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๐ด๐ต๐ถ๐ท is a rectangle, in which ๐ด๐ต equals 45 centimeters, ๐ต๐ถ equals 55 centimeters, and ๐ท๐ธ equals 28 centimeters.
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Forces of magnitudes 225, 275, 265, and 135 newtons act along ๐ด๐ต, ๐ต๐ถ, ๐ถ๐ธ, and ๐ธ๐ด respectively.
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If the system of forces is equivalent to a couple, determine the magnitude of the moment of the forces.
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Okay, so here we have this rectangle ๐ด๐ต๐ถ๐ท.
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And weโre told that side length ๐ด๐ต equals 45 centimeters, ๐ต๐ถ is 55 centimeters, and that side length ๐ท๐ธ is 28 centimeters.
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Weโre Also told the magnitude of four forces acting on this rectangle and that this system of forces is equivalent to a couple.
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This means we can effectively replace the four forces with two equal and opposite forces that donโt lie along the same line of action.
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Knowing all this, we want to determine the magnitude of the moment of these four forces.
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To start doing that, letโs clear a bit of space on screen.
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And letโs consider what it means that these four forces are equivalent to a couple.
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It means if we were to draw the lines of action of these four forces, then if we were to pick two intersection points of these lines so that these points included all four lines of action, then we could effectively model all four forces as though they originated from those two points.
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So for example, if we pick the points ๐ถ and ๐ด in our rectangle, then from point ๐ด we can consider our 225-newton force to be acting down and our 135-newton force to be acting to the right, likewise with point ๐ถ where we say our 275-newton force acts to the left and our 265-newton force acts up and to the right.
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Because our system of forces is equivalent to a couple, we can say that the total force acting at point ๐ด is equal and opposite that total force acting at point ๐ถ.
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And actually we can go further than this.
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Since forces can be separated into independent vertical and horizontal components, we can say that the net horizontal force at point ๐ด is equal and opposite that at point ๐ถ, and similarly for the net vertical force at these two points.
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As a side note, we didnโt need to use the points ๐ด and ๐ถ as a model for the origins of our four forces.
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Taken together, these two points meet our condition of overlapping all four lines of action, but, looking back to our rectangle, so did the points, say, ๐ธ and ๐ต.
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Either of these pairs of points would work for analyzing this scenario.
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But anyway, knowing that the total forces at ๐ถ and ๐ด form a couple, our goal is to solve for the moment created by that couple.
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We can recall that the moment created by a couple of forces is equal to two times the force component perpendicular to the distance between where the force is applied and where the axis of rotation is located.
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The idea here is that each of the forces in the couple contributes equally to the moment overall.
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Thatโs why if we solve for this perpendicular component of one of those two forces and then multiply it by the distance ๐, we need only multiply that result by two to solve for the overall moment.
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Using ๐ด and ๐ถ as the points of origin of the two forces in our couple, we can think of those forces as acting on a line like this that joins at the two points.
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So the moment weโre solving for acts about this midpoint of that line.
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In our rectangle, that line and midpoint would look like this.
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So hereโs the idea.
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If we can find the net force thatโs perpendicular to this line acting either at point ๐ด or point ๐ถ, then we will have solved for ๐น perpendicular in this equation.
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We could choose either point ๐ถ or point ๐ด at which to solve for this force.
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And since the forces at point ๐ด act in what we could call the purely vertical and purely horizontal directions, letโs choose that point.
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Our goal then, is to solve for the components of the 225- and 135-newton forces that are perpendicular to this dashed orange line.
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Adding those together will give us ๐น perpendicular here.
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Looking at the right triangle created by our 225-newton force, we can call this interior angle of that triangle ๐ sub one.
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Weโre interested in this angle because the sin of ๐ sub one times 225 equals the perpendicular component of this force.
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Now, if we go back to our original diagram, this angle in that diagram is also equal to ๐ sub one.
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And we see that indeed this is an interior angle in this right triangle drawn in orange.
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The shorter two sides of the triangle have lengths of 45 and 55 centimeters, respectively.
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And by the Pythagorean theorem, we can say then that the hypotenuse has a length of the square root of 45 squared plus 55 squared.
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Thatโs equal to the square root of 5050.
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At this point, we can recall that given a right triangle where ๐ is another interior angle of that triangle, then the sin of ๐ equals the ratio of the opposite side length to the hypotenuse.
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And that means when it comes to our angle of interest, the sin of ๐ sub one, this is equal to 55 divided by the square root of 5050.
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For ๐ sub one, thatโs the opposite side length to the hypotenuse length ratio.
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So then we now have an expression for the perpendicular component of this 225-newton force.
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And we can start to keep a tally of this using a variable weโll call ๐น perpendicular.
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This is exactly that variable we see in our equation for ๐ sub c.
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And we now know that itโs equal to this term plus a second term weโll figure out soon.
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That second term is equal to this component of our 135-newton force.
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To solve for it, we can use a similar approach.
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Letโs say that this angle interior in this right triangle is ๐ sub two.
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On our rectangle, the angle ๐ sub two would look like this.
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And notice that this is the same as this angle here in our triangle.
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Once more, to calculate the component of this force perpendicular to the dashed line, weโll want to use the sin of this angle.
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We want to calculate 135 times the sin of ๐ sub two.
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The sin of this angle equals the opposite side length, 45 centimeters, divided by the hypotenuse.
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We now know the component of the 135-newton force thatโs perpendicular to our dashed orange line.
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This then is the second and final term in our equation for ๐น perpendicular.
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All right, so to find ๐ sub c, the only thing that remains for us now is to solve for this distance ๐ and then multiply it by ๐น perpendicular and two.
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Looking at our original diagram, that distance is half the distance from point ๐ถ to point ๐ด.
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In other words, itโs this distance here.
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We see right away that that equals one-half, our hypotenuse length.
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In other words, ๐ is equal to the square root of 5050 over two.
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Knowing all this, weโre now ready to calculate ๐ sub c.
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Plugging in the values we calculated for ๐น perpendicular and ๐, notice that this leading factor of two will cancel out with this denominator of two and that also this denominator of the square root of 5050 that appears in both terms in ๐น perpendicular will be multiplied by that same value and therefore become equal to one.
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We find then that ๐ sub c equals 225 times 55 plus 135 times 45.
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And this equals 18450.
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We recall that the units of our forces are newtons and the units of our distances in this example are centimeters.
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The moment then created by this system of forces, which is equivalent to a couple, equals 18450 newton centimeters.