In this explainer, we will learn how to use Newtonβs law of gravitation to calculate the force due to gravity between two massive objects.
Newtonβs 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 objectsβ centers of massβan objectβs 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 massesβthat is, theyβre 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, .
Newtonβs law of gravitation can be used to calculate the magnitude of the gravitational force of attraction that acts between any two bodies with mass. Letβs start by considering two bodies with masses and , separated by a distance .
Newtonβs 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 βconvertedβ 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 m3β kgβ1β sβ2. Using this constant, we can write the following equation.
Equation: Newtonβs 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.
Letβs 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 m3β kgβ1β sβ2. 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 smallβfar 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, letβs 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 Newtonβs law of gravitation using , , and gives us
Even though this force is 10βββ000 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!
Letβs 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 30βββ000 kg and object B has a mass of 55βββ000 kg. What is the magnitude of the gravitational force between them? Use a value of m3/kgβ s2 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 objectsβ centers of mass.
Newtonβs 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 massesβsuch as stars, planets, and moonsβthe 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 384βββ000 km. What is the magnitude of the gravitational force between Earth and the Moon? Use a value of m3/kgβ s2 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 objectsβ 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 m3/kgβ s2 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 Newtonβs 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 Titanβs mass, equal to kg, and is Saturnβs 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 Newtonβs law of gravitation:
Since the question makes it clear we are talking about βtwo objects,β we can assume that they each have constant mass. Letβs 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 graphβthe purple, red, and black linesβshow 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 βundefinedβ. 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, Newtonβs law of gravitation tells us that the magnitude of the gravitational force Β 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 m3β kgβ1β sβ2. 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 objectsβ centers of mass.
