Let’s talk about vectors, and how we can add force vectors to figure out the resultant force.
You probably know that a vector is simply a physical quantity that has both a magnitude and a
direction. We show vectors using arrows like this. So in this example, we are showing the velocity
of a car. The length of the arrow represents the magnitude of the vector, so the longer the arrow,
the bigger the magnitude and vice versa. The angle is what gives the direction. So in this example,
we see that the arrow is 30 degrees away from the x-axis. In other words, it gives us the direction
of it’s line of action. Now let’s move onto forces. Forces are also shown using arrows.
For example, here, we see 2 forces applied to this object. When we have multiple forces
applied to an object, we can figure out something called the resultant force.
All it means is that it’s the addition of all the vectors represented by a single arrow. So really,
we are taking multiple arrows and showing all of them as 1 arrow. The resultant force shows us
the same effect on the object that the original forces do. We can do this in a few ways but in
this video, we’re going to use something called the parallelogram law of addition. In the next
few examples, we will go over how to use it. But before we do that, to use the parallelogram law
of addition, you should know how to use the sine law and the cosine law, since pretty much all of
it is just solving triangles. Without further ado, let’s get started with some examples.
Let’s take a look at this problem. There are 2 forces applied to the object, and we need
to find the magnitude of the resultant force and its direction. We will use the parallelogram law
of addition to solve this problem. The first step is to draw a line that is parallel to
the 700 N force. Next, a line that’s parallel to the 450 N force. Notice how this creates
a parallelogram. You can see where the name for this law comes from. Now if we draw another line,
starting from the origin to where these 2 lines intersect, that’s our resultant force vector.
It is the diagonal of the parallelogram. Notice also that we have 2 triangles formed. You can
solve this problem using either of the triangles. I am going to pick the top one and move it out.
So one side of the triangle is 450 N and the other side is 700 N. To find the remaining side, we can
use the law of cosines. But to use it, we actually need an angle inside the triangle. The easiest one
to figure out is actually the top angle. If we draw a line parallel to the x-axis at the top,
then that whole angle has to be 60 degrees, because it’s an alternate interior angle. Now we
know that the 700 N force is 15 degrees below the x-axis, which means the angle between the x-axis
parallel line and the 700 N parallel line is 15 degrees at the top as well. So if we subtract
one from the other, we get 45 degrees. Now we can apply the law of cosines. Let’s solve. Just this
value isn’t a complete answer, we need an angle to show the direction of the resultant force
since it’s a vector. So what we need is to find the angle it makes with respect to the x-axis.
So first, let’s find the bottom angle in our triangle. For that, we can again use the law
of cosines. Solving gives us the angle inside the triangle. To get the full angle from the x-axis,
all we need to do is add the 60 degrees. So the answer to our question is both of these parts.
Let’s take a look at this question where we need to figure out the angle between the 2 forces so
that the resultant force would have a magnitude of 800 N. So the first step is to draw the parallel
lines, starting with the 400 N force. Now for the 600 N force. Next we draw our resultant
force to the point of intersection. The goal of this problem is to figure out this theta value.
But notice how this theta value plus the top angle must equal 180 degrees since these are
consecutive interior angles. In other words, the angle at the top is 180 degrees minus theta. Now
let’s bring our triangle out. We can figure out the theta value by writing the law of cosines.
Let’s simplify. If we take the inverse of cosine, then we can solve for theta.
So for the resultant force to have a magnitude of 800 N, the angle has to be 75 degrees.
Let’s take a look at one last example. In this question, we need to figure out
the magnitude of the resultant force and the angle theta if the resultant force is directed
vertically upwards. In other words, we need the resultant force to point up on the y-axis.
So the first step is to draw our parallel lines, starting with our 500 N force. Next,
the 600 N force. Now we can draw our resultant force, which is straight upwards.
We can easily figure out the top angle since the 30 degree angle is an alternate interior angle.
Now we can bring our triangle out. For this example, let’s use the law of sines. Let’s
write it down. Now we can solve for theta. Let’s figure out the last angle left inside
the triangle. Using this,
we can write another law of sines to figure out the resultant force and that’s our answer.
That should cover the types of problems you will face when it comes to finding resultant forces
and the addition of forces. In the next video, we will cover how to break forces in the x and
y plane into x and y components. Thanks for watching and best of luck with your studies!