Einstein to Newton
In the early years of the 20th century the name of Albert Einstein achieved fame in popular culture. Few who recognized the name understood the theories that he developed. The ideas which caught the popular imagination concerned the relativity of space and time. Time and space, he said, were not absolute, but relative to the frame of reference of the observer. In other words, a person moving with respect to another would experience different space and time.
These new theories started to erode the classical, clock-work universe formulated during the 17th and 18th centuries by Galileo, Newton, Descartes and others. Their solid, predictable mechanics could be easily reconciled with common sense perceptions of space and time. Einstein said that these common-sense ideas were faulty.
Einstein's relativity seemed bizarre. But his conclusions were not mere flights of fancy. They followed from a rigorous analysis of observed facts. Einstein was forced to his conclusions by logic much like a detective might be forced by the evidence to suspect his best friend of a major crime.
The road from Newton to Einstein took little more than 200 years to traverse. This was barely 1/10th the time from the first attempts at rationally understanding the physical world until the first steps along the path of modern physics were taken.Once these first steps were found, progress along the path was probably inevitable. The methods of measuring and analysing space, time and motion were the key that unlocked the gate to modern physics and a completely new vision of the universe. Galileo Galilei is usually credited with establishing the science of measuring motion called Kinematics, though, of course, many others have contributed to its development
Let's see what it was.
The measure of space
We must first establish precise definitions of terms to be used if we are to understand each other. In physics this means defining in terms of a process by which one can measure a quantity: a list of operations leading to a number. This kind of definition is called an operational definition.
A Distance Standard
Pick up a stick and call it your standard of length. Count how many times you can lay it down, end to end, along a straight line between two points. This count is called the distance between the two points.
Let me warn you now about a common problem: the circular definition. In my definition of distance I haven't said what I mean by a straight line. This is often defined as the shortest distance between two points. Now go back and reread both my definition of distance. Do you see the problem!A straight line is defined in terms of distance and distance is defined in terms of a straight line. It's like a cat chasing it own tail--she won't go anywhere doing that. To avoid this circularity (which is often found in poorly written physics textbooks) one must use another definition for straight line. Perhaps one could define it as the path followed by a string tightly stretched between the points.
Getting back to distance, the defining procedure would give the distance to the nearest "stick". Say we were measuring the length of a table. We might get something like, "the length of the table is more than three sticks but less than 4 sticks". Next take a stick 1/10th the length of the first and count how many "decisticks" fit between the end of the third stick and the end of the table. If we got, for example, between six and seven of those we could say the the table was 3.6 or 3.7 "decisticks" long depending on which seemed closer. Then we could take another stick 1/100 the length of the original and continue the process. Thus we can refine the measurement to more and more significant figures until we reached the practical limit of our equipment.
[Significant figures: We make it a point to only write digits of which we are fairly sure. Thus if we only measured to the nearest decistick, it wouldn't make any sense to write 3.654. That would imply we knew something which we don't actually know. We would be writing four significant figures when we knew only two.
Sometimes the number of significant digits in a number is not obvious when zeros are at the beginning of at the end of a number. The following rules will help avoid confusion: Leading zeroes don't count as significant digits. Thus 0.036 would have only two significant figures. Trailing zeroes before the decimal point don't count unless the decimal point is explicitly written after them. Thus 3600 has two significant figures but 3600. has four. Trailing zeroes after the decimal point count. Thus 3600.0 has five significant figures.]
The length standard we happen to use doesn't affect the physical reality that we measure. However, once we pick a standard, we have to stick with it! The laws of physics are the same if they are expressed in metres, yards, chains or furlongs. Before the French revolution every country and often every city had its own system of measurements. Kings and rulers would measure the extent of their power by how many cities and villages adopted his (or, less often, her) system of measure. The makers of the French revolution wanted no more of this kind of nonsense. Let logic rule! So they established a unit of length independent of any particular person. The metre was defined to be one-ten-millionth of the distance from the north pole to the equator along the meridian that passes through Paris. This definition was rather difficult to put into operational practice. So they also put a metre-long platinum-iridium bar in Sèvres, a town just outside Paris, for people to come and compare their rulers to.
A word on spelling: The official spelling of the unit of length in Canada and many other countries is metre. This has the added advantage of being correct French too. However, our westcoast culture seems to have adopted the American spelling, meter, to such an extent that the offical spelling looks bizarre. The name of a gadget that tells us a number is always meter in both American and British usage; e.g., voltmeter.
Powers of Ten
Smaller and larger units were defined as follows
decimetre 0.1 metre kilometre 1000 metres centimetre 0.01 metre megametre 1 000 000 metres millimetre 0.001 metre gigametre 1 000 000 000 metres micrometre 0 .000001 metre
nanometre 0.000000001 metre
picometre 0.0000000000001 metre
As you can see, writing numbers with lots of zeros in it gets a little confusing. One looses count of the zeroes. In order to span the distance from microscopic to cosmic with great notational ease another method of writing numbers is often used. One writes numbers several factors of ten larger or smaller than 1 with a number close to one (usually between 0.1 and 10) multiplied by 10 taken to an integer power. Here are some examples:
For large numbers
100 = 1 × 102
1000 = 1 × 103
10 000 = 1 × 104
100 000 = 1 × 105
1 000 000 = 1 × 106
3465.3 = 3.4653 × 103
For small numbers
0.01 = 1 × 10–2
0.0000001 = 1 × 10–7
0.0034653 = 3.4653 × 10–3
The prefixes used on the metre are also used for other units to be defined. They correspond to a power of ten multiplying the number. The most common prefixes and the corresponding integer powers are as follows:
(Units marked with an * are not quite official.)
Units smaller than one metre
prefixed unit power-of-ten abbreviation decimetre* 10-1 m dm centimetre* 10-2 m cm millimetre 10-3 m mm micrometre 10-6 m µm nanometre 10-9 m nm picometre 10-12 m pm femtometre 10-15 m fm attometre 10-18 m am Units larger than one metre
prefixed unit power-of-ten abbreviation dekametre* 101 m dam hectometre* 102 m hm kilometre 103 m km megametre 106 m Mm gigametre 109 m Gm terametre 1012 m Tm petametre 1015 m Pm exametre 1018 m Em
We will illustrate methods for measuring very small and very large lengths and distances with two demonstrations. In the first we measure the diameter of a hair viewed through a microscope. The method used depends on first calibrating the measurement system by measuring something which can be measured by both the new and old methods. The old method is known to be reliable, the new method can take us down to smaller dimensions.
