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In the figure, find the algebraic moment about point π, given that the force has a magnitude of 14 newtons.
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We recall that the moment of a force is a measure of its tendency to cause a body to rotate about a specific point.
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In order to calculate the moment, we multiply the force by the perpendicular distance from the point at which it is trying to rotate.
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In this question, we will begin by calculating the distance π.
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We notice that this forms a right triangle and we can therefore use the Pythagorean theorem.
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The two shorter sides of our triangle have length four root three centimeters and 10 centimeters.
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Since π is the hypotenuse, we have π squared is equal to four root three squared plus 10 squared.
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Four root three squared is 48, and 10 squared is 100.
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This means that π squared is equal to 148.
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We can then take the square root of both sides.
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And since π must be positive, π is equal to the square root of 148.
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This can be rewritten as root four multiplied by root 37.
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π is therefore equal to two root 37.
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The perpendicular distance from π to where the force acts is two root 37 centimeters.
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We know that the force and this distance must be perpendicular.
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And from our diagram, it appears unlikely that the 14-newton force is perpendicular to this distance.
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In order to calculate the component of this force that does act perpendicular to the line, we begin by calculating the angle π.
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This is the angle between the 14-newton force and the perpendicular force.
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We will begin by calculating the angle πΌ from our right triangle as πΌ plus 90 degrees plus π plus 30 degrees must equal 180 degrees, as these four angles lie on a straight line.
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Subtracting 90 degrees and 30 degrees from both sides, we have πΌ plus π is equal to 60 degrees, which means that π is equal to 60 degrees minus πΌ.
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Using our knowledge of the trigonometric ratios in right trigonometry, we see that the opposite side has length four root three centimeters and the adjacent has length 10 centimeters.
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We know that the tan of πΌ is equal to the opposite over the adjacent.
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So in this question, we have tan πΌ is equal to four root three over 10.
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We can then take the inverse tangent of both sides of this equation.
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Ensuring that our calculator is in degree mode, we can type this in, giving us πΌ is equal to 34.715 and so on degrees.
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We can now use this value to calculate angle π.
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Subtracting πΌ from 60 degrees gives us π is equal to 25.2849 and so on degrees.
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We can now use this to create a right triangle with a 14-newton force.
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This will enable us to work out the perpendicular component of the force.
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Once again, we can use the trigonometric ratios.
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We know the hypotenuse, and weβre trying to calculate the adjacent.
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We will therefore use the cosine ratio where the cos of π is equal to the adjacent over the hypotenuse.
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The cos of our angle π is equal to π¦ over 14.
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We can then multiply through by 14 such that the component of the force weβre looking for is equal to 14 cos π.
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We note that this force acts in a clockwise direction about π.
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However, we are told that the positive direction is counterclockwise.
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The moment of this force about π is therefore equal to negative 14 multiplied by the cos of 25.2849 degrees multiplied by two root 37.
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Ensuring that we used the exact value of the angle on our calculator, this gives us negative 154.
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The moment of the force about point π is negative 154 newton meters.