Unit Energy Module Work page 1 of 2 The Work Done by a Variable Force In one dimension the work done by a variable force Fx is W In two dimensions the work done by a variable force F is W xf x i Fx dx rf ri F id r The graph on the left shows the force exerted on an object as a function of position The force varies as the object moves What value of force should you use to calculate work If you break the displacement into smaller increments x the force is relatively constant You can sum up the work done over small increments of displacement to approximate the work done from the initial to the final position Your approximation will improve as x becomes smaller For a variable force work is equivalent to the area under the curve Fx between xi and xf Work can be expressed as an integral of the force with respect to displacement between xi and xf W xf x i Fx dx In two dimensions the work done by a variable force rf F is W F id r ri Variable forces are often expressed as polynomials To integrate a polynomial use the formula on the left Remember that when you calculate work you must include the endpoints of the displacement You may also need to integrate simple trigonometric functions such as sine and cosine as shown in the box on the left www thinkwell com info thinkwell com Copyright 2001 Thinkwell Corp All Rights Reserved 1937 doc rev 03 30 2001 Unit Energy Module Work page 2 of 2 The Work Done by a Variable Force The force on a spring is given by Fx kx where k is a constant The farther you pull one end of a spring a greater force is needed to continue to displace it You would like to calculate the work done on a spring as it is stretched from xi to xf The formula for the force on a spring is a polynomial where k is the coefficient and the exponent of the variable x is one According to the formula for integrating a polynomial you add one to the variable s exponent so the new exponent is two You also divide by this new exponent The end points of the integration are xi and xf www thinkwell com info thinkwell com Copyright 2001 Thinkwell Corp All Rights Reserved 1937 doc rev 03 30 2001
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