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Sometimes we have the x and y components of a force, and we want to find the magnitude and direction of the force.
Let's see how we can do this.
There are three possible cases to consider:
• The two components are both different from zero
If a force F has the x and y components both different from zero, in order to find F we start by roughly representing the components on an xy-plane (based on the magnitude of the components and their sign).
If, for instance, both the components are positive, with the x component slightly larger in magnitude, we would represent them something like this:
Then we draw the rectangle with Fx and Fy as two of the sides:
The diagonal of the rectangle that goes from the origin is the force F:
We can find the magnitude of F, by applying Pythagoras' Theorem:
F = √Fx2 + Fy2
And what about the direction of F?
The direction is often expressed by the direction angle, i.e. the counterclockwise angle that F makes with the positive x axis.
Let's see how we can find it:
First we find θ, the angle F makes with its component Fx:
According to trigonometry:
| θ = tan-1 | Fy |
| Fx |
After we have found θ, we can easily determine the direction angle.
Sometimes θ will already be the direction angle, other times you will need to add θ to 180° or subtract it from 180° etc., it depends in what quadrant your force is.
Check out the exercises below to see some examples.
• One of the two components is equal to zero
Often a force has either the x or y component equal to zero and the other component different from zero.
In that case, the magnitude and direction of the force is equal to the magnitude and direction of the non-zero component:
For example let's assume that a force F has y component zero, and x component >0:
Fx > 0
Fy = 0
If we represent the two components graphically, we should see something like this:
Fy is zero, so we can't actually see it.
It is clear that F will be in the direction of the positive x axis and have the same magnitude as Fx:
F = Fx
On the other hand, if the x component of F is negative,
Fx < 0
Fy = 0
F will be in the negative direction of the x axis, and the magnitude will be the same as that of Fx.
And since Fx is negative, the magnitude will be −Fx (remember a magnitude is always positive), therefore:
F = −Fx
So if Fx is −10N, then F has magnitude 10N.
The same can be shown for a force that has the x component equal to zero, and the y component different from zero.
• The two components are both equal to zero
If both the components are equal to zero, then the force is also equal to zero:
Fx = 0
Fy = 0
↓
F = 0
To test your understanding, make sure to do the exercises below.
Exercises
#1
The x component of a force is −7.0N, the y component is 0N. Find the magnitude and direction of the force.
Solution
Fx = −7.0 N
Fy = 0 N
F will be in the negative x direction, and have the same magnitude as the x component:
F = −Fx = 7.0 N
It is −Fx because Fx is negative, and the magnitude must be positive.
#2
Find a force knowing that its x and y components are 50.0N and 21.2N respectively.
Solution
We first roughly represent Fx and Fy on an xy-plane, and from that we draw the rectangle and F.
Fx = 50.0 N
Fy = 21.2 N
Let's find the magnitude of F applying Pythagoras' Theorem:
F = √Fx2 + Fy2
F = √50.02 + 21.22 N
F = √2949 N
F = 54.3 N
Next we find θ:
| θ = tan-1 | Fy |
| Fx |
| θ = tan-1 | 21.2 N |
| 50.0 N |
θ = tan-1 0.424
θ = 23.0°
In this case θ is already the direction angle of F. Indeed θ is the counterclockwise angle that F makes with the positive x axis.
Therefore the force has magnitude 54.3N and the direction angle is 23.0°.
#3
Assuming that a force has the x component −387N and the y component −532N, find magnitude and direction of the force.
Solution
Fx = −387 N
Fy = −532 N
Let's determine the magnitude of F:
F = √Fx2 + Fy2
F = √(−387)2 + (−532)2 N
F = 658 N
And then θ:
| θ = tan-1 | Fy |
| Fx |
| θ = tan-1 | −532 N |
| −387 N |
θ = tan-1 1.37
θ = 53.9°
We need the direction angle of F, i.e. the counterclockwise angle F makes with the positive x axis.
Looking at the xy-plane above, we see that we just need to add 180° to θ. Therefore the direction angle of the force will be 53.9°+180°=233.9°:
Hence, the magnitude is 658N and the direction angle is 233.9°.
#4
Find F knowing that Fx is −9.48N and Fy 5.67N.
Solution
Fx = −9.48 N
Fy = 5.67 N
The magnitude of F will be:
F = √Fx2 + Fy2
F = √(−9.48)2 + (5.67)2 N
F = √122 N
F = 11.0 N
And θ:
| θ = tan-1 | Fy |
| Fx |
| θ = tan-1 | 5.67 N |
| −9.48 N |
θ = tan-1 (−0.598)
θ = −30.9°
Since the tangent is negative, θ came out negative. But we are just interested in the magnitude, so we ignore the minus sign:
θ = 30.9°
The direction angle of F will be 180°−θ (look at the figure above), i.e. 180°−30.9°=149.1°:
#5
F has the following components: 0N in the x direction, and 8.3×102N in the y direction. Determine magnitude and direction of F.
Solution
Fx = 0 N
Fy = 8.3 × 102 N
F will be in the direction of the positive y axis, and have the same magnitude as Fy:
F = Fy = 8.3 × 102 N
#6
Fx is 0.41N, Fy is −0.80N. Find F.
Solution
Fx = 0.41 N
Fy = −0.80 N
Let's first determine the magnitude of F:
F = √Fx2 + Fy2
F = √(0.41)2 + (−0.80)2 N
F = √0.808 N
F = 0.90 N
Next let's find θ:
| θ = tan-1 | Fy |
| Fx |
| θ = tan-1 | −0.80 N |
| 0.41 N |
θ = tan-1 (−1.95)
θ = −63°
θ is negative because the tangent is negative. But we are just interested in the magnitude of the angle, so we can ignore the minus sign:
θ = 63°
Looking carefully at the xy-plane above, we can see that the direction angle of F is θ subtracted from 360°, i.e. 360°−63°=297°:
You may also want to read:
- How to decompose a force into x and y components
- Solving problems which involve forces, friction, and Newton's Laws: A step-by-step guide
- 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)
