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It is often useful to decompose a force into x and y components, i.e. find two forces such that one is in the x direction, the other is in the y direction, and the vector sum of the two forces is equal to the original force.
Let's see how we can do this.
Suppose we have a force F that makes an angle of 30° with the positive x axis, as shown below:
And we want to decompose F into x and y components.
The first thing we need to do is to represent the two components on the xy-plane. We do this by dropping two perpendiculars from the head of F: one to the x axis, the other to the y axis.
Like this:
And we join the origin of the xy-plane with the x-intercept to represent the x component of F:
And again, we join the origin with the y-intercept to represent the y component of F:
Fx and Fy are two vectors, i.e. they both have a magnitude and a direction. However, since Fx and Fy are in the directions of the x and y axes, they are commonly expressed by the magnitude alone, preceded by a positive or negative sign: positive when they point in the positive directions, and negative when they point in the negative directions of the x and y axes.
In our example Fx and Fy are positive because both point in the positive directions of the x and y axes.
The positive values of Fx and Fy can be found using trigonometry:
Fx = F cos 30°
Fy = F sin 30°
To keep it simple, just remember that if a component is adjacent to the angle, then it is cos, otherwise it is sin.
Often Fx will be the component adjacent to the angle, so it will be cos, and Fy will be sin.
Let's now consider a force that has one of its components negative:
In this case Fx is negative because it points in the negative direction of the x axis.
Therefore:
Fx = −F cos 15°
Fy = F sin 15°
Notice the minus sign before Fcos15° which we have added to make Fx negative.
You have to be very careful if your angle is not between 0° and 90°, because the sin or(and) cos of that angle may be already negative, so the product is also negative and you don't need to add a minus sign.
To be on the safe side, we recommend to always work with angles between 0° and 90°, so that the sin and cos are always positive, and therefore the product is also always positive.
The bottom line
We can summarize the process of decomposing a force F, as follows:
- Represent the x and y components of the force on the xy-plane by dropping perpendiculars from the head of the force to the x and y axes, and then joining the origin of the xy-plane with the two intercepts (the goal of graphically representing the components is to help you see which component is adjacent to the angle and what the signs of the two components are).
- Find the values of the x and y components: the component adjacent to the angle will be Fcosθ and the other will be Fsinθ. Components that point in the negative directions of the x and y axes are negative, therefore you will need to add a minus sign (given that you are working with θ between 0° and 90°, so that Fcosθ and Fsinθ are always positive).
Forces with tail not in the origin
Sometimes a force does not have the tale in the origin of the xy-plane.
For example:
In cases like this, we draw two straight-lines parallel to the x and y axis that pass through the tail of the force, and then we drop two perpendiculars from the head of the force to the straight-lines:
Fx = F cos 30°
Fy = F sin 30°
Forces that are already in the x or y direction
Often we deal with forces that are already in the x or y direction. In that case we can determine the x and y components in a simpler and more intuitive way, without using trigonometry.
If, for example, we have a force F that is in the direction of the positive x axis:
It is obvious that the y component of F is 0, and the x component is positive with magnitude equal to the magnitude of F:
Fx = F
Fy = 0
On the other hand, if we have a force F in the direction of the negative x axis:
Then the y component is again 0, and the x component is negative (because it points in the negative direction of the x axis) and has the same magnitude as F:
Fx = −F
Fy = 0
The same can be shown for forces in the y direction: They will always have x component 0, and y component either positive or negative with magnitude equal to the magnitude of the force.
To verify your understanding of the concept, do the exercises below.
Exercises
#1
A force of 19N is in the direction of the negative x axis. Find the x and y components of the force.
Solution
Fx is negative, and has the same magnitude as the force (19N). Fy is zero.
Fx = −19 N
Fy = 0 N
#2
A force of 114N makes an angle of 67° with the positive x axis. Decompose the force into x and y components.
Solution
Both the components are positive.
Fx = F cos 67° = 44.5 N
Fy = F sin 67° = 105 N
#3
A force makes an angle of 221° with the positive x axis. Assuming the force has magnitude 3.1×103N, find the x and y components.
Solution
Instead of dealing with the 221° angle, we want to deal with the 41° angle (221°−180°) that the force makes with the negative x axis:
As you can see, both Fx and Fy are negative:
Fx = −F cos 41° = −2.3 × 103 N
Fy = −F sin 41° = −2.0 × 103 N
#4
A force of 4.5×105N has the direction of the positive y axis. Determine its components.
Solution
This one is simple. Fx is zero. Fy is positive and has the same magnitude as the force.
Fx = 0 N
Fy = F = 4.5 × 105 N
#5
A 90.0N force makes an angle of 33° with the positive y direction. Calculate the x and y components of the force.
Tip: Since the angle is +33°, it goes counterclockwise from the positive y axis.
Solution
In this case Fy is adjacent to the angle, therefore its magnitude is the force times the cos of the angle, while the magnitude of Fx is the force times the sin.
Also notice that Fx is negative:
Fx = −F sin 33° = −49.0 N
Fy = F cos 33° = 75.5 N
Alternatively you could have considered the 57° angle (90°−33°) which F makes with Fx. That way Fx would have been adjacent to the angle.
#6
A force that has magnitude 3.21×104N makes an angle of −50° with the positive x axis. Determine the components.
Tip: The angle is negative, meaning it goes clockwise from the positive x axis.
Solution
Fx = F cos 50° = 2.06 × 104 N
Fy = −F sin 50° = −2.46 × 104 N
Problems with solutions
To further verify your understanding of force decomposition, see our force problems, which include problems where you need to decompose forces acting on objects that move horizontally, move up an incline, and hang from ropes. For each problem, we provide a step-by-step guide on how to solve it.
You may also want to read:
- How to find the magnitude and direction of a force given the x and y components
- What is the Resultant Force and How to Find it (with Examples)
- What is a Free-Body Diagram and How to Draw it (with Examples)
