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How to decompose a force into x and y components

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:

A force F that makes an angle of 30 degrees with the positive x axis

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:

Representation of the two perpendiculars that go from the head of F to the x and y axis

And we join the origin of the xy-plane with the x-intercept to represent the x component of F:

Representation of the x component of F

And again, we join the origin with the y-intercept to represent the y component of F:

Both the x and y components of F are represented

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 cos30°
Fy = F sin30°

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:

A force F that has a negative x component

In this case Fx is negative because it points in the negative direction of the x axis.

Therefore:

Fx = −F cos15°;
Fy = F sin15°

Notice the minus sign before F cos15° 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:


1. 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).


2. Find the values of the x and y components: the component adjacent to the angle will be F cos θ and the other will be F sin θ. 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 F cos θ and F sin θ 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:

A force with the tail in the 3rd quadrant

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:

Decomposition into x y components of the force with tail in the 3rd quadrant
Fx = F cos30°
Fy = F sin30°

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:

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:

The x component of the force F in the positive x direction
Fx = F
Fy = 0

On the other hand, if we have a force F in the direction of the negative x axis:

A force F that is 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:

The x component of the force F in the negative x direction
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. Then, if you want to see how useful force decomposition is in practice for solving real problems, head to the step-by-step guide for solving force problems.


Exercises


#1

A force of 19N is in the direction of the negative x axis. Find the x and y components of the force.


#2

A force of 114N makes an angle of 67° with the positive x axis. Decompose the force into x and y components.


#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.


#4

A force of 4.5×105N has the direction of the positive y axis. Determine its components.


#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.


#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.


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