Chapter
Gravitation
7
In this chapter you will:
Learn the nature of
gravitational force.
Solve problems using
Newton’s law of gravitation.
Learn about the two kinds
of mass.
HW 7: Handout
Section
7.1
Planetary Motion and Gravitation
Newton’s Law of Universal Gravitation
In 1666, Sir Isaac Newton began his studies of planetary motion.
Newton found that the magnitude of the force, F, on a planet
due to the Sun varies inversely with the square of the distance,
r, between the centers of the planet and the Sun.
That is, F is proportional to 1/r2. The force, F, acts in the
direction of the line connecting the centers of the two objects.
Section
7.1
Planetary Motion and Gravitation
Newton’s Law of Universal Gravitation
The sight of a falling apple
made Newton wonder if the
force that caused the apple to
fall might extend to the Moon,
or even beyond.
He found that both the apple’s
and the Moon’s accelerations
agreed with the 1/r2
relationship.
Section
7.1
Planetary Motion and Gravitation
Newton’s Law of Universal Gravitation
According to his own third law, the force Earth exerts on the
apple is exactly the same as the force the apple exerts on Earth.
The force of attraction between two objects must be proportional
to the objects’ masses, and is known as the gravitational force.
Section
7.1
Planetary Motion and Gravitation
Newton’s Law of Universal Gravitation
The law of universal gravitation states that objects attract
other objects with a force that is proportional to the product of
their masses and inversely proportional to the square of the
distance between them.
The gravitational force is equal to the universal gravitational
constant, times the mass of object 1, times the mass of object 2,
divided by the square of the distance between the centers of the
objects.
Section
Planetary Motion and Gravitation
Newton’s Law of Universal Gravitation
Force Vs. Mass
According to Newton’s
equation, F is directly
proportional to m1 and
m2.
1
0.9
0.8
0.7
0.6
F  m1m2
Force (N)
7.1
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Mass (kg)
4
5
Section
7.1
Planetary Motion and Gravitation
Inverse Square Law
According to Newton’s
equation, F is inversely
related to the square of the
distance.
Section
7.1
Planetary Motion and Gravitation
Universal Gravitation
Newton’s law of universal gravitation has both direct and inverse
relationships.
F  m1m2
F  1/r2
Change
Result
Change
Result
2m1m2
2F
2r
1/4 F
3m1m2
3F
3r
1/9 F
4m1m2
4F
1/2 r
4F
½m1m2
1/2 F
1/3 r
9F
Section
7.1
Gravitation
Measuring the Universal Gravitational Constant
How large is the constant, G?
The force of gravitational attraction between two objects on
Earth is relatively small. In fact, it took 100 years from the time
of Newton’s work for scientists to develop an apparatus that was
sensitive enough to measure the force kg2of gravitational
attraction.
G = 6.67 x
-11
10
2
Nm
2
kg
Section
7.1
Planetary Motion and Gravitation
Measuring the Universal Gravitational Constant
Video clip: Ch7_4_movanim
Section
7.1
Planetary Motion and Gravitation
Importance of G
Cavendish’s experiment often is called “weighing Earth,”
because his experiment helped determine Earth’s mass. Once
the value of G is known, not only the mass of Earth, but also the
mass of the Sun can be determined.
In addition, the gravitational force between any two objects can
be calculated using Newton’s law of universal gravitation.
Section
7.1
Planetary Motion and Gravitation
Example:
The attractive gravitational force, Fg, between two bowling balls
of mass 7.26 kg, with their centers separated by 0.30 m, can be
calculated as follows:
Section
Gravitation
7.1
Practice Problems
1. What is the gravitational force between two 15 kg packages that
are 35 cm apart?
2. The gravitational force between two electrons that are 1.00 m
apart is 5.54 x 10-71 N. Find the mass of the electron.
3. The Moon’s mass is 7.34 x 1022 kg, and it is 3.8 x 105 km away
from Earth. Earth’s mass is 5.97 x 1024 kg. Calculate the
gravitational force of attraction between Earth and the Moon.
Section
7.1
Planetary Motion and Gravitation
Cavendish’s Experiment
Determined the value of G.
Confirmed Newton’s
prediction that a gravitational
force exists between two
objects.
Helped calculate the mass of
Earth.
Section
7.2
Using the Law of Universal Gravitation
Two Kinds of Mass
Mass is equal to the ratio of the net force exerted on an object to
its acceleration.
Mass related to the inertia of an object is called inertial mass.
Inertial mass is equal to the net force exerted on the object
divided by the acceleration of the object.
Section
7.2
Using the Law of Universal Gravitation
Two Kinds of Mass
The inertial mass of an object is measured by exerting a force
on the object and measuring the object’s acceleration using an
inertial balance.
The more inertial mass an object has, the
less it is affected by any force – the less
acceleration it undergoes. Thus, the
inertial mass of an object is a measure of
the object’s resistance to any type of
force.
Section
7.2
Using the Law of Universal Gravitation
Two Kinds of Mass
Mass as used in the law of universal gravitation determines the
size of the gravitational force between two objects and is called
gravitational mass.
The gravitational mass of an object is equal to the distance
between the objects squared, times the gravitational force, divided
by the product of the universal gravitational constant, times the
mass of the other object.
Section
7.2
Using the Law of Universal Gravitation
Two Kinds of Mass
ch7_6_mov.anim (no audio)
Section
7.2
Using the Law of Universal Gravitation
Two Kinds of Mass
Newton made the claim that inertial mass and gravitational mass
are equal in magnitude. This hypothesis is called the principle of
equivalence. All experiments conducted so far have yielded data
that support this principle. Albert Einstein also was intrigued by
the principle of equivalence and made it a central point in his
theory of gravity.
Section
Section Check
7.1
Question 1
According to the law of universal gravitation, how are force and
distance related?
A. The gravitational force is inversely proportional to mass.
B. The change in gravitational force with distance follows the
inverse square law.
C. Gravitational force and distance are directly proportional.
D. Gravitational force does not depend on distance.
Section
Section Check
7.1
Answer 1
Answer: B
Reason: F = G m1m2
r2
Section
Section Check
7.1
Question 2
If the Earth suddenly lost half of its mass, what would happen to the
gravitational force between it and the moon?
A. The gravitational force would remain the same.
B. The gravitational force would double.
C. The gravitational force would be cut in half.
D. The gravitational force would be four times smaller.
Section
Section Check
7.1
Answer 2
Answer: c
Reason: F = G m1m2
r2
Section
Section Check
7.1
Question 3
Which of the following helped calculate Earth’s mass?
A. Inverse square law
B. Cavendish’s experiment
C. Kepler’s first law
D. Kepler’s third law
Section
Section Check
7.1
Answer 3
Answer: B
Reason: Cavendish's experiment helped calculate the mass of
Earth. It also determined the value of G and confirmed
Newton’s prediction that a gravitational force exists
between two objects.
Section
Section Check
7.2
Question 4
The inertial mass of an object is measured by exerting a force on the
object and measuring the object’s __________ using an inertial
balance.
A. gravitational force
B. acceleration
C. mass
D. force
Section
Section Check
7.2
Answer 4
Answer: B
Reason: The inertial mass of an object is measured by exerting a
force on the object and measuring the object’s acceleration
using an inertial balance.
Section
Section Check
7.2
Question 5
Your weight __________ as you move away from Earth’s center.
A. decreases
B. increases
C. becomes zero
D. does not change
Section
Section Check
7.2
Answer 5
Answer: A
Reason: As you move farther from Earth’s center, the acceleration
due to gravity reduces, hence decreasing your weight.
Circular Motion and Gravitation Test Information
The test is worth 41 points.
True/False – 5 questions, 1 point each
Multiple Choice – 3 questions, 1 point each
Short Answer – 4 questions, 2 points each
Problems – 3 questions with multiple parts, total of 25 points