Chapter 6 Work Done by a Variable Force   So far we have used a constant force for our work calculations, but we can also use a force that varies over time to calculate work. To do this we need to use a little bit of calculus. For a one-dimensional variable force we need to look at the Force vs. Distance graph. We then have to find the area under the curve. An estimate could be found using rectangles and adding them up to find the area. The more rectangles the greater the accuracy. But to get the exact answer you need to find the integral. This equation works for a one-dimensional variable force. If the force is two or three dimensional you must add appropriate integrals.   Example 12 A 5.0 kg block moves in a straight line on a horizontal frictionless surface under the influence of a force that varies with position as shown in the figure. How much work is done by the force as the block moves from the origin to x = 8.0 m? Solution   Example 13 Force ,with x in meters, acts on a particle, changing only the kinetic energy of the particle. How much work is done on the particle as it moves from coordinates (2 m, 3 m) to (3 m, 0 m)? Does the speed of the particle increase, decrease, or remain the same? Solution   Example 14 What work is done by a force ,with x in meters, that moves a particle from a position to a position ? Solution Potential Energy