In this explainer, we will learn how to use Newton’s law of gravitation to calculate the force due to gravity between two massive objects.

Newton’s law of gravitation is a foundational law of classical physics that describes how objects with mass
are attracted to each other by gravity. This law is a universal principle, meaning it applies to *all* masses in
the universe: it tells us that the forces that govern how an object falls to the ground from a short distance
are the same as the forces that govern the motion of stars and planets.

One of the first things we should note about gravitation is that it is always an *attractive* force. Gravity
causes all masses in the universe to be attracted to all other masses. If we just consider two masses in
isolation, gravity exerts a force on each of them:

We can see that the forces acting on each object point in opposite directions, pulling the two objects directly
together. In other words, the force vectors acting on each mass both lie on a line between the two masses. Technically, we say that the force vectors point along the straight line connecting the two objects’ *centers
of mass*—an object’s center of mass is the average position of all of its mass, and in the case of a spherical
object, this lies in the center. We can also see that the forces acting on each object both have the same
magnitude . In all cases, when considering two masses, gravity exerts
*equal and opposite* forces on both
objects.

### Example 1: Finding the Directions of a Pair of Gravitational Forces

Each of the following figures shows two rocks in outer space. Which figure correctly shows the direction of the gravitational force exerted on each rock?

### Answer

In order to answer this question, we just need to remember that gravity is
*always attractive*. This
means that any pair of masses are attracted to each other by force vectors acting along the line between
the masses—that is, they’re pulled directly together. Looking at the available answer options, we
can see that the only option where this is the case is option D. We can also notice that even though
the rocks look different and may have different masses, the gravitational forces acting on each rock
have the same magnitude, .

Newton’s law of gravitation can be used to calculate the magnitude of the gravitational force of attraction that acts between any two bodies with mass. Let’s start by considering two bodies with masses and , separated by a distance .

Newton’s law of gravitation tells us that the magnitude of the gravitational force acting on each body is proportional to both and and inversely proportional to the square of the distance between them. This relationship can be expressed by the following statement of proportionality:

This proportionality can be “converted” into an *equation* by the
introduction of a constant of proportionality. This constant is known as the
universal gravitational constant, , and it has a value of
m^{3}⋅kg^{−1}⋅s^{−2}. Using this constant, we can write the following equation.

### Equation: Newton’s Law of Gravitation

The constant plays two important roles in this equation. Firstly, it scales the right-hand side of the equation so that the value of the right-hand side matches the size of the gravitational force produced (measured in newtons). Secondly, it ensures that the units on each side of the equation are the same.

Let’s see how we can use this equation to calculate the gravitational force between two objects.

Again, we are going to consider two bodies separated by a certain distance as above. This time we will say each body has a mass of 1 kg, and the two are separated by a distance of 1 m.

Here, we have , , and . So, the gravitational force produced, , is given by

We can pay special attention to how the units work out to give us
newtons. The unit of is m^{3}⋅kg^{−1}⋅s^{−2}. When we multiply by two masses (expressed in kilograms) and divide by a
squared distance (expressed in square metres), we have

We can also notice that the force we have calculated here is very small—far too small to be noticeable in everyday life. In fact, generally, we will only notice significant gravitational forces when we are dealing with relatively large masses separated by relatively small distances. So, let’s try another example with some larger masses.

This time we will consider two people, each with a mass of 100 kg, separated by a distance of 1 m.

Applying Newton’s law of gravitation using , , and gives us

Even though this force is 10 000 times larger than that in the previous example, it is still far too small to be noticeable. This fits with our everyday experience of the world: we would not expect there to be a noticeable gravitational force between two people!

Let’s look at another example with even larger masses.

### Example 2: Calculating the Magnitude of the Gravitational Force between Two Objects

Two objects, A and B, are in deep space. The distance between the centers of mass of the two objects
is 20 m. Object A has
a mass of 30 000 kg
and object B has a mass of
55 000 kg. What is the
magnitude of the gravitational force between them? Use a value of
m^{3}/kg⋅s^{2} for the
universal gravitational constant. Give your answer in scientific notation to two decimal places.

### Answer

We know that gravity will exert force on both objects so that they are attracted together. Specifically, these force vectors will be equal in magnitude but pointing in opposite directions along the straight line connecting the two objects’ centers of mass.

Newton’s law of gravitation tells us that the force exerted on each object is given by

In this case we have , , and . So the gravitational force produced, , is given by

Rounding this to 2 decimal places gives us a final answer of .

Even this force is relatively small despite the objects having masses of several tons each. However, when dealing with objects with very large masses—such as stars, planets, and moons—the forces produced by gravity are much more apparent.

### Example 3: Calculating the Magnitude of the Gravitational Force between Earth and the Moon

Earth has a mass of kg, and the Moon
has a mass of kg. The average distance
between the center of Earth and the center of the Moon is
384 000 km. What is the magnitude of
the gravitational force between Earth and the Moon? Use a value of
m^{3}/kg⋅s^{2} for the
universal gravitational constant. Give your answer in scientific
notation to two decimal places.

### Answer

We know that gravity exerts forces of equal magnitude on both Earth and the Moon. These forces act along the straight line connecting the two objects’ centers of mass, that is, between the center of Earth and the center of the Moon.

We can use this equation to calculate the magnitude of the force :

The question tells us the values , , and . Note that we should convert into the standard unit of length before substituting this into our equation: . So, the gravitational force produced, , is given by

Rounding this to 2 decimal places gives us a final answer of .

In the next example, we see how the equation can be rearranged to find quantities other than the force .

### Example 4: Calculating the Magnitude of the Gravitational Force between Saturn and Titan

Titan is the largest moon of Saturn. It has a mass of
kg. Saturn has a mass of kg. If the magnitude of the gravitational force between them is
N, what is the distance
between the centers of mass of Saturn and Titan? Use a value
of m^{3}/kg⋅s^{2} for the
universal gravitational constant. Give your answer in scientific notation to two decimal places.

### Answer

Here we are thinking about the gravitational force that acts between Saturn and its moon Titan. This
time we have been *given* the magnitude of the gravitational force that attracts the two bodies, and we
want to work out the distance between them.

For this problem, we can still use the same equation that describes Newton’s law of gravitation:

Since we want to find the value of , we need to rearrange the equation to make the subject. We can start by multiplying both sides by to give us

Then, we divide both sides by :

Finally, we take the square root of each side, leaving us with on the left-hand side:

Since we know the force as well as each mass, we now just need to substitute these values into the right-hand side of the equation. In this case, we will say is Titan’s mass, equal to kg, and is Saturn’s mass, equal to : which, rounded to two decimal places, gives us a final answer of .

In the next example, we think about how gravitational force varies with distance, and how this relationship can be represented with a graph.

### Example 5: Recognizing How Gravitational Force Varies with Distance on a Graph

Which of the lines on the graph shows how the magnitude of the gravitational force between two objects varies with the distance between their centers of mass?

### Answer

We are looking for the line that shows us how the gravitational force between two objects varies with distance.

This relationship is given by Newton’s law of gravitation:

Since the question makes it clear we are talking about “two objects,” we can assume that they each have constant mass. Let’s think about what this equation shows us when we keep and constant but change the distance, .

Note that we have in the denominator on the right-hand side of the equation. If we were to
*increase* the size of , then the size of would increase too. Increasing the denominator in this way
will *decrease* the size of . Since this is equal to ,
we know that will decrease. In other
words, *increasing* the distance between
two objects will *decrease* the magnitude of the gravitational
force, , between them.

This tells us that we are looking for a line that always has a negative
gradient, that is, one that shows
the force getting smaller as distance increases. This means we can rule out the **blue** and **green** lines,
which both have positive gradients.

All the other lines on the graph—the **purple**, **red**, and **black** lines—show the force decreasing as
the distance increases. We can figure out which line is correct by asking
ourselves what happens to as
approaches zero.

Looking at the equation above, we can see that decreasing will increase . But, more than this, we can see that if we set equal to zero, we run into a problem:

We are dividing by zero! This means that the result
(i.e., ) is “undefined”. So, we know that a graph
of this equation will not show a finite value of when = 0. Looking at the graph again, we can
see that, of our remaining options, the **purple** and **black** lines do not follow this rule: each of these
lines simply intercepts the vertical axis at some finite value of . This leaves us with only one available
answer option: the **red** line.

In fact, Newton’s law of gravitation tells us that the magnitude of the
gravitational force *tends
to infinity* (i.e., it gets bigger and bigger with no limit) as the distance
between two objects gets
closer and closer to zero. Indeed, this is represented by the red line on
the graph: as decreases, the
gradient of the red line gets steeper and steeper, and it never intersects
the -axis. So, we know that
the **red** line correctly shows how gravitational force
changes with distance .

### Key Points

- All masses attract all other masses.
- If we have two objects with masses and
, whose centers of mass are separated by a distance
, the magnitude of the gravitational force of attraction
between them is
given by the equation
where is the universal gravitational constant, equal to
m
^{3}⋅kg^{−1}⋅s^{−2}. Both objects in the pair experience a force of this magnitude. - The gravitational forces between two objects
- always attract,
- act along the straight line connecting the two objects’ centers of mass.