Tutorial: Gravity
(From Dr. Toby Smith, University of Washington)
Gravity is one of the four fundamental forces that exist in nature. While it is the weakest of the four forces, it is the
dominate force over the large distances in our solar system, the Galaxy, the Universe! Gravity dictates the orbits of
objects (moons, planets, stars), constrains their internal structure, determines their shape, controls their evolution. The
force of gravity is ubiquitous in astronomy, hardly a lecture will go by without our discussing it in some form. This
tutorial is a simple attempt to introduce gravity and show how we will use it most often in this class.
The classic way to introduce gravity is to imagine a universe
consisting only of two spherical objects with masses M and m. These
masses are separated by a distance d (the distance is measured from
the center-of-mass of the spheres).
These two spheres are attracted to each other by the force of gravity.
The force of gravity is always attractive (there is no anti-gravity). The
force of gravity between two objects (M) and (m) depends of their
respective mass and the inverse of the square of the distance (d)
between them. The force is very strongly dependent on the distance
between the objects.
In a more compact form the force of gravity can be expressed by the formula
shown on the right. The symbol in the middle that looks like a Greek letter alpha
means “proportional to.” This formula can be expressed in English to read: “The
force of gravity if proportional to the product of the two mass divided by the
distance between them squared.” Learn how to work with this equation!
The force of gravity can be more correctly expressed by the equation shown on the
left. The only difference between this equation and the formula above is that the
“proportional to” has been changed to an “equals to” and the constant G has been
added. This constant G is called the universal gravitational constant. G is just a
number (shown on the left), so it does not change the relationships in the equation.
This is the equation you would use if you wanted to calculated the force of gravity
in Newtons (kg m/s2). In this class we will mostly use the simpler formulation for
gravity shown above.
While we will have occasion to talk about two masses separated by a distance in this class, the
format we will most often encounter gravity is show in the figure on the right. A person of mass m
is standing on a planet of mass M. The planet has a radius d. You can then calculate the force of
gravity between the planet and the person. This calculation is very familiar to us. The force of
gravity between you and Earth is called your weight. When you step on the bathroom scale you are
measuring the force of gravity between you and Earth.
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One interesting consequence of this is that your weight depends on the mass of the
Earth and the distance between you and the center of the Earth (again, all objects
should be centered). For example, imagine that you doubled the size of the Earth
but kept the mass of the Earth the same. How would your weight change? Well you
can determine that by using the formulation shown above and a little algebra.
As you can see, increasing your distance from the center of the Earth, decreases your weight. In this case of doubling
the Earth’s radius, your weight would decrease by a factor of four. For example, a 200 pound person on the normal
Earth would weigh 50 pounds on the resized Earth. As an exercise, you should plug other numbers into the gravity
equation and see how your weight would change. Note: You should expect us to ask a questions like this on any
exams.
A recipe for calculating how the relative force of gravity will change given changes in the mass of either (or both)
objects and the distances between them (or the size of the radius) based on the formula given just above:
If a value changes, substitute the factor for the letter in the formula:
Fgravity 
Mm
.
d2
If a value does not change, then substitute a 1 (to keep things straight).
Examples:
1. The mass of the Earth changes by a factor of 4. How does the force between you and the
Earth change relative to what it was originally?
2. The radius of the Earth shrinks to 1/2 of its current value. How does the
force between you and the Earth change?
3. Here’s one for you: Jupiter has approximately 300 times the mass of the
Earth and is 5 times farther away from the Sun. How does the force of gravity
between Jupiter and the Sun compare to the force between the Earth and the Sun?
Notice we put in a “1” for the mass of the Sun since it didn’t change.
Frelative 
Frelative 
1 1
2

4 1
4
12
1
4
 1  4
1
1
4
1
 
2
1 300 300
Frelative 

6
52
25
4. Trick question: Scientists succeed in moving the current-day Earth magically out to the distance of Jupiter.
Assuming you survive the trip, how does your weight on the more distant Earth compare to what it is now? (Your mass
does not change.)
Now work on the two gravity quizzes that follow.
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Two Little Gravity Quizzes
You and the Earth
Your weight is a measure of the force that the Earth exerts on you and that you exert on the Earth. In terms of Newton's
Second Law:
F = mg
where g is the acceleration due to gravity, approximately 9.8 m/s2. In terms of the mass of the Earth, M, your mass, m,
and the radius of the Earth, d (ignoring "big G"), this equation can be written as:
Fgravity 
Mm
where d here represents the radius of the Earth.
d2
This equation is read as the "force is proportional to the mass of the Earth times the mass of the object (you), and
inversely proportional to the radius of the Earth squared. Interestingly, solid spherical bodies act as if all of their mass is
concentrated right at the center! Figure out what would happen in a relative sense in the following scenarios (answers
given at the end of the quiz):
1.
The Earth suddenly doubles in size (radius)
a. Your weight stays the same
b. Your weight doubles
c. You weigh 1/4 as much
d. You weigh 1/2 as much
2.
You eat so much that you now have twice as much mass (Earth stays the same)
a. You weigh twice as much
b. You weigh 4 times as much
c. You weigh the same
d. You weigh 1/2 as much
3.
The Earth is now twice as massive, but your diet leaves you with only 1/2 your original mass
a. You weigh exactly the same as before the changes
b. You weigh 4 times as much
c. You weigh 1/2 as much
d. You weigh 1/4 as much
4.
The Earth balloons to twice its size (radius) and also gains twice its mass
a. Your weight doesn't change
b. You weigh twice as much
c. You weigh 1/2 as much
d. You weigh 4 times as much
5.
The Earth is 1/2 its current size, and twice its current mass.
a. Your weight doesn't change
b. You weigh 8 times as much
c. You weigh 1/8 as much
d. You weigh 1/4 as much
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The Sun and the Earth
We can calculate the gravitational force between the Sun and the Earth by using the mass of each object, Msun and mearth
and the distance between them, d. Again, ignoring "big G," the force is represented by:
Fgravity 
Mm
, where d here represents the distance between the Sun and Earth.
d2
Figure out what would happen in a relative sense in the following scenarios (answers given at the end of the quiz):
1.
The Sun suddenly moves two times farther away
a. The force stays the same
b. The force doubles
c. The force is 1/4 as much
d. The force is1/2 as much
2.
The Sun stays in the same place, but the Earth suddenly has twice as much mass
a. The force is twice as much
b. The force is 4 times as much
c. The force is the same
d. The force is 1/2 as much
3.
The Sun is now twice as massive, but the Earth has only 1/2 its original mass
a. The force is exactly the same as before the changes
b. The force is 4 times as much
c. The force is 1/2 as much
d. The force is 1/4 as much
4.
The Earth moves two times farther away and also gains two times its original mass
a. The force doesn't change
b. The force is twice as much
c. The force is1/2 as much
d. The force is 4 times as much
5.
The Sun moves to 1/2 its current distance, and also gains twice its current mass.
a. The force doesn't change
b. The force is 8 times as much
c. The force is1/8 as much
d. The force 1/4 as much
The answers to both sets of questions: c, a, a, c, b
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