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Many examples exist of work done with varying force. These include but are not limited to: a rocket leaving a planet with the force of gravity decreasing as it get farther from the planet, pushing on a spring where the force must increase as the spring is depressed, and a full-back scoring a touchdown as more and more opposing players jump on his back. We can graph the Force parallel to the displacement versus the displacement in the example shown below.
We can divide the displacement into small sections and find the work done in each section. The example above shows the average parallel force in the 10th section and the change in displacement for that section. Their product is an approximation of the work for that section. This can be repeated in all the other sections. The sum of all these sections will be the work for the total displacement. The smaller the /\d, the closer the sum will be to the actual value. If /\d approaches zero, the sum will have no measurable error. If the equation for parallel force is known, calculus allows us to integrate to get the area under the curve.
Math tells us that the area in the above graph is equal to the product of the parallel force and the displacement. Since work is the product of the parallel force and the displacement, the area must equal the work. |
