What is the Resultant Force and How to Find it (with Examples)
In this article, you will learn what the resultant force (also known as net force) is, and how to find it when an object is subject to parallel forces as well as non-parallel forces with the help of examples.
What is the resultant force?
When an object is subject to several forces, the resultant force is the force that alone produces the same acceleration as all those forces.
For example, if 4 forces act on a block and cause it to accelerate 1 m/s^{2} south, then the resultant force is the force that, if applied alone to the block, will also make it accelerate 1 m/s^{2} south.
The reason why the resultant force is useful is that it allows us to think about several forces as though they were a single force. This means that to determine the effect that several forces have on an object, we only need to determine the effect that a single force has.
How to find the resultant force?
If we know the mass m of an object and the acceleration a produced by the forces that act on it, we can find the resultant force using Newton's Second Law. Indeed, according to Newton's Second Law, the force F that alone produces the acceleration a on an object of mass m is:
This force F is our resultant force. So, we can write:
Which indicates that the resultant force R has the same direction as a, and has magnitude equal to the product ma.
For example, if a box of 1.5kg is subject to 5 forces which make it accelerate 2.0m/s^{2} north-west, then the resultant force is directed north-west and has the magnitude equal to 1.5kg × 2.0m/s^{2} = 3.0N.
Often, however, we know the forces that act on an object and we need to find the resultant force.
Experiments show that when an object is subject to several forces, F_{1}, F_{2}, ..., the resultant force R is the vector sum of those forces:
Notice that this is not a mere sum of the magnitudes of the forces, but the sum of the forces taken as vectors, which is more involved because vectors have both a magnitude and a direction that we need to consider when doing the sum.
According to the above equation, if an object is subject to no forces, then the resultant force is zero, and if an object is subject to only one force, then the resultant force is equal to that force. These two cases are pretty simple, but what about an object subject to two or more forces? How do we perform the vector sum then?
To explain this clearly, we will now go through all the cases that can happen, from simple ones in which all the forces are parallel, to more complex ones in which the forces are not parallel, and show how to find the resultant force in each of them with the help of examples.
• Two forces acting in the same direction
Let's start with the simple case in which an object is subject to two forces that act in the same direction:
The resultant force is in the same direction as the two forces, and has the magnitude equal to the sum of the two magnitudes:
• Two forces acting in opposite directions
Let's consider the case in which an object is subject to two forces that act in opposite directions.
If the two forces are equal in magnitude:
The resultant force will be zero because two opposite forces cancel each other out.
On the other hand, if the two forces are not equal in magnitude:
The resultant force will be in the same direction as the force with the larger magnitude (the 5N force in the example), and have the magnitude equal to the difference between the magnitudes of the two forces (in the example that would be 2N):
• More than 2 forces parallel to one another
Let's now consider the case in which an object is subject to more than two parallel forces:
To find the resultant force in this case, we first sum all the forces that go in one direction, and then all the forces that go in the other direction:
At this point, we have two forces that are in opposite directions, which is a case that we already know how to solve: the resultant force has the same direction as the force with the larger magnitude (the 11N force), and its magnitude is equal to the difference between the two magnitudes (4N):
• Two forces that are not parallel
In the previous cases, we have forces that are all parallel to one another. It's time to consider the case in which an object is subject to two forces that are not parallel.
For example, let's assume that we have a block subject to two forces, F_{1} and F_{2}.
F_{1} has magnitude 50N and is applied at a 45° angle, whereas F_{2} has magnitude 60N and is applied horizontally, as shown in the free-body diagram below:
How do we find the resultant force R in this case?
The first step is to draw coordinate axes on our free-body diagram.
Since one of the two forces is horizontal, for convenience, we choose the x-axis horizontal, and the y-axis vertical, and we place the origin at the center of our block:
The next step is to determine the x and y components of all the forces that act on the block:
Now comes the important part:
If we sum all the x components, we will get the x component of the resultant force:
Similarly, if we sum all the y components, we will get the y component of the resultant force:
At this point, we know the x and y components of R, which we can use to find the magnitude and direction of R:
The magnitude of R can be calculated by applying Pythagoras' Theorem:
The angle θ that R makes with R_{x} can be calculated using trigonometry:
θ = tan^{-1} | R_{y} |
R_{x} |
θ = tan^{-1} | 35N | = 20° |
95N |
Thus, the resultant force R has magnitude 100N and direction angle of 20°.
• More than 2 non-parallel forces
Finally, let's examine the case in which an object is subject to more than two non-parallel forces.
For example, suppose we have an object that is subject to three forces, F_{1}, F_{2}, and F_{3}.
The magnitude of each force is shown below:
The free-body diagram of the object looks like this:
We can find the resultant force R using the same process that we used in the previous case of two non-parallel forces.
First, we draw the coordinate axes on our free-body diagram:
Then, we determine the x and y components of the individual forces:
Again, the x component of the resultant force R is the sum of all x components:
Similarly, the y component of R is the sum of all y components:
Finally, let's calculate the magnitude and direction of R using its two components R_{x} and R_{y}:
θ = tan^{-1} | R_{y} |
R_{x} |
θ = tan^{-1} | −15N | = 56° |
−10N |
To express the direction of R, we need to calculate the direction angle (i.e. the counterclockwise angle that R makes with the positive x-axis), which in our case is 180° + θ, i.e. 236°.
The process that we used in this case and in the previous one to find the resultant force when the forces are not parallel can also be used when all the forces are parallel. In fact, it can be used in any case – it's a generic process. However, in the cases of parallel forces, we recommend using the much simpler processes that we described before.
Here's a quick summary of the generic process:
- Draw a free-body diagram of the object
- Draw coordinate axes on the free-body diagram
- Decompose the forces acting on the object into x and y components
- Calculate the x and y components of the resultant force by adding the x and y components of all forces
- Finally, find the magnitude and direction of the resultant force by using its x and y components
A note on drawing coordinate axes on a free-body diagram: we recommend you to draw them so that one of the axes is in the same direction as the acceleration of the object. For example, if you have an object accelerating up a ramp, you should draw tilted coordinate axes with the x-axis uphill. Sometimes, however, your object may be at rest or you may not know the direction of the acceleration. In that case, place the coordinate axes so that as many forces as possible are parallel to them since this will simplify the expressions for their components.
To test your understanding, do the exercises below.
Exercises
#1
John and Rob are engaged in a tug of war. John is pulling with a force of 230N, and Rob is pulling with a force of 215N. Determine the magnitude and direction of the resultant force.
Solution
15N towards John
#2
A car of 1400kg is subject to multiple forces which produce an acceleration of 3.5m/s^{2} directed north. Find the net force.
Solution
The direction of F_{net} is the same as that of a (north), and the magnitude is:
#3
A block is pulled by two forces of 15N and 25N to the left, and by three forces of 10N, 20N, 30N to the right. Find the magnitude and direction of the resultant force.
Solution
If you sum the forces pulling to the left, you get 40N to the left, and if you sum the forces pulling to the right, you get 60N to the right.
Thus, the resultant force is 20N to the right.
#4
An apple is subject to two vertical forces: one of 40N pulling upward, and the other of 10N pulling downward. What is the net force acting on the apple?
Solution
30N upward
#5
A box of 1.0kg is in free fall (i.e. moving subject only to the force of gravity). Calculate the net force.
Solution
The net force is equal to the force of gravity because the box is subject only to that one force. Therefore, the direction is downward, and the magnitude is:
#6
A tugboat is horizontally pulled by two forces of 1450N, each making an angle of 20° with the long axis of the tugboat, as shown in the figure (the view is from the above):
Assuming there is no friction, what is the magnitude and direction of the resultant force acting on the tugboat?
Solution
#7
A ball is subject to two forces F_{1} and F_{2}. The magnitudes of the two forces are 45.0N and 70.0N respectively. In the figure below you can see the free-body diagram of the ball:
Find the magnitude and direction of the resultant force acting on the ball.
Solution
#8
An empty box is pulled by two men with horizontal forces, as shown below (the view is from the above):
Assuming that F_{1} is 345N and F_{2} is 458N, and there is no other horizontal force acting on the box, find the magnitude and direction of the resultant force.
Solution
Problems with solutions
To further test your understanding of resultant forces, see our force problems, which include problems where you need to find the resultant force acting on objects that move horizontally, move up an incline, and hang from pulleys. For each problem, we provide a step-by-step guide on how to solve it.