calculate how long it takes
(period) for planets to orbit if semimajor axis is known
KEPLER’S
THIRD
LAW
Astronomical Unit
HOME
 Astronomical unit – AU
 AU is the mean distance between
Earth and the Sun
 1 AU ≈ 1.5 x 108 km ≈ 9.3 x 107 miles
KEPLER’S
THIRD
LAW
Examples of 3rd Law
HOME
 Calculating the orbital period of 1AU
 T² = a³
 T² = (1)³ = 1
 T = 1 year
 Calculating the orbital period of 4AU
 T² = a³
 T² = (4)³ = 64
 T = 8 years
KEPLER’S
THIRD
LAW
The planet Saturn is located 9.6 AU
from the sun. What is the length of a
year on Saturn?
A. 885 years
B. 4.5 years
C. 30 years
D. 45 years
Comets
 Although Kepler’s
laws were intended
to describe the
motion of planets
around the sun, the
laws also apply to
comets
 Comets are good
examples because
they have very
elliptical orbits
HOME
So what would happen is fell in hole
through the center of the earth?
 Well Lets see!!!
So what would happen is fell in hole
through the center of the earth?
So lets recap
 The deeper you get into the earth, the weaker
the gravity because there is less mass pulling
you down because some is pulling you up.
 The center of the earth has no gravity because
the mass is surrounding you on all sides.
 Once you reach the other side of the earth,
you’d stop and be pulled back toward the
center.
Gravity
Recap:
Newton’s universal law of gravitation:
F = GMm
R2
Acceleration due to gravity:
Acceleration that an object experiences when it a
distance r from the center
g = GM
R2
GPE in a uniform field
When we do vertical work on a book, lifting it onto a
shelf, we increase its gravitational potential energy
(Ug). If the field is uniform (e.g. Only for very short
distances above the surface of the Earth) we can
say...
GPE gained (Ug) = Work done = F x d
= Weight x Change in height
so...
ΔUg = mg∆h
E.g. In many projectile motion questions we assume
the gravitational field strength (g) is constant.
GPE in non-uniform fields
However, as Newton’s universal theory of gravity
says, the force between two masses is not constant
if their separation changes significantly. Also, the
true zero of GPE is arbitrarily taken not as Earth’s
surface but at ‘infinity’.
Ep = 0
If work must be done to
“lift” a small mass from
near Earth to zero at
infinity then at all points
GPE must be negative.
(This is not the same as
change in GPE which
can be + or -)
Lots of positive
work must be
done on the
small mass!
Ep = negative
‘Infinity’
The gravitational potential energy of a mass at any
point is defined as the work done in moving the mass
from infinity to that point.
The GPE of any mass will always be due to another
mass (after all, what is attracting it from infinity?)
Strictly speaking, the GPE is thus a property of the
two masses.
Ug = - GMm
r
E.g. Calculate the potential energy of a 5kg mass at
a point 200km above the surface of Earth.
( G = 6.67  10-11 N m2 kg-2 , mE= 6.0  1024 kg, rE= 6.4  106 m )
E.g. Calculate the potential energy of a 5kg mass at a
point 200km above the surface of Earth.
( G = 6.67  10-11 N m2 kg-2 , mE= 6.0
1024 kg, rE= 6.4  106 m )

Escape speed
If a ball is thrown upwards, Earth’s gravitational field does
work against it, slowing it down. To fully escape from Earth’s
field, the ball must be given enough kinetic energy to enable
it to reach infinity.
The escape speed is the minimum launch speed needed
for a body to escape from the gravitational field of a larger
body (i.e. to move to infinity).
Loss of KE
= Gain in GPE
½ mv2 = GMm
r
So...
but…
(Note this also = Vm)
so…