Work Done by a Variable Force

Where F(x) is the force as a function of distance x and a and b are

the limits of integration of x.

Example: Find the work done to lift a safe weighing 500lb. The cable weighs 50lb/ft and is 100 ft long, and the safe is lifted 80 feet.

The equation for work is above. What we need is an expression for the total weight for F(x). The weight changes, of course, as the cable is wound up during the lifting, so the force is variable.

See if you can write an equation to describe the total weight. You may want to draw a quick sketch and label x first.

F(x) = 500lb + 5lb/ft*(100-x)ft = 1000-5x

Now we put this into the equation for work. What are the limits?