Help on Calculations with gravity
Calculating the Force of Gravity
Normally we use Fg=mg to calculate the force of gravity where m is mass and g is the gravitational field strength. This
equation is always true BUT g is NOT ALWAYS equal to 9.81 N/kg. The gravitational field strength only equals
approximately 9.81 when you are close to the surface of the earth. When you move farther away form the earth the
gravitational field weakens and so does the force of gravity. In the past we have said that things in space have no force
of gravity but this is actually NOT true. There still is a force of gravity form the earth on objects in spaces. It’s just that as
that object gets farther and farther away from earth the strength of the field g and there for the force of gravity
decreases. So how do you calculate the force of gravity on an object when it is far away from the surface if you do not
know the new gravitational field strength? Answer:
The size of the force of gravity is given by this equation:
mm
F  G 12 2
r
Newton’s law of gravity
G is the universal gravitational constant.
G  6.67 x 1011
N  m2
kg 2
m1 is the mass of one of the bodies and m2 is the mass of the other body.
r is the distance between the center of mass of the two bodies.
The value of G is the same everywhere throughout the universe.
Useful tips when calculating Force of gravity:
 Remember Newton’s third says for every force there is an equal and opposite force so when you calculate the
force of gravity on a dog by a planet that same amount of force is on the planet by the dog just in the other
direction
 To enter really big or small numbers in your calculator use the “EE” button for example If want to enter the
universal gravitational constant 6.67x10-11 into my calculator I would press the buttons 6 . 6 7 EE - 1 1. When
you are done your calculator should read “6.67ᴇ-11”
 Don’t forget to square your radius! It’s the #1 calculation mistake with Fg.
 This equation assumes masses are point masses. A point mass is an object that has all it’s mass concentrated at a
single point which will be the center of mass. Assuming this makes the calculations WAY, WAY easier and also
give us the exact same answer as reality (non-point masses)
EXAMPLE Fg Calculation
 A girl, Brandy (42.5 kg), sits 1.50 m from a boy (63.0 kg), George. What is the force of gravity between them?
(This will tell us how attracted they are to each other.)
mm
F  G 12 2
r

N  m 2   42.5 kg  63.0 kg 
F   6.67 x 1011


kg 2 
1.50 m 2

F  7940 x 1011 N

7.94 x 108 N
Defining and calculating Gravitational Field Strength (g)
What the heck is the gravitational field strength? Gravitational field strength is defined as the amount of Force of gravity
at a location per unit mass at that location. It is calculated by the equation:
Drawing field Lines
The gravitational field is a vector and always point’s inward toward the mass
causing it. Conventionally we do not show strength of the field by the length
of the arrow but by how closely packed the arrows are together. You will
notice in the picture below the close to the planet the lines are closer
together because the field is strongest close to a planet and weakens the
farther you get
Point A
Conceptually, what is the gravitational field
Point C
There is a gravitational around all objects with mass. This field can be
Point B
thought of as the object attempt to pull things in but if there is nothing
there then there is nothing to pull on. Look at point B on the picture to
the right. If there is no object at point B then there is no force of
gravity but we still say there is a gravitational field ready to begin the pulling once an object is place there. The
field strength tells you how much force you would have at a location per unit of mass if the was a mass there.
If the field strength at point B is 4.5 N/kg and a 1 kg object is placed there then the force of gravity would be
4.5 N. If a 2 kg object is placed at B then the force is 9N. Even if an object is at point B there is still a
gravitational field.
Useful tips about Gravitational Fields:
 g fields cannot be shielded. (Placing a wall between point B and the planet does nothing to affect the
gravitational field strength from the planet.)
 Multiple g fields form multiple objects can add up to give you a resulting g field . You add these g fields like any
other vector quantity. Therefore it is possible to get a net g field of zero by adding two opposing ones.
 Gravitational fields strength around a single point is radial. This means that all points that are the same distance
from the center of mass have the same field strength. (example: the g field at points B and C are the same but
the field a point A is stronger)
 The gravitational field strength at a location happens to equal an objects rate of acceleration at that location if
AND ONLY IF the gravitational force is the only force acting one the mass at that location
Calculating g fields
You can calculate a g field in two ways find the Fg and then divide by mass
 A 340 kg satellite orbits the Earth (Mearth = 5.97 x 1024 kg) 6.2 x 107 m from the center of the planet. Calculate
the gravitational field strength at the location of the satellite.
satellite
First find the force of gravity on the satellite
Next find the field strength by using
BUT which mass do you divide by; the earth or the satellite?
NOTE: YOU DIVIDE BY THE MASS THAT IS AT THE LOCATION YOU ARE FINDING YOUR G FIELD.
So…
But how do you solve a problem if there is only one mass? ANSWER: do the same thing you did above but just do it
where one mass remains a variable and hopefully it will just go away. For example:
 What is the gravitational field strength 9.7 x 106 m from the center of Earth (Mearth = 5.97 x 1024 kg)?
First find the force of gravity on the satellite
Next find the field strength
You will notice that the answer is real obtained by using the Fg equation but with only one mass. Or we can derive the
equations
This is a useful equation but they
do not give it to you on a test. It is
easy enough to derive on your
own.
Where the mass that disappears is the mass that would normally be at the location of the g field and the mass that
remains in the equation is the mass that is causing the g field.
G field and acceleration
It is instructive to note that in the very special (and common) case where the force of gravity is the ONLY force acting on
an object the gravitational field strength g just happens to equal the acceleration as derived below
or
NOTE: this is ONLY true if the Fg is
the only force on the object.