Simple Nature - Light and Matter

Checking units

A useful technique for finding mistakes in one’s algebra is to

analyze the units associated with the variables.

Checking units example 2

.Jae starts from the formulaV=^13 Ahfor the volume of a cone,

whereAis the area of its base, andhis its height. He wants to

find an equation that will tell him how tall a conical tent has to be

in order to have a certain volume, given its radius. His algebra

goes like this:

V=

1

3

[ 1 ] Ah

[ 2 ] A=πr^2

V=

1

3

[ 3 ] πr^2 h

h=

πr^2

3 V

[ 4 ]

Is his algebra correct? If not, find the mistake.

.Line 4 is supposed to be an equation for the height, so the units

of the expression on the right-hand side had better equal meters.

The pi and the 3 are unitless, so we can ignore them. In terms of

units, line 4 becomes

m =

m^2

m^3

=

1

m

This is false, so there must be a mistake in the algebra. The units

of lines 1, 2, and 3 check out, so the mistake must be in the step

from line 3 to line 4. In fact the result should have been

h=

3 V

πr^2

.

Now the units check: m = m^3 /m^2.

Discussion Question

A Isaac Newton wrote, “... the natural days are truly unequal, though

they are commonly considered as equal, and used for a measure of

time... It may be that there is no such thing as an equable motion, whereby

time may be accurately measured. All motions may be accelerated or re-

tarded... ” Newton was right. Even the modern definition of the second

in terms of light emitted by cesium atoms is subject to variation. For in-

stance, magnetic fields could cause the cesium atoms to emit light with

a slightly different rate of vibration. What makes us think, though, that a

pendulum clock is more accurate than a sundial, or that a cesium atom

is a more accurate timekeeper than a pendulum clock? That is, how can

one test experimentally how the accuracies of different time standards

compare?

26 Chapter 0 Introduction and Review