Video Transcript
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.
