### Video Transcript

In this lesson, weโll learn how to
solve problems about the equilibrium of a particle under the action of three forces
using the triangle of forces method. By this stage, youโll probably have
worked with systems of forces fairly extensively using the Pythagorean theorem and
right-angled trigonometry to find the magnitude and direction of these forces and
also to split them into their component parts. You might recall that a particle is
in equilibrium if the vector sum of these forces, in other words, their resultant,
is equal to zero. Weโre now interested in what
happens when there are exactly three forces acting at a point.

The key point to remember here is
that when three coplanar forces, in other words, three forces that lie in the same
two-dimensional plane, acting at a point are in equilibrium, they can be represented
in magnitude and direction by the adjacent sides of a triangle taken in order. With that process complete, we can
then use facts about triangles such as trigonometry and similar triangles to solve
problems involving these forces. For instance, letโs imagine we have
three forces ๐น sub one, ๐น sub two, and ๐น sub three acting at a point as
shown. We can represent these
alternatively using a triangle, remembering to take them in order and consider the
direction of each force. We begin by adding the force ๐น sub
one as the first side in our triangle.

Then we move on to force ๐น sub
two. This begins at the terminal or
endpoint of ๐น sub one and travels in the same direction. Finally, we repeat this process
with the third force, ๐น sub three. This time, that begins at the
terminal point of ๐น sub two and travels in the same direction as in our first
diagram. And there we have it, a triangle of
forces. Weโre now going to see how we can
solve problems using this triangle of forces method.

Three coplanar forces ๐น sub one,
๐น sub two, and ๐น sub three are acting on a body in equilibrium. Their triangle of forces forms a
right triangle as shown. Given that ๐น sub one is equal to
five newtons and ๐น sub two is equal to 13 newtons, find the magnitude of ๐น sub
three.

Remember, when three coplanar
forces acting at a point are in equilibrium, they can be represented in magnitude
and direction by the adjacent sides of a triangle taken in order. We actually have the triangle of
forces drawn for us, and we know the magnitude of two of the forces. ๐น sub one is equal to five newtons
and ๐น sub two is equal to 13 newtons. This triangle now represents the
relative magnitude of each of our forces. And since it forms a right
triangle, we can find the magnitude of the third force by using the Pythagorean
theorem. This tells us that, in a right
triangle, the sum of the squares of the two shorter sides is equal to the square of
the hypotenuse. If we let the hypotenuse be equal
to ๐, then we say that ๐ squared plus ๐ squared equals ๐ squared.

In this case, the longest side in
our triangle is the side represented by the 13-newton force. And so, using the magnitudes weโve
been given and letting the magnitude of ๐น sub three be equal to ๐ newtons, we can
say that five squared plus ๐ squared equals 13 squared. That is, 25 plus ๐ squared equals
169. And subtracting 25 from both sides
of this equation, we find ๐ squared is equal to 144. To solve for ๐, we simply need to
find the square root of both sides of this equation. Now, usually we would find both the
positive and negative square root of 144. But since this represents a
magnitude, we know it absolutely must be positive. And so ๐ is equal to the square
root of 144, which is 12. Given that ๐น sub one is five
newtons and ๐น sub two is 13 newtons then, we can say the magnitude of ๐น sub three
is 12 newtons.

In our next example, weโll look at
how to take three coplanar forces acting at a point and form a triangle of
forces.

In the figure, three forces of
magnitudes ๐น sub one, ๐น sub two, and ๐น sub three newtons meet at a point. The lines of action of the forces
are parallel to the sides of the right triangle. Given that the system is in
equilibrium, find the ratio of ๐น sub one to ๐น sub two to ๐น sub three.

We know that when three coplanar
forces acting at a point are in equilibrium, they can be represented in magnitude
and direction by the adjacent sides of a triangle taken in order. So weโre going to begin by
representing the three forces in our question using a triangle. Weโre going to take these in order,
so letโs begin with force ๐น sub one. Then, ๐น sub two is perpendicular
to ๐น sub one. So we can add that force to our
diagram, noting that we must start at the terminal point of ๐น sub one. Then, we add ๐น sub three starting
at the terminal point of ๐น sub two to complete our triangle. This is a right triangle since we
said that ๐น sub one and ๐น sub two are perpendicular to one another.

We might also notice that the force
๐น sub one is parallel to the side in our original triangle measuring 87
centimeters. ๐น sub two is colinear to the side
measuring 208.8 centimeters. And then thereโs a shared side
represented by this ๐น sub three force. Since this is the case, we can say
that the two triangles, that is, the force triangle and the one whose dimensions we
know, must be similar. Theyโre proportional to one
another. We can therefore say that the
magnitudes of the forces in our triangle of forces will be directly proportional to
the lengths of the respective sides in that original triangle.

And so to find the ratio of ๐น sub
one to ๐น sub two to ๐น sub three, weโre going to find the ratio of the lengths of
the sides in this triangle. Letโs find the length of the third
side then. Weโll label it ๐ฅ centimeters. Since this is a right triangle, we
can use the Pythagorean theorem to find the length of ๐ฅ. For a right triangle whose longest
side is ๐ units, the Pythagorean theorem says that ๐ squared plus ๐ squared
equals ๐ squared. In this case, our hypotenuse is ๐ฅ
centimeters. So the Pythagorean theorem gives us
87 squared plus 208.8 squared equals ๐ฅ squared. Evaluating the left-hand side of
this equation and we find that thatโs equivalent to 51166.44. To find the value of ๐ฅ then, we
find the square root of both sides. The square root of 51166.44 is
226.2. And so the length of the third side
in our triangle is 226.2 centimeters.

Remember though, weโre trying to
find the ratio of the forces ๐น sub one to ๐น sub two to ๐น sub three. And we said that that will be the
same as the ratio of the relevant sides. Listing these in the relevant order
and we find the ratio of ๐น sub one to ๐น sub two to ๐น sub three to be equivalent
to 87 to 208.8 to 226.2. Dividing each of these numbers by a
rather unusual shared factor, thatโs 17.4, and we get five to 12 to 13. Alternatively, if we had calculated
87 divided by 226.2 and 208.8 divided by 226.2, we would have found five thirteenths
and twelve thirteenths, respectively. The ratio of ๐น sub one to ๐น sub
two to ๐น sub three is five to 12 to 13.

In this example, we had some
diagram to go from. In our next example, weโll need to
draw the triangle of forces completely.

A body is under the effect of three
forces of magnitudes ๐น sub one, ๐น sub two, and 36 newtons, acting in the
directions of line segments ๐ด๐ต, ๐ต๐ถ, and ๐ด๐ถ, respectively, where triangle
๐ด๐ต๐ถ is a triangle such that ๐ด๐ต equals four centimeters, ๐ต๐ถ equals six
centimeters, and ๐ด๐ถ equals six centimeters. Given that the system is in
equilibrium, find ๐น sub one and ๐น sub two.

We know that when three coplanar
forces acting at a point are in equilibrium, they can be represented in magnitude
and direction by the adjacent sides of a triangle taken in order. So weโre going to represent the
three forces given using a triangle. But weโre told that they act in the
directions of the various sides of triangle ๐ด๐ต๐ถ. So weโll sketch triangle ๐ด๐ต๐ถ
first. Triangle ๐ด๐ต๐ถ looks a little
something like this, and we notice that the sides ๐ด๐ถ and ๐ต๐ถ are both six
centimeters in length. So itโs actually an isosceles
triangle.

Weโll now use this triangle to
sketch a triangle of forces representing ๐น sub one, ๐น sub two, and the 36-newton
force. The force with magnitude ๐น sub one
newtons acts in the direction of line segment ๐ด๐ต. Then the force with magnitude ๐น
sub two acts in the direction of line segment ๐ต๐ถ. Notice that the force with
magnitude ๐น sub two begins at the terminal point of our previous force. And so we have to begin our third
force at the terminal point of ๐น sub two. But we were told that this
36-newton force acts in the direction of the line segment ๐ด๐ถ, not the line segment
๐ถ๐ด. However, since we know that the
magnitude of this force is 36 newtons, we can label it as shown. If we were considering the
direction of the force, we would need to consider that this would be the negative
direction of our original force. But for magnitudes which just
represent size, this is absolutely fine.

Weโre now ready to compare our
triangles. Since each of our forces acts in
the same direction as each side in our triangle ๐ด๐ต๐ถ, the two triangles must in
fact be similar. And so we can say that the
magnitudes of each of our forces must be directly proportional to the lengths of the
sides in triangle ๐ด๐ต๐ถ. So we can find force ๐น sub two
really easily. We know that the sides ๐ด๐ถ and
๐ต๐ถ are equal in length. So this force and this force must
be equal in magnitude. And so ๐น sub two must be equal to
36 newtons. And then we have two different ways
that we can calculate the magnitude ๐น sub one.

One way is to say that the ratio of
the line segment ๐ด๐ต to the ratio of line segment ๐ด๐ถ will be equal to the ratio
of the magnitude ๐น sub one to the magnitude 36 newtons. In other words, four divided by six
will give us the same outcome as ๐น sub one divided by 36. And whilst we could simplify the
fraction four-sixths, it doesnโt make a lot of sense to do this because weโre going
to multiply both sides of this equation by 36. Then we spot that 36 and six have a
common factor of six. So ๐น sub one will be equal to six
times four over one, which is simply equal to 24. And so ๐น sub one is 24
newtons. Itโs worth noting at this point
that we couldโve used scale factor to calculate the value of ๐น sub one.

Since the two triangles are
similar, we can deduce that one is an enlargement or a dilation of the other. And thus, the scale factor for
enlargement would be 36, thatโs one of the dimensions on our force triangle, divided
by six, the corresponding dimension on triangle ๐ด๐ต๐ถ. 36 divided by six is six. And so we can transform any
measurement on our triangle ๐ด๐ต๐ถ onto the dimensions of our force triangle by
multiplying by six. This means ๐น sub one would be
equal to four times six, which is once again 24. ๐น sub one is 24 newtons and ๐น sub
two is 36 newtons.

In our final example, weโll look at
how to use a similar technique to solve a problem involving tension.

A uniform rod of length 50
centimeters and weight 143 newtons is freely suspended at its ends from the ceiling
by means of two perpendicular strings attached to the same point on the ceiling. Given that the length of one of the
strings is 30 centimeters, determine the tension in each string.

Weโre going to begin by drawing a
free-body diagram of this scenario. Letโs define the ends of our rod to
be ๐ด and ๐ต. And so we have our rod suspended
from two perpendicular strings, one of which measures 30 centimeters. Now, in fact, we can work out the
measurement of the second piece of string. Itโs 40 centimeters. And we can calculate that using the
Pythagorean theorem or just recognizing that we have a multiple of a Pythagorean
triple โ three, four, five. Since the rod is uniform, we can
say that the downwards force of its weight acts at a point exactly halfway along the
rod. Now, since the rod exerts a
downwards force on the pieces of string, there will be an opposite reaction force of
tension. Letโs call that ๐ sub ๐ด for the
tensional force in the first bit of string and ๐ sub ๐ต for the tensional force in
the second.

We have our free-body diagram, but
thereโs still an awful lot going on here. So how do we simplify it
further? Well, we know that when three
coplanar forces acting at a point are in equilibrium, they can be represented in
magnitude and direction by the adjacent sides of a triangle taken in order, since
the systems in equilibrium will lay the force vectors end to end to make a
triangle. ๐ sub ๐ด and ๐ sub ๐ต will be
perpendicular to one another, since the pieces of string along which they run are
also perpendicular to one another. But in fact, with a little bit of
inspection, we can see that these are similar triangles. Weโll define the angle between the
rod and the 30-centimeter piece of string to be ๐ฅ degrees. This is equal to the angle between
the 143-newton force and the tensional force at ๐ต.

This isnโt hugely intuitive, but we
can convince ourselves that this is true by adding a line parallel to the 143-newton
force on our first diagram and then using the fact that angles in a triangle sum to
180. Since the angles in our force
triangle and the angles in the triangle represented by the lengths of each item are
equal, we know these triangles are similar. And so we can use scale factor or
ratios to find the magnitudes of ๐ sub ๐ด and ๐ sub ๐ต. Letโs consider the ratio of ๐ sub
๐ด to 143. This must be equal to the ratio of
the 40-centimeter length to the 50-centimeter length. And we chose these pairs of sides
in both our diagrams because weโre interested in the pairs of sides that are
adjacent to the angle measuring 90 minus ๐ฅ degrees.

Multiplying both sides of this
equation by 143 and we find that ๐ sub ๐ด is 40 over 50 or four-fifths times 143,
which is 114.4. So the magnitude of ๐ sub ๐ด is
114.4 newtons. In a similar way, the ratio of ๐
sub ๐ต to 143 newtons will be equal to the ratio of the 30-centimeter length to the
50-centimeter length. To solve for ๐ sub ๐ต once again,
we multiply by 143, which gives us 85.8 or 85.8 newtons. The tensional forces in each string
are 85.8 newtons and 114.4 newtons.

Weโll now recap the key points from
this lesson. Weโve seen that when three coplanar
forces acting at a point are in equilibrium, they can be represented in magnitude
and direction by the adjacent sides of a triangle taken in order. Once we have this triangle of
forces, we can solve problems by using similarity, the Pythagorean theorem, and even
trigonometry.