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. Find the work done by the force.
Remember, work is the measure of energy transfer when a force 𝐹 moves an object through a distance 𝑑. We calculate work done by multiplying the force by the distance covered. Now, we should be a little bit careful with the units. Work done is measured in joules. This corresponds to when the force is measured in newtons and the distance is measured in meters. Now, we actually have a force measured in newtons and a distance measured in meters. So, our work done will indeed be in joules.
The problem is, at the moment, we see from our diagram that the force and the displacement are not in the same direction. And so, we’re going to split the force 𝐹 into its horizontal and vertical components. We add a line perpendicular to the displacement, as shown. 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. We’re told that the magnitude of the force is two newtons. So, the measure of the hypotenuse in this triangle is two newtons.
We want to calculate the component of the force that acts parallel to the displacement. So, that’s the opposite side in our triangle. 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. 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. Of course, sin 30 is equal to one-half. So, one-half is equal to 𝑥 over two. Then, by observation or multiplying both sides of our equation by two, we see that 𝑥 is equal to one. And so, the component of the force that is acting parallel to the direction of the displacement is one newton.
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. It’s acting in the opposite direction to which the body is traveling. And so, we can say that the force must be negative one and the distance is 16 or 16 meters. We’re now ready to calculate the work done. It’s negative one times 16, which is negative 16 or negative 16 joules. The work done is negative 16 joules.