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The figure shows a force 𝐹 of magnitude two newtons and the displacement 𝑠 equal 16 meters covered by a body acted on by the force.
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Find the work done by the force.
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Remember, work is the measure of energy transfer when a force 𝐹 moves an object through a distance 𝑑.
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We calculate work done by multiplying the force by the distance covered.
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Now, we should be a little bit careful with the units.
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Work done is measured in joules.
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This corresponds to when the force is measured in newtons and the distance is measured in meters.
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Now, we actually have a force measured in newtons and a distance measured in meters.
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So, our work done will indeed be in joules.
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The problem is, at the moment, we see from our diagram that the force and the displacement are not in the same direction.
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And so, we’re going to split the force 𝐹 into its horizontal and vertical components.
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We add a line perpendicular to the displacement, as shown.
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And since we know that angles on a straight line sum to 180 degrees, we see that this included angle in our right-angled triangle is 180 minus 60 minus 90 which is 30 degrees.
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We’re told that the magnitude of the force is two newtons.
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So, the measure of the hypotenuse in this triangle is two newtons.
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We want to calculate the component of the force that acts parallel to the displacement.
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So, that’s the opposite side in our triangle.
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Since we’re trying to calculate the value of the opposite side and we know the hypotenuse, we’re going to use the sine ratio; that is, sin 𝜃 is opposite over hypotenuse.
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If we define the opposite side in this triangle to be 𝑥 newtons, then we can say that sin 30 must be equal to 𝑥 divided by two.
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Of course, sin 30 is equal to one-half.
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So, one-half is equal to 𝑥 over two.
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Then, by observation or multiplying both sides of our equation by two, we see that 𝑥 is equal to one.
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And so, the component of the force that is acting parallel to the direction of the displacement is one newton.
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We need to be really careful though, since the force is acting down and away from the direction of the displacement, this force, one newton, must be acting to the left.
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It’s acting in the opposite direction to which the body is traveling.
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And so, we can say that the force must be negative one and the distance is 16 or 16 meters.
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We’re now ready to calculate the work done.
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It’s negative one times 16, which is negative 16 or negative 16 joules.
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The work done is negative 16 joules.