Chapter 8 Universal Gravitation
Quiz 8
Chapter 8 Objectives
• Relate Kepler's laws of planetary motion to
Newton's law of universal gravitation.
• Calculate the periods and speeds of orbiting
objects
• Describe the method Cavendish used to
measure G and the results of knowing G
Chapter 8 Objectives
• Solve problems involving orbital speed and
period
• Relate weightlessness to objects in free fall
• Describe gravitational fields
Chapter 8 Objectives
• Distinguish between inertial mass and
gravitational mass
• Contrast Newton's and Einstein's views about
gravitation
Warm Up
• How do trains steer?
Kepler’s Laws of Planetary Motion
• 1. The paths of the planets are ellipses
• 2. Imaginary line from sun to planet sweeps
out equal areas in equal times.
• 3. Square of the ratio of the periods is equal
to the cube of the ratio of average distances
 TA

T
 B
2

 rA 
 


r 

 B 
3
Ellipses: Two Center of Focus
(One Empty)
Kepler’s 3rd Law
• An asteroid revolves around the sun with a
mean (average) orbital radius twice that of
Earth’s. Predict the period of the asteroid in
earth years.
– Take note that mass doesn’t matter for orbital
period
– Period = 1 complete revolution
Kepler’s 3rd Law
• The moon has a period of 27.3 days and has a
mean distance of 3.90x105 km from the center
of the earth. Predict the mean distance from
Earth’s center that an artificial satellite that
has a period of 1.00 day would have.
– Satellites that are in geostationary orbit have this
period (TV, Weather)
– Military uses satellites not in geostationary orbits
Newton’s Law of Universal Gravitation
• All objects which have mass exert a
gravitational pull and are pulled in turn by
others of mass. The force is equal to
 m AmB 
F  G

2
 d

• Where m is mass (kg) and d is distance (m)
and where G = 6.67 E-11 N*m2/kg2
Newton’s Law of Universal Gravitation
• Force is then proportional to the inverse
square of distance
• Distance
Force
– 1m
– 2m
– 0.5 m
1N
¼N
4N
 m AmB 
F  G

2
 d

• Also note: Distance is between Center of
Masses
Newton’s Law of Gravitation
• If dealing with gravitational field strength
(acceleration of object) is
– Large Mass = Large g
– Small Radius = Large g
g 
Gm E
d
2
• Because g is gravitational field (N/kg)
– All objects accelerate at same rate
– Bigger mass, more force
g 
F
m
Newton’s Law of Universal Gravitation
• If earth began to shrink, buts its mass
remained the same, what would happen to
the value of g on Earth’s surface?
• If earth began to lose mass, but radius stayed
the same, what would happen to the value of
g on Earth’s surface?
Newton’s Law of Universal Gravitation
• Funny Shirt
Newton’s Law of Universal Gravitation
• An astronaut is on the moon.
• a) Can the astronaut pick up a rock with less
effort on the moon?
• b) How will the weaker gravitational force on
the moon’s surface affect the path of the rock
if the astronaut throws the rock?
• C) If the astronaut drops the rock, and it lands
on their toe, will it hurt more or less than on
earth? Explain
Satellites and their Speed
• Satellites are constantly falling towards the
planet they orbit (just as Earth is constantly
falling towards the Sun)
• Inertia (horizontal velocity) + Falling
(centripetal acceleration) leads to orbit
Satellites and their Speed
• To slow, Ac to
large = fall to
Earth
• To fast, Inertia to
large = Leaves
orbit
Satellites and their Speed
• Where v is speed in orbit, r is radius away
from center, G is constant and m is mass of
said planet.
v 
Gm E
r
• How does the mass of the planet and your
radius influence the speed of V necessary to
orbit?
Satellites and their Speed
• Calculate the speed that a satellite shot from a
cannon must have in order to orbit Earth 150
km above the Earth’s surface.
– Radius of Earth = 6.38 x106 meters
v 
Gm E
r
Satellites and their Speed
• Find the speed of Mercury and Saturn around
the sun. Does it make sense that Mercury is
named after a speedy messenger of the gods
and Saturn is named after the father of
Jupiter? Mercury is 5.79 E10 meters from the
sun, Saturn 1.43 E12 meters, the mass of the
sun is 1.99 E30 kg.
v 
Gm E
r
Satellites and their Speed
• The sun is considered to be a satellite of our
galaxy, the Milky Way. The sun revolves around
the center of the galaxy with a radius of 2.2 E20
meters. The period of one revolution is 2.5 E8
years.
• a) Find the mass of the galaxy
• b) Assuming that the average star has the same
mass as the sun, find the number of stars.
• c) Find the speed with which the sun moves
around the center of the galaxy.
Gravity in the Planets
• Explain the trend
in of gravity field
strength seen in
the interior of
planets
Mass revisited again
• Two types of mass
– Inertial Mass : The inertial mass of an object is
measured by applying a force to the object and
measuring its acceleration
• Example: Put a block of ice in the back of a truck.
When you accelerate forward, the ice will slide to the
back of the truck as a result of its inertial mass
(resistance to acceleration)
Mass revisited again
• Gravitational Mass: A measurement of the
attractive force between objects of mass (a
scale can measure this)
– Example: Drive up a hill with that ice at a constant
rate (no acceleration), yet the block slides, due to
gravitational mass.
• Both masses are always in agreement and are
a central point in Einstein’s general theory of
relativity
Mass revisited again
• List three examples of examples of the effects
of gravitational mass and inertial mass.
Mass revisited again
• Einstein said that gravity is not a force, but an
effect on space itself. A mass changes the
space around it. It causes space to be curved,
and other bodies are accelerated by following
the curve of space.
Mass revisited again
• Black holes: So massive, that even light can’t
escape the curve
Question
• The Earth has a radius of 149.6 million km on
average from the sun. Find the following
• a) Period of the revolution of the earth around
the sun
b) Tangential speed of the earth around the sun
• c) Period of the rotation of the earth
• d) Tangential speed of the rotation of the earth
(at the equator). The radius of the earth is 6.38
E6 m.