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Question Video: Finding the Ratio between Three Forces Acting Parallel to the Sides of a Right-Angled Triangle given the System Is in Equilibrium Mathematics

In the figure, three forces of magnitudes ๐นโ‚, ๐นโ‚‚, and ๐นโ‚ƒ 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 ๐นโ‚ : ๐นโ‚‚ : ๐นโ‚ƒ.

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Video Transcript

In the figure, three forces of magnitudes ๐น sub one, ๐น sub two, and ๐น 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 ๐น sub one to ๐น sub two to ๐น 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 weโ€™re going to begin by representing the three forces in our question using a triangle. Weโ€™re going to take these in order, so letโ€™s begin with force ๐น sub one. Then, ๐น sub two is perpendicular to ๐น sub one. So we can add that force to our diagram, noting that we must start at the terminal point of ๐น sub one. Then, we add ๐น sub three starting at the terminal point of ๐น sub two to complete our triangle. This is a right triangle since we said that ๐น sub one and ๐น sub two are perpendicular to one another.

We might also notice that the force ๐น sub one is parallel to the side in our original triangle measuring 87 centimeters. ๐น sub two is colinear to the side measuring 208.8 centimeters. And then thereโ€™s a shared side represented by this ๐น 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. Theyโ€™re 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 ๐น sub one to ๐น sub two to ๐น sub three, weโ€™re going to find the ratio of the lengths of the sides in this triangle. Letโ€™s find the length of the third side then. Weโ€™ll label it ๐‘ฅ centimeters. Since this is a right triangle, we can use the Pythagorean theorem to find the length of ๐‘ฅ. For a right triangle whose longest side is ๐‘ units, the Pythagorean theorem says that ๐‘Ž squared plus ๐‘ squared equals ๐‘ squared. In this case, our hypotenuse is ๐‘ฅ centimeters. So the Pythagorean theorem gives us 87 squared plus 208.8 squared equals ๐‘ฅ squared. Evaluating the left-hand side of this equation and we find that thatโ€™s equivalent to 51166.44. To find the value of ๐‘ฅ 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, weโ€™re trying to find the ratio of the forces ๐น sub one to ๐น sub two to ๐น 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 ๐น sub one to ๐น sub two to ๐น sub three to be equivalent to 87 to 208.8 to 226.2. Dividing each of these numbers by a rather unusual shared factor, thatโ€™s 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 ๐น sub one to ๐น sub two to ๐น sub three is five to 12 to 13.

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