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In the figure, three forces of magnitudes ๐น sub one, ๐น sub two, and ๐น sub three newtons meet at a point.
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The lines of action of the forces are parallel to the sides of the right triangle.
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Given that the system is in equilibrium, find the ratio of ๐น sub one to ๐น sub two to ๐น sub three.
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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.
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So weโre going to begin by representing the three forces in our question using a triangle.
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Weโre going to take these in order, so letโs begin with force ๐น sub one.
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Then, ๐น sub two is perpendicular to ๐น sub one.
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So we can add that force to our diagram, noting that we must start at the terminal point of ๐น sub one.
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Then, we add ๐น sub three starting at the terminal point of ๐น sub two to complete our triangle.
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This is a right triangle since we said that ๐น sub one and ๐น sub two are perpendicular to one another.
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We might also notice that the force ๐น sub one is parallel to the side in our original triangle measuring 87 centimeters.
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๐น sub two is colinear to the side measuring 208.8 centimeters.
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And then thereโs a shared side represented by this ๐น sub three force.
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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.
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Theyโre proportional to one another.
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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.
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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.
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Letโs find the length of the third side then.
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Weโll label it ๐ฅ centimeters.
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Since this is a right triangle, we can use the Pythagorean theorem to find the length of ๐ฅ.
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For a right triangle whose longest side is ๐ units, the Pythagorean theorem says that ๐ squared plus ๐ squared equals ๐ squared.
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In this case, our hypotenuse is ๐ฅ centimeters.
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So the Pythagorean theorem gives us 87 squared plus 208.8 squared equals ๐ฅ squared.
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Evaluating the left-hand side of this equation and we find that thatโs equivalent to 51166.44.
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To find the value of ๐ฅ then, we find the square root of both sides.
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The square root of 51166.44 is 226.2.
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And so the length of the third side in our triangle is 226.2 centimeters.
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Remember though, weโre trying to find the ratio of the forces ๐น sub one to ๐น sub two to ๐น sub three.
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And we said that that will be the same as the ratio of the relevant sides.
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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.
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Dividing each of these numbers by a rather unusual shared factor, thatโs 17.4, and we get five to 12 to 13.
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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.
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The ratio of ๐น sub one to ๐น sub two to ๐น sub three is five to 12 to 13.