# Work

The work required to raise a weight of P pounds a distance of y feet is foot-pounds. (In m-k-s units, one would say that a force of k newtons exerted over a distance of y feet does newton-meters, or joules, of work.)

Example. If a 100 pound weight is lifted a distance of 3 feet, the work done is 300 foot-pounds.

Example. A 6-foot cable weighing 0.6 pounds per foot is suspended by one end from a reel. How much work is done in winding the entire cable onto the reel?

Consider a small piece of the cable of length which is y units above the bottom end (before the cable is wound up). The weight of the piece is , and it must be raised a distance of feet to get to the reel. The work done in lifting this piece is .

The total work required is

Example. Find the work done in pumping all the water out of the top of a filled cylindrical tank of radius 3 feet and height 6 feet.

The picture on the left shows a side view of the cylinder. I'll cut the cylinder of water up into circular slices; a typical slice is shown on the right.

I'll make the simplifying assumption that to pump the water in the slice out of the tank, I just need to raise the whole thing vertically to the top of the tank.

Water has a density of 62.4 pounds per cubic foot. A typical circular slice of the water weighs

It is raised a distance of feet.

The work done in raising such a slice is foot-pounds. Thus, the total work done in emptying the tank is

Example. Find the work done in pumping all the water out of a filled sphere of radius 2 feet. Assume that the water is pumped out of the top of the sphere.

The picture on the left shows a side view of the sphere. I'll cut the cylinder of water up into circular slices; a typical slice is shown on the right.

In the picture, the slice has radius . The weight of the slice is

The slice is raised a distance of feet, so the work done is

The total work done is

Consider a spring attached at one end to a stationary object, such as a wall. Stretch the spring so that its free end moves a distance x from its unstretched position. The force exerted by the spring is given by Hooke's law:

where k is the spring constant. (The negative sign reflects the fact that the force is a restoring force: It always points toward the origin.)

To stretch the spring, I must exert a force equal to and opposite in direction to the force exerted by the spring: . Suppose I do so, and the end of the spring is moved by a small amount . Then the work done is . Hence, the total work done in stretching the spring from to is

Example. A force of 12 pounds is required to stretch a spring 1.5 feet beyond its unstretched length.

(a) What is the spring constant for this spring?

Using , I have

(b) How much work is done in stretching the spring from 2 feet beyond its unstretched length to 4 feet beyond its unstretched length?

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Copyright 2005 by Bruce Ikenaga