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How to find the resultant force acting on an object

First of all let's answer the question: What is the resultant force?


When an object is subject to several forces, the resultant force is the force that alone would produce the same effect as all those forces.


How can the resultant force be found?


Experiments show that when an object is subject to several forces F1, F2, ..., the resultant force is the vector sum of those forces. In symbols:

R = F1 + F2 + ...

Therefore, it is not a mere sum of magnitudes of all the forces as you may think, but the sum of all the forces taken as vectors, which is more complex since vectors have both a magnitude and a direction that we need to consider when doing the sum.


In the following sections we will go over the process of finding the resultant force in different cases, starting from simple ones and moving to more complex ones.


• Two forces acting in the same direction

Suppose we have an object subject to two forces that act in the same direction:

Two forces of 3N and 4N pulling to the right

The resultant force will be in the same direction as the two forces, and have magnitude equal to the sum of the two magnitudes:

Resultant force is 7N to the right

• Two forces acting in opposite directions

Let's assume that an object is subject to two forces that act in opposite directions.

In the case where the two forces are equal in magnitude:

Two opposite forces

The resultant force will be zero since the two opposite forces cancel each other out.


On the other hand, if the two forces are not equal in magnitude:

Two forces: 3N to the left, 5N to the right

The resultant force will be in the same direction as the larger force (the 5N force in the example), and have the magnitude equal to the difference between the magnitudes of the two forces (in our example that would be 2N):

Resultant force is 2N rightwards

• More than 2 forces parallel to one another

Let's now consider more than two forces:

Four parallel forces: 3N and 4N to the left, 5N and 6N to the right

To find the resultant force, we first need to sum all the forces that go in one direction, and then all the forces that go in the other direction:

Two forces: 7N to the left and 11N to the right

We have now two forces that are in opposite directions, which is a case we already know how to solve: the resultant force is in the direction of the larger force (the 11N force) and has magnitude equal to the difference between the two magnitudes (4N)

The resultant is 4N to the right

• Two forces that are not parallel

In the previous cases, we had forces that were parallel to one another, but let's now consider a more complex case with two forces that are not parallel.


Suppose we have a block subject to two forces, which we'll indicate with F1 and F2.

F1 has magnitude 50N and is applied at a 45° angle, whereas F2 has magnitude 60N and is applied horizontally, as shown in the free-body diagram below:

Free-body diagram of the block

So 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, it is convenient to choose the x axis in the horizontal direction, the y axis in the vertical direction, and place the origin right in the center of our object:

The free-body diagram of the block with coordinate axes.

Our next step is to determine the x and y components of the individual forces:


All the forces acting on the block have been decomposed into x and y components
F1x = F1 cos45°
F2x = F2
F1y = F1 sin45°
F2y = 0

Pay attention now:

If we sum all the x components, we will get the x component of the resultant force:

F1x + F2x = Rx
F1 cos45° + F2 = Rx
(50N)(cos45°) + 60N = Rx
Rx = 95N

And if we sum all the y components, we will get the y component of the resultant force:

F1y + F2y = Ry
F1 sin45° + 0 = Ry
F1 sin45° = Ry
(50N)(sin45°) = Ry
Ry = 35N

At this point, knowing the x and y components of R, it is easy to find the magnitude and direction of R:

Rx = 95N
Ry = 35N
Components of the resultant force acting on the block, and the force itself

The magnitude of R can be calculated by applying Pythagoras' Theorem:

R = Rx2 + Ry2
R = 952 + 352 N = 100N

The angle θ that R makes with Rx can be calculated using trigonometry:

θ = tan-1 Ry
Rx
θ = tan-1 0.368
θ = 20°

So the resultant force R has magnitude 100N and direction angle 20°.


• More than 2 non-parallel forces

Finally, let's examine the most generic case in which we have more than two non-parallel forces.


Let's assume we have an object that is subject to three forces F1, F2, and F3.

The magnitudes of the three forces are shown below:

F1 = 10N
F2 = 20N
F3 = 40N

And the free-body diagram of the object looks like this:

Free-body diagram of the object

We can find the resultant force R in the same way as we did for the object that was subject to two forces.


First, we need to draw the coordinate axes on our free-body diagram:

The free-body diagram of the object with coordinate axes.

Then, we determine the x and y components of the individual forces:

All the forces acting on the object have been decomposed into x and y components
F1x = F1
F2x = 0
F3x = −F3 cos60°
F1y = 0
F2y = F2
F3y = −F3 sin60°

Again, the x component of the resultant force R is the sum of all x components:

Rx = F1x + F2x + F3x
Rx = F1 − F3 cos60°
Rx = 10N − (40N)(cos60°)
Rx = −10N

And the y component of R is the sum of all y components:

Ry = F1y + F2y + F3y
Ry = F2 − F3 sin60°
Rx = 20N − (40N)(sin60°)
Ry = −15N

Finally, let's calculate the magnitude and direction of R using the two components Rx and Ry:

Rx = −10N
Ry = −15N
Components of the resultant force R, and R itself
R = Rx2 + Ry2 = 18N
θ = tan-1 Ry = 56°
Rx

To express the direction of R, we need the direction angle (i.e. the counterclockwise angle that R makes with the positive x axis), which in this case is 180° + θ, i.e. 236°.


We're done. It wasn't that hard, was it?


Summary

To find the resultant force acting on an object, follow these steps:


  1. Draw a free-body diagram of the object
  2. Draw coordinate axes on the free-body diagram
  3. Decompose the forces acting on the object into x and y components
  4. Calculate the x and y component of the resultant force by adding the x and y components of all forces
  5. Finally, find the magnitude and direction of the resultant force by using its x and y component

A note on drawing coordinate axes: it is good practice to draw them so that one of the axis is in the same direction as the motion of the object. So if you have an object moving 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 motion, in that case place the coordinate axes in the way you think is best (in most cases it's best to have the x axis horizontal and y axis vertical).


To test your understanding, do the exercises below. When you're done, make sure to check our step-by-step guide for solving problems with forces.


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


#2

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.


#3

An apple is subject to two vertical forces: one of 40N pulling upward, and the other of 10N pulling downward. What is the total force acting on the apple?


#4

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):

Tugboat's free-body diagram

Assuming there is no friction, what is the magnitude and direction of the resultant force acting on the tugboat?


#5

A ball is subject to two forces F1 and F2. 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:

FBD of the ball

Find magnitude and direction of the resultant force acting on the ball.


#6

An empty box is pulled by two men with horizontal forces, as shown below (the view is from the above):

The free-body diagram of the empty box

Assuming that F1 is 345N and F2 is 458N and there is no other horizontal force acting on the box, find the magnitude and direction of the resultant force.