The Direction of Forces in Two Dimensions

When adding forces in two dimensions, it is easy to find the magnitude of the resultant force using the Pythagorean Theorem. The resultant is the hypotenuse of a triangle made by the two forces.

It is also easy to find the direction of the two forces. First, look at the image to the right.

Force A is up and Force B is at a right angle to it (also known as being orthogonal).

Because of the nature of vectors, and because forces are vectors, the image can be redrawn so that the tail of one force originates at the head of the other.

The result of that is to the left, below.

When viewed in this way, it is easy to see that the resultant force is the hypothesis of the two sides, represented by the blue line.

The angle, which is the direction of the force is represented as ø and the sides are labeled O for opposite the angle, and A for adjacent to the angle.

So the green side is adjacent to the angle (it touches) and the gold side is opposite the angle (it does not touch).

Now, through the magic of trigonometry, we can use the fact that we have a right angle to calculate the angle.

The end product of all the math is that you need only divide the magnitude of opposite by the magnitude of the adjacent, and then hit the equal sign, and then hit one more button on your calculator.

Since it is soooo easy, here's the math!

By definition the tangent is a number. Because of the magic of math, for any triangle with one right angle, the sides are always in the same proportion, and are represented as a decimal, such as .7832 or .3231. There used to be books with pages and pages of tables so that the angle could be looked up for any given decimal. (IKR?)

So, the way that gets written is this:

tan(ø) = O/A

Wow.

So if you divide O by A you get a decimal decimal number. Then you go look that up and find the angle. Except, no.

There is a button on your calculator for that!

So, more math. We can solve for ø by doing the division, and then doing the un-tangent process. The actual name is the inverse tangent or (so to be a fuzz more confusing) arctangent. 

So, look at this:

http://www.mathsisfun.com/algebra/trig-inverse-sin-cos-tan.html (2017)

Okay, now for the calculator!

To find the direction of a resultant force, resolve all the forces into two orthogonal forces. Then draw out the triangles. Draw the hypotenuse. Then, you are ready.

First, divide the opposite by the adjacent and hit the equal button.

Now the fun! Find the inverse tangent button! Good luck with that! This is what it looks like on the laptop calculator (You MUST hit the "2nd" button first!):

On the TI-30, you also have to hit the "2nd" button first:

So, after you find the button, you are pretty much there.

Example:

Force A has a magnitude of 5 N Up

Force B has a magnitude of 7 N Right

Find the direction in degrees of the resultant force.

tan(ø) = 5/7

Divide 5 by 7 = .71428

Push the inverse tan button. The angle is 35.54 degrees.

Done.

The method for finding the direction of the resultant force when two orthogonal forces is relatively easy. Once the button is found, it takes only two steps on the calculator. While the theory and foundations come from trigonometry, it is not necessary to fully understand the math to use it.

Adding Forces

When more than one force acts on an object, they magnitude and direction of the forces can be added up. The sum of all the forces on an object is called the net force.

Finding the net force is fairly intuitive for many cases. For instance, if something is pulling with 4 newtons to the left and something else is pulling with 4 newtons to the right, it is easy to visualize that the object will just sit in place, unmoved because the forces are equal.  However, if the force to the left is (for instance) 5 newtons, and the force to the right is less—3 newtons, then it follows logically that the object would move to the left. Just taking a wild guess, you would say that the net force on the object is 2 newtons to the left, and that would be right.

Whenever the forces acting on an object are in the same or opposite direction, adding up the forces is very easy. One direction is considered to be negative, and the other is considered to be positive. Then, the magnitudes are added up. Bam! Done.

So, in order to make it less easy, we need to come up with some fancy science words, so…

When you decide which direction is negative and which is positive, you are actually creating a frame of reference. A frame of reference is the orientation of everything so that you can give numerical values to distances and directions. Of course, there has to be a more complex (and more accurate and complete) definition:

frame of reference

noun

a set of criteria or stated values in relation to which measurements or judgments can be made: the observer interprets what he sees in terms of his own cultural frame of reference.

• (also reference frame)a system of geometric axes in relation to which measurements of size, position, or motion can be made.

At an introductory level, only a simple frame of reference is needed for adding forces. Basically, the only considerations needed are for up versus down and left versus right.  If it is necessary to make things even more complex, then toward versus away or in versus out could be considered.

Now, back to adding forces…

Forces in a Single Direction

When the forces are only in a single dimension (up/down or left/right), then finding the net force is a matter of assigning a positive or negative value to the direction and adding them up. This is so easy, an example will suffice.

EXAMPLE 1:

Force 1: 3 N left

Force 2: 8 N right

Since left and right have opposite directions, they have opposite signs in our simple, 1 dimensional frame of reference (yeah, that got worked in!) Assume that left is negative, because every number line since elementary school.

So, to find the net force, add up all the forces using the appropriate sign based on left and right. (Just in case you missed something, left is negative.)

Fnet = F1 + F2

Fnet = 8N + (-3N)

Remember that if you add a negative it is just subtraction! Also remember that the N is just the symbol for newton and is not a variable or anything annoying.

Fnet = 8N -3N

Fnet = 5N 

Now, back to that frame of reference thing… Since the answer is positive, then the direction of the force is to the right. 

Fnet = 5N Right

EXAMPLE 2:

Force 1: 3 N left

Force 2: 8 N right

Force 3: 4 N right

Since left and right have opposite directions, they have opposite signs in our simple, 1 dimensional frame of reference (yeah, that got worked in!) Assume that left is negative, because every number line since elementary school.

So, to find the net force, add up all the forces using the appropriate sign based on left and right. (Just in case you missed something, left is negative.)

Fnet = F1 + F2 + F3 

Fnet = 8N + (-3N) + 4N

Fnet = 8N -3N + 4N

Fnet = 9N

Now, back to that frame of reference thing… Since the answer is positive, then the direction of the force is to the right. 

Fnet = 9N Right

EXAMPLE 3:

Force 1: 1 N left

Force 2: 8 N left

Force 3: 4 N right

Since left and right have opposite directions, they have opposite signs in our simple, 1 dimensional frame of reference (yeah, that got worked in!) Assume that left is negative, because every number line since elementary school.

So, to find the net force, add up all the forces using the appropriate sign based on left and right. (Just in case you missed something, left is negative.)

Fnet = F1 + F2 + F3 

Fnet = (-1N) + (-8N) + 4N

Fnet = -8N -1N + 4N

Fnet = -5N

Now, back to that frame of reference thing… Since the answer is negative, then the direction of the force is to the left. 

Fnet = 5N left

The same principle apples for up and down. It generally works out to have up be positive and down to be negative, but that is not always true. The frame of reference concept allows you to define how everything works out in a way that best meets the needs of the situation.

Forces in a Two Dimensions

When adding forces in two dimensions, the math becomes much more fun. Or not fun, depending on if you like Pythagoras or not. As in Pythagorean theorem. And if the forces that happen to be in two dimensions are not at right angles, then the math becomes much, more fun. Or less fun, again depending on your point of view (which is somewhat like a frame of reference.)

So, for introductory physics students, keeping the forces at right angles is a reasonable way to dive in. When forces are at right angles, the magnitude of the net force becomes the hypotenuse of a triangle with the legs equal to the forces in the perpendicular directions.

What?

If one force is going up at 3 and one is going right at 4, then the net force (also know as the resultant force) is 5 going up and to the right. For introductory purposes, it will suffice to find the magnitude and forego finding the direction, which would require trigonometry.

 So, how does this look as an example.

Force 1: 3 N right

Force 2: 4 N up

In this case, let the frame of reference say that up and right are positive and down and left are negative.

So, to find the net force, since the two forces are at right angles, it is necessary to use the Pythagorean theorem.

So, the net force becomes:

Fnet=F21+F22‾‾‾‾‾‾‾‾√${\displaystyle }$

NOTE:  

Raising something to the 1/2 (that is .5) power is the same as finding the square root So,

(16)^.5 = 4

or using borrowed symbols,

Fnet = (F1² + F2²)^.5

Fnet = (3² + 4²)^.5

Fnet = (9 + 16)^.5

Fnet = (25)^.5

Fnet = 5

So the magnitude, ignoring the direction is 5 N.

If you have more than one force in the up/down direction and more than one force in the left/right direction—If you have more than one force on any direction line, you need to find the net force in each of those directions first, then use the Pythagorean theorem to find the magnitude of the combined forces that are at right angles.

To say that in a more "mathy" way, combine all the forces until you have only one force on each axis, then use the Pythagorean theorem to find the resulting magnitude of the combined forces.

EXAMPLE:

In the following picture…

B and C would have to be combined first. Then, the resulting force on the "left - right" line would be combined with A using the Pythagorean theorem.

For forces such that:

F1 = F(left-right)
F2 = F(up-down)

A = 3 N Up

B = 5 N Left

C = 9 N , Right

Find the resulting force.

F1 = F(left-right)

F(left-right) = -B + C  

F(left-right) = 5 Left + 9 Right 

F(left-right) = -5 Right + 9 Right 

F(left-right) = 4 Right

F1 = 4 Right

F2 = F(up-down)

F(up-down) = A
F(left-right) = 3 Up

F2 = 3 Up

Find the magnitude of the resulting net force (The units, Newtons in this case, are ignored, but the math works out such that the final answer is in Newtons):

Fnet = (F1² + F2²)^.5

Fnet = (4² + 3²)^.5

Fnet = (16 + 9)^.5

Fnet = (25)^.5

Fnet = 5 N

That's it! No matter how many forces you have on any given axis (up/down or left/right) just combine the forces given until you have one force on each axis, then combine the two perpendicular forces using the Pythagorean theorem

Finding the direction of the resulting net force is simply a matter of using the inverse tangent relationship, and that is covered in another article.

___________________

Content and definitions from:

New Oxford American Dictionary

https://en.wikipedia.org/wiki/Pythagoras