A lot of the time, we will be given forces in the x and y plane,
and we will need to know how to break those forces into x and y components.
When we break them into such components, they are called rectangular components. So let’s draw
a coordinate system like this and draw a force applied 30 degrees away from the positive x-axis.
We can draw the x and y components of this force like this. Now if we want to calculate what these
values are, we can move one of the components like this to form a right-angle triangle. After that,
you just need to keep in mind trigonometric ratios. So sine is opposite over hypotenuse
and cosine is adjacent over hypotenuse. The hypotenuse becomes the magnitude of the force,
while the sides become the components of the force. When we have multiple forces,
we can break all them into x and y components. Then if we add up each individual x components
and y components, we can figure out the x and y component of our resultant force. In simple terms,
let’s say we have an object like this and many different forces are applied to it.
We can take all of those forces, break it into components, and then we can represent all of these
forces with 2 component forces. These 2 components will have the same exact effect as did all
of those other forces. We can go even further by finding the magnitude of the resultant force. That
way, we can show one single force which accounts for all the other forces that effected the object.
To do that, notice how our x and y components create a right-angle triangle. The magnitude
is simply the hypotenuse of this triangle, so we can use the Pythagorean theorem to figure it out.
All of this will make more sense when we do examples so let’s get started.
Let’s take a look at this problem where we have 2 forces applied to an object. We need to find the
resultant force and it’s direction measured from the positive x-axis. The first step is to break
down all the forces into x and y components. Let’s focus on the 400 N force. Let’s draw the x and y
components of this force. Now we can move one of these components to create a right-angle triangle
like this. You can see that the x-component is the adjacent length and the y-component
is the opposite length with respect to the angle that’s given. So remember, the 400 N force is the
hypotenuse of this triangle. Since we want to figure out the adjacent length, we need to use
cosine, because that’s adjacent over hypotenuse. Let’s write it down. Next, the y-component. For
that, we will use sine since that’s opposite over hypotenuse. So again, what we are looking for is
with respect to the angle. Since we’re trying to find the y component, we see that it’s opposite to
our angle, that is why we used sine. This is not always the case, do not think that y components
are always sine and x components are always cosine, this is a pitfall many students fall into.
Now let’s focus on the 800 N force. Let’s draw the components. First the x-component of the force.
Notice how with the 45 degree angle that’s given, the x-component is actually opposite to the angle.
Which means we need to use sine. So let’s write it down. Next, the y-component. This time, it’s
the adjacent length, so we need to use cosine. The next step is to sum all the x-axis components
and the y-axis components. We will start with the x-axis components, and we will pick up and to the
right to be positive. So we have both x-components of our 2 forces. Since both of the components are
pointing to the right, they are both positive. Let’s simplify. Next, the y-axis components.
Notice how one of our y-components is facing up while the other one is facing down. So our 800 N
force has a negative y-component. Let’s simplify. You can see that we got a negative value. All
it means is that this vector is pointing down, instead of up since we picked up to be positive.
Now the 2 values we found are the x and y components of the resultant force.
To clarify this, let’s draw these vectors. So for the x-component, we have a vector that’s pointing
to the right, and for the y-component, we have a force pointing straight down. Now if we draw our
parallel lines, you can see that the diagonal is indeed our resultant force. To find it,
all we need to do is use this equation, its just the Pythagorean theorem. Let’s plug in our values.
Solving gives us the magnitude of our resultant force. Now we need the angle from the positive
x-axis. That’s easy to figure out using tan. Tan is opposite over adjacent. So let’s write
it down. Opposite is the y-component and the adjacent is the x-component. Let’s solve,
and those are our answers. Let’s take a look at this problem
where we have to find the resultant force and the angle. So as before,
we can draw the components. Now let’s move them so that each forms a triangle. We will first focus
on just force F1. This is exactly like before, so the x-component is opposite to the angle, so
that will be sin. Let’s write it down. Next, the y-component, which is adjacent, so we use cosine.
Now let’s look at force F2. So we aren’t given an angle, instead, we’re give this little slope
triangle to figure out the components. The key here is to realize that this slope triangle
is a similar triangle to the right-angle triangle we drew, so we can use the ratios given to figure
out the components. For the x-components, we use 5/13. Easiest way to see this is to look
for corresponding sides. So notice how on the mini triangle, the side length of 5 corresponds
to the same side as our big triangle. Now for the y-component, we use 12/13. Again, notice how the
12-length side corresponds to the y-component side of our big triangle. If you keep that in mind,
using the slope triangles becomes second nature. Now that we have all the components, let’s find
the resultant force by adding up the x and y components. We will pick up and to the right to
be positive. Both x-components point to the left, so they are both negative since we picked right
to be positive. Next, the y components. Notice how for force F1, we have the y component facing down,
while the y component of force f2 faces up. So remember to use proper positive and negative signs
based on which sides you pick to be positive. Now that we have the components of our resultant
force, we can figure out the magnitude. To find the angle, let’s draw the vector
components. Since both components we found were negative, they face opposite to the way
we picked to be positive. In other words, the x-component faces left and the y component faces
down. Now let’s calculate the angle using tan. This angle we found is measured from the negative
x-axis, and the question wants it to be measured counterclockwise from the positive x-axis.
To do that, all we need to do is add 180 degrees which gives us our answers.
Let’s take a look at this problem where we need to figure out force F and its angle,
so that the resultant force would have a magnitude of 8 kn, and be directed along the x
prime axis. The first step is to break down the forces given to us into x and y components.
Let’s focus on the 4kN force first. So the x component is opposite to the angle,
so it’ll be sine. The y component is adjacent to the angle, so it’ll be cosine.
Now for force F. We don’t have a magnitude or an angle, but that’s alright. We can still break it
into components. The x component is opposite to the angle so it’ll be sine, and the y component
is adjacent to the angle, so it’ll be cosine. We have one more force, which is the 6 kN force,
but that only has an x component since it’s directed along the x-axis.
So in the question, we are told that the magnitude of our resultant force
is 8 kN and it’s 30 degrees above the positive x-axis. If we draw it separately like this,
we can figure out it’s x and y components. The x component is adjacent to the length,
so it’ll be cosine, and the y component is opposite to the angle so it’ll be sine.
Now we can write the sum of our components. Let’s start with the x components.
We will pick up and to the right to be positive. You will notice that unlike the other questions
we covered before, we are working backwards. We already have the x and y components of
our resultant force. So that’s what’s on the left side. On the right side,
we have all the x components of the other forces. Let’s simplify. Now for the summation of the y
components. Let’s simplify this as well. We now have 2 equations with 2 unknowns,
you can solve them any way you like. That gives us our answer.
So remember, we didn’t find the resultant in this problem, what we found was force F so
that our resultant force would have a magnitude of 8 kN and be directed along the x prime axis.
That should cover the types of problems you will face in this chapter. In the next video, we cover
how to break forces in 3 dimensions. Thanks for watching and best of luck with your studies.