## UNITS Definitions and How to Convert Between Units ## Why Do We Care about Units?

In science, when quantities ore measured or calculate, they must be given proper units. A measurement without a unit specification really does not make much sense. Imagine if someone told you that Mt. Everest is 104 tall. Without a unit specification this number should mean nothing to you.

There is a certain degree of arbitrariness in agreeing on a system of units, but there are a set of fundamental physical quanities - some of which you might already have some experience with - which form a sort of "building block" for measurements and calculation. More specifically, there are THREE fundamental/standard "building blocks" that are needed for specifying the relative magnitude of certain physically observable attributes of our universe. These three are: Length, Mass, and Time.

## What Are the Basic Units?

Length: This is a geometrical quantity understood from our common experience with distances and lengths of objects.

Mass: In principle this requires some elaboration, but we can just simply define it as the quantity of matter, and we need some set of units for measuring this quantity.

Time: Again, in principle this can get complicated, but we all of have an idea of the flow of time. We need a unit to measure this as well.

You are probably familiar with the fundamental units of length, mass and time in the American system: the yard, the pound, and the second. The other common units of the American system are often strange multiples of these fundamental units such as the ton (2000 lbs), the mile (1760 yds), the inch (1/36 yd) and the ounce (1/16 lb). Most of these units arose from accidental conventions, and so have few logical relationships.

Most of the world uses a much more rational system known as the metric system (the SI, Systeme International d'Unites, internationally agreed upon system of units) with the following fundamental units:

• The meter for length. Abbreviated "m".

• The kilogram for mass. Abbreviated "kg". (Note: kilogram, not gram, is the standard.)

• The second for time. Abbreviated "s".

Since the primary units are meters, kilograms and seconds, this is sometimes called the 'mks system'. Some people also use another metric system based on centimeters, grams and seconds, called the 'cgs system'.

## Base 10 System of the MKS Units

All of the unit relationships in the metric system are based on multiples of 10, so it is very easy to multiply and divide. The SI system uses prefixes to make multiples of the units. All of the prefixes represent powers of 10. The table below gives prefixes used in the metric system, along with their abbreviations and values.

Metric Prefixes

 Prefix Abbreviation Value Prefix Abbreviation Value deci d 10-1 decka da 101 centi c 10-2 hecto h 102 milli m 10-3 kilo k 103 micro m 10-6 mega M 106 nano n 10-9 giga G 109 pico p 10-12 tera T 1012 femto f 10-15 atto a 10-18

## Other Systems of Units and Relations to MKS System

The United States, unfortunately, is one the few countries in the world which has not yet made a complete conversion to the metric system. (Even Great Britain has adopted the SI system; so what we used to call "English" units are no more - they are strictly "American"!) As a result, you are forced to learn conversions between American and SI units, since all science and international commerce is transacted in SI units. Fortunately, converting units is not difficult. Although you can find tables listing seemingly endless conversions, you can do most of the lab exercises (as well as most conversions you will ever need in science, business, etc.) by using just the four conversions between American and SI units listed below (along with your own recollection of the relationships between various American units).

Units Conversion Table

 American to SI SI to American 1 inch = 2.54 cm 1 m = 39.37 inches 1 mile = 1.609 km 1 km = 0.6214 mile 1 lb = 0.4536 kg 1 kg = 2.205 pound 1 gal = 3.785 liters 1 liter = 0.2642 gal

Strictly speaking, the conversion between kilograms and pounds is valid only on the Earth since kilograms measure mass while pounds measure weight. However, since most of you will be remaining on the Earth for the foreseeable future, we will not yet worry about such details. (If you're interested, the unit of weight in the SI system is the newton, and the unit of mass in the American system is the slug.)

Converstions: Using the "Well-Chosen 1"

Many people have trouble converting between units because, even with the conversion factor at hand, they aren't sure whether they should multiply or divide by that number. The problem becomes even more confusing if there are multiple units to be converted, or if you need to use intermediate conversions to bridge between two sets of units. We offer a simple and foolproof method for handling the problem, which will always work if you don't take shortcuts!

We all know that any number multiplied by 1 equals itself, and also that the reciprocal of 1 equals 1. We can exploit these rather trivial properties by choosing our 1's carefully so that they will perform a unit conversion for us.

Suppose we wish to know how many kilograms a 170 pound person weighs. We know that 1 kg = 2.205 pounds , and can express this fact in the form of 1's: Note that the 1's are dimensionless; the quantity (number with units) in the numerator is exactly equal to the quantity (number with units) in the denominator. If we took a shortcut and omitted the units, we would be writing nonsense: neither 1 divided by 2.205, nor 2.205 divided by 1, equals "1"! Now we can multiply any other quantity by these 1's, and the quantity will remain unchanged (even though it will look considerably different).

In particular, we want to multiply the quantity "170 pounds" by 1 so that it will still be equivalent to 170 pounds, but will be expressed in kg units. But which "1" do we choose? Very simply, if the unit we want to "get rid of" is in the numerator, we choose the "1" that has that same unit appearing in the denominator (and vice versa) so that the undesired units will cancel. Hence we have Note that you do not omit the units, but multiply and divide them just like ordinary numbers. If you have selected a "well-chosen" 1 for your conversion your units will nicely cancel, which will assure you that the numbers themselves will also have been multiplied or divided properly. That's what makes this method foolproof: if you used a "poorly-chosen" 1, the expression itself will immediately let you know about it: Strictly speaking, this is not really incorrect: 375 lbs2/kg is the same as 170 lbs, but it's not a very useful way of expressing it, and it's certainly not what you were trying to do!

Example: As a passenger on the Space Shuttle, you note that the inertial navigation system shows your orbital velocity at 7,000 meters per second. You remember from your astronomy course that a speed of 17,500 miles per hour is the minimum needed to maintain an orbit around the Earth. Should you be worried? Because of your careful analysis using "well-chosen 1's", you can conclude that you will probably not survive long enough to have to do any more unit conversions.

## A Few Other Important Units in Astronomy

1. Temperature Scales

Scales of temperature measurement are tagged by the freezing point and boiling point of water. In the U.S., the Fahrenheit (F) system is the one commonly used; water freezes at 32 °F and boils at 212 °F (180° hotter). In Europe, the Celsius system is usually used: water freezes at 0 °C and boils at 100 °C. In scientific work, it is common to use the Kelvin temperature scale. The Kelvin degree is exactly the same "size" as the Celsius degree, but is based on the idea of absolute zero, the temperature at which all random molecular motions cease. 0 K is absolute zero, water freezes at 273 K and boils at 373 K. Note that the degree mark is not used with Kelvin temperatures, and the word "degree" is often not even mentioned: we say that "water boils at 373 kelvins".

To convert between these three systems, recognize that 0 K = -273 °C = -459 °F and that the Celsius and Kelvin degree is larger than the Fahrenheit degree by a factor of 180/100 = 9/5. The relationships between the systems are:

K = °C + 273     °C = 5/9 (°F - 32)     °F = 9/5 K - 459 .

2. Energy and Power: Joules and Watts

The SI unit of energy is called the joule. Although you may not have heard of joules before, they are simply related to other units of energy with which you probably are familiar. For example, 1 food Calorie (which actually is 1000 "normal'' calories) is 4,186 joules. House furnaces are rated in btus (British thermal units), indicating how much heat energy they can produce: 1 btu = 1,054 joules. Thus, a single potato chip (having an energy content of about 9 Calories) could also be said to possess 37,674 joules or 35.7 btu's of energy.

The SI unit of power is called the watt. Power is defined to be the rate at which energy is used or produced, and is measured as energy per unit time. The relationship between joules and watts is: For example, a 100-watt light bulb uses 100 joules of energy (about 1/42 of a Calorie or 1/10 of a btu) each second it is turned on. Weight-watchers might be more motivated to stick to their diet if they realized that one potato chip contains enough energy to operate a 100-watt light bulb for over 6 minutes!

Another common unit of power is the horsepower; one hp equals 746 watts, which means that energy is consumed or produced at the rate of 746 joules per second. (In case you're curious, you can calculate (using unit conversions) that if your car has fifty "horses" under the hood, they need to be fed 37,300 joules, or the equivalent energy of one potato chip, every second in order to pull you down the road.)

To give you a better sense of the joule as a unit of energy (and of the convenience of scientific notation, our next topic), here are some comparative energy outputs:

 Energy Source Energy (joules) Big Bang 1068 Radio galaxy 1055 Supernova 1043 Sun's radiation for 1 year 1034 Volcanic explosion 1019 H-bomb 1017 Thunderstorm 1015 Lightening flash 1010 Baseball pitch 102 Hitting keyboard key 10-2 Hop of a flea 10-7