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Two parallel forces ๐น one and ๐น two have the same direction, and the distance between their lines of action is 90 centimeters.
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Given that the magnitude of their resultant is 49 newtons and it is 60 centimeters away from ๐น two, find the magnitudes of the two forces, rounding your answer to two decimal places.
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Okay, so in this example, we have these two forces ๐น one and ๐น two.
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And we know that theyโre parallel to one another.
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Here, weโve drawn ๐น one as having a shorter length and therefore a shorter magnitude than ๐น two.
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But strictly speaking, we donโt know that.
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We donโt know which of these two forces is greater, or they might have the same magnitude, but just that they act in the same direction and their lines of action are separated by 90 centimeters.
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Along with this, weโre told that their resultant, the sum of these forces, is 49 newtons.
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And if we were to draw this resultant, weโll call it ๐
, in our sketch, we also know that its line of action is 60 centimeters away from the line of action of ๐น two.
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Given all this, we want to find the magnitudes of ๐น one and ๐น two.
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Because there are two unknowns to solve for, weโll need two independent equations.
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We have one such equation here, and weโll need to find one more.
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This first equation, we can say, has to do with the linear sum of our forces.
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Itโs possible though to also consider the rotational effects of these forces, in other words, the moments they create.
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In general, any force can create a moment ๐ about some point called the axis of rotation.
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The magnitude of that moment equals the component of the force that is perpendicular to a distance between where the force is applied and to that axis.
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In a given system, we can choose an axis of rotation to be at any point.
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For the case of these two parallel forces, letโs say that we locate our axis of rotation right here at the base of the resultant force ๐
.
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Because ๐
is the resultant force, we can say that ๐
times the perpendicular distance from this forceโs line of action to our axis of rotation is equal to the sum of the moments about that same point created by ๐น one and ๐น two.
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If we set up the convention that a counterclockwise moment is positive and therefore a clockwise one is negative, then we can say that our resultant force ๐
multiplied by the perpendicular distance between this forceโs line of action and our axis of rotation is equal to ๐น two times 60 minus ๐น one times 30.
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The reason this value in the left zero is because the line of action of ๐
passes through our axis of rotation.
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We can imagine that axis coming into and out of the screen at this X.
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The whole left-hand side of this equation then is equal to zero.
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On the right, we have ๐น two multiplied by the perpendicular distance between this forceโs line of action and our axis โ and note that this moment is positive because it is counterclockwise around our axis โ minus ๐น one times the perpendicular distance from this forceโs line of action to our axis of rotation.
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Because ๐น one tends to create a clockwise rotation about our axis, its moment is negative.
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This then is our second independent equation involving our two unknowns.
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If we add ๐น one times 30 to both sides of the equation, then we have 30๐น one equaling 60๐น two.
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Then, dividing both sides of the equation by 30, we find that ๐น one equals 60 over 30 or two times ๐น two.
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Looking back at our original sketch, we can now see that the relative sizes of our two force vectors arenโt correct.
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๐น one should actually be twice as long as ๐น two, like this.
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In any case, now that we know that ๐น one equals two ๐น two, we can replace ๐น one in this equation with two ๐น two.
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And so we find that three ๐น two equals 49, or ๐น two equals 49 over three.
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And then since ๐น one is twice this value, we can write that itโs two times 49 over three.
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And these are the magnitudes of our two forces.
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And before we finish, we want to round these answers to two decimal places.
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Entering these two fractions on our calculator, to two decimal places, ๐น two equals 16.33, while ๐น one equals 32.67.
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And both of these forces have units of newtons.
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Our final answer then is that the magnitude of ๐น one is 32.67 newtons and the magnitude of ๐น two is 16.33 newtons.