Measurement

Basic Units

The universal basic units are length, mass, and time.

Time

Anything that repeats itself can be used as a measurement of time.

Anciently, heartbeats, the cycles of the moon, the rotation of the earth, both yearly and daily, were used, giving us units of the second, the day, the month and the year.

As instruments have improved, we've improved our accuracy in measuring time. Pendulums gave way to quartz crystals.

Today, the second is based off of the behavior of Cesium atoms. (See wikipedia.)

Length

Over history, various units were used for measuring length, and corresponding quantities such as area (length x length) and volume (length x length x length).

Today, the international standard is the meter, which is defined as the distance light travels in a vacuum in <math>{1 \over 299,792,458}</math> seconds. All other lengths are defined in proportion to this fundamental measurement.

Mass

Weight is the pull of gravity, and is proportional to mass and the acceleration due to gravity. Thus, the same object in different places will weight differently because the acceleration due to gravity changes in proportion to the masses of the objects and inversely to the distance squared.

Of all the international standards, mass is the least precise, defined as the mass of a particular weight.

Perhaps one day we will be able to express mass in terms of time and length, or some other more precise quantity.

Converting Units

In order to convert similar units, you only have to know how much of one will give you how much of the other. As an example, 3 feet makes 1 yard.

So to convert feet to yard:

<math>x \text{ feet} = {x \over 3} \text{ yards}\,</math>

Or yards to feet:

<math>y \text{ yards} = 3y \text{ feet}\,</math>

You can simply think of treating the units as a number, and then things become clearer:

<math>

\begin{align} y \text{ yards} &= y \text{ yards} \cdot {3 \text{ feet} \over 1 \text{ yards}} \\ &= {3y \text{ yards} \text{ feet} \over \text{ yards}} \\ &= {3y \cancel{\text{ yards}} \text{ feet} \over \cancel{\text{ yards}}} \\ &= 3y \text{ feet} \\ \end{align} </math>

Scientific Notation

Since no measurement is precise in the real world, the way we write numbers that reflect reality must demonstrate this fact.

We use scientific notation. This is simply:

<math>1.23 \times 10^{14}</math>

This form beautifully conveys some important information about the number we are using.

First, we know it's significant digits. This has 3 significant figures, while <math>3 \times 10^9</math> only has 1.

Second, we know the order of magnitude---the exponent. Just looking at that number alone, we can get a good sense of how incredibly large or small the number really is. Each increment of the exponent is a multiple of 10; each increment of 3 is a multiple of 1,000. Conversely, 1 order of magnitude less is 1/10th, while 3 less is 1/1000th.

When doing math with scientific notation, keep in mind the following rules:

Rule 1: Counting. Don't count leading zeros as significant. Don't count following zeros unless it is in scientific notation or has a decimal point somewhere. 500 has 1 significant figure, but 500. has 3.

Rule 2: Multiplying and dividing. When multiplying and dividing, the answer can only have as many significant digits as the fewest significant digits of the factors.

Rule 3: Adding and subtracting. When adding and subtracting, the answer's least significant figure is the same as the left-most significant figure of the parts. In other words, line up the numbers you are adding vertically, and find the left-most least significant figure. This will be the least significant figure of the answer.

Unfortunately, this means that <math>5.00 \times 10^3</math> plus <math>5.00 \times 10^0</math> is <math>5.00 \times 10^3</math>. If we added the smaller number a lot, it still wouldn't change. We'd have to multiply it by a number greater than 10 to see the answer change.

Orders of Magnitude

If you drop all significant figures and simply do math with orders of magnitude, a lot of problems can be solved with surprising accuracy. This is because unless you are extremely precise with your measurements, the error will explode so large that your measurements could have been ball-park anyway.

For instance, how long would it take to travel to Mars in a spacecraft that travels at the same speed as the spacecraft that went to the moon? The distance to the moon is about 400,000 km (or <math>10^5</math> km. It took a few days, about a half-week to go to the moon, or about <math>10^{-2}</math> years.. Let's use weeks for our time scale. The distance to Mars when it is closest is about 60 million km, which is <math>10^8</math> km. Divide that by the distance to the moon and you get <math>10^3</math>, which works out to be about ten years, as opposed to centuries or months. Indeed, accurate predictions of the time to travel to Mars is about a year, so it shows we are pretty close. (When doing this kind of analysis, being within a factor of 10 is as good as it gets.)

This is a very useful tool for analyzing all sorts of real phenomena and making reasonable predictions. A physicist who wanted to measure the total power output of the first atomic bomb dropped pieces of paper as the bomb exploded, and noted how far away they fell. This, combined with an understanding of how the two measurements are related, and some quick orders-of-magnitude analysis as I've shown above, gave him a very good hint at how much power was released.

Dimensional Analysis

It may be useful to analyze a problem simply by using the units of measurement, without numbers.

At the very least, you'll know that the answer is proportional to the square or the square root or the inverse or whatnot of the units involved. This is often a "good enough" answer to get things done in the real world, or to write down an equation that you're trying to find.

Source

• Chapter 1 of HRK