f/Example 1.
1.1.1 Problem-solving techniques
How do we use a conservation law, such as conservation of mass,
to solve problems? There are two basic techniques.
As an analogy, consider conservation of money, which makes it
illegal for you to create dollar bills using your own inkjet printer.
(Most people don’t intentionally destroy their dollar bills, either!)
Suppose the police notice that a particular store doesn’t seem to
have any customers, but the owner wears lots of gold jewelry and
drives a BMW. They suspect that the store is a front for some kind
of crime, perhaps counterfeiting. With intensive surveillance, there
are two basic approaches they could use in their investigation. One
method would be to have undercover agents try to find out how
much money goes in the door, and how much money comes back
out at the end of the day, perhaps by arranging through some trick
to get access to the owner’s briefcase in the morning and evening. If
the amount of money that comes out every day is greater than the
amount that went in, and if they’re convinced there is no safe on the
premises holding a large reservoir of money, then the owner must
be counterfeiting. This inflow-equals-outflow technique is useful if
we are sure that there is a region of space within which there is no
supply of mass that is being built up or depleted.A stream of water example 1
If you watch water flowing out of the end of a hose, you’ll see
that the stream of water is fatter near the mouth of the hose, and
skinnier lower down. This is because the water speeds up as it
falls. If the cross-sectional area of the stream was equal all along
its length, then the rate of flow (kilograms per second) through a
lower cross-section would be greater than the rate of flow through
a cross-section higher up. Since the flow is steady, the amount
of water between the two cross-sections stays constant. Conser-
vation of mass therefore requires that the cross-sectional area of
the stream shrink in inverse proportion to the increasing speed of
the falling water.
self-check A
Suppose the you point the hose straight up, so that the water is rising
rather than falling. What happens as the velocity gets smaller? What
happens when the velocity becomes zero? .Answer, p. 1054
How can we apply a conservation law, such as conservation of
mass, in a situation where mass might be stored up somewhere? To
use a crime analogy again, a prison could contain a certain number
of prisoners, who are not allowed to flow in or out at will. In physics,
this is known as aclosed system. A guard might notice that a certain
prisoner’s cell is empty, but that doesn’t mean he’s escaped. He
could be sick in the infirmary, or hard at work in the shop earning
cigarette money. What prisons actually do is to count all their
prisoners every day, and make sure today’s total is the same as58 Chapter 1 Conservation of Mass