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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