# Units

In a Physics problem, when you are told that you have a quantity of “things” — the numerical part tells* how much,* and the units tell *of what.* So in order to add quantities together, the units must match. This often requires changing units. I will show you a foolproof method in the section **Unit** **Conversion.** Next, there are different systems and types of units. I will explain these in **Unit ****Systems.** Units can also help you figure out how to do a calculation, for example, “do I divide by area and multiply by velocity??” I will teach you how this works in **Dimensional Analysis.** Units are also scaled using prefixes. I explain all that in **Prefixes and Powers of 10.** Finally, there is a style to writing units, both abbreviated and long form. I will explain the proper way and include some history in **Notation.** Let’s get started!

## Unit Conversion

What follows is the method that I use to do unit conversions. This will make conversion easy, automatic, and you won’t have to think about whether to multiply or divide.

### Technique

Let’s start with a simple length conversion. Convert 6 ft to meters. Start by drawing a bracket:

We look up in a conversion table and find that 1m = 3.28 ft. How do we use this in a conversion? In the bracket above, we have units of feet, and we want units of meters. So we need to cancel the “ft” and replace with “m.” That means that in the conversion spot (right side in the above), “ft” needs to be in the bottom so it will cancel with the “ft” we already have, like so:

Now just collect all the numbers and the un-cancelled units — this gives us

Notice that we have arranged them so that miles, feet, and meters cancel. Also notice that the “ones” have no effect, so we leave them out of the calculation.

### Examples

Here, let’s convert 15 mph to meters-per-second (m/s). We will leave out the “ones” so it doesn’t get cluttered. We use 60 min = 1 h, 60 s = 1 min, 5280 ft = 1 mi, and 3.28 ft = 1 m. Arrange so that all the units cancel except meters and seconds.

Of course, you could shorten this with “grouped conversions,” such as: 3600 s/hr and 1609.34 m/mi.

Here, I’ve worked it out in multiple steps so you can get a feel for the technique. The benefit of this technique is that you never again have to guess, “do I multiply or divide?” or, “how do I use this conversion here?” It also helps later on when we get to compound units like newtons or joules.

Next, let’s do a time conversion. Ever wonder, “How many seconds in a day?” Here’s the calculation:

Note that each foot measure (3 of them) had to be converted, not just one. So we end up with m

^{3}, a unit of volume. Liter is also a unit of volume (1 liter = 0.001 m

^{3}), so we can easily convert to liters:

or 136/0.001 = 136000 liters. Divide by 2 and we get 68000 2-liter bottles to fill a typical backyard pool!

## Units systems — MKS, cgs, English/Imperial

There are different systems of units. The “international standard” system is MKS, also referred to as SI. What does that mean? MKS refers to the units of length, mass and time in that system — meter, kilogram, and second. There is another system called CGS that uses the centimeter, gram, and second as its basis. The label SI comes from the French term, Syst`eme International d’Unit´es, or in English, International System of units. The International Standards Organization chose the MKS system as its standard.

There are units for every measurable quantity — length, mass, time, velocity, acceleration, force, energy, power, etc. But most units can be de-composed into various combinations of length, mass and time (LMT). There are some that cannot be expressed in LMT; e.g., temperature, electric charge, light intensity, and from chemistry, the mole.

Those of us in the U.S. are used to the English or “Imperial” system of units. Lengths are inches, feet, yards, miles. The main difference between Imperial units and MKS (metric) is the mass/weight units. The Imperial system uses “weights” instead of masses. A weight is a unit of force equal to mg (mass × the acceleration of gravity). So a pound (lb) is a unit of force. To get mass, you must divide by g, or 32 ft/s2. The unit of mass in the Imperial system is called a “slug” or sometimes lb-m (pound-mass) and is referred to as w/g (weight ÷ gravity).

## Dimensional Analysis

Dimensional analysis is a technique that will allow you to figure out “where things go” in a formula that, perhaps you can’t remember, or you are trying to figure out. Let’s start with an example.

Velocity is the measurement of how your position changes with time. In the MKS system, its unit is meters-per-second, or m/s. Acceleration is the measurement of how your velocity changes with time — i.e., speeding up or slowing down. Its unit is meters-per-second-per-second, or meters-per-second-squared, written as m/s2. Now let’s use this and show how the unit for force is composed.

Newton’s law tells us that force is equal to mass × acceleration, or F = ma. So the units are: mass in kilograms and acceleration in meters-per-second-squared. This looks like the following:

which can be written as kg-m/s^{2}. This equals the unit of force, not surprisingly, called a newton (N).

Similarly, for energy, let’s look at Einstein’s famous equation, E = mc^{2}. The unit of mass is kilogram (kg), the units of c are meters-per-second (m/s), so the units of c^{2} are m^{2}/s^{2}. This gives us:

which can also be written as kg-m^{2}/s^{2}. This is defined as a unit of energy, called a joule (J), after James Prescott Joule.

Here’s an example of using Dimensional Analysis. Say that you are asked to calculate the amount of heat (energy), Q, needed to increase the temperature of 0.4 liters of water by 2°C. You are told that the heat capacity of water is c = 4190 (J/kg-°C). You are also told that the mass of 0.4 liters of water is m = 0.4 kg. So start with the units of specific heat, like so:

Now, arrange mass and ΔT so that all the units cancel except the unit of energy, J. Since mass (kg) is in the denominator, we need mass in the numerator (multiply by mass, m) to cancel the kg unit and the same for temperature change (ΔT) to cancel the °C unit, giving

which looks like this as a formula

and the numbers work out to:

which equals \(0.4 \times 4190 \times 2 = 3352\,{\rm J.}\)

One more example of Dimensional Analysis. You are given two plates of metal that have an area A = 10 m^{2} and they are separated by a distance d = 0.02 m. You are told that the permittivity (a relevant constant) is ε_{0} = 8.85 × 10^{−12} F/m. For this configuration, calculate the capacitance in farads (F). The only way to use all the givens and balance the units is like so

which we write as

and the numbers give

which equals \((8.85 \times 10^{−12})(10)/(0.02)\,{\rm F} = 4.43 \times 10^{−9}\,{\rm F.}\)

## Notation

When writing quantities with units, convention (standard practice) says that if the unit is named after a person, then the abbreviation is capitalized. However, since they are named for a person, when you write the full word in a sentence, don’t capitalize it, so it is clear that you are not talking about the person. For example: newton (N), joule (J), watt (W), amp (A), volt (V), etc.

## Prefixes and Powers of 10

### Scientific Notation

Scientific notation is a way to write very large or very small numbers in a compact way. It dates back to the use of slide rules where precision was limited. Scientific notation is a tool to keep track of where the decimal point is. Recipe: if you move the decimal point left *n* spots, multiply by 10^{+n}. If you move the decimal point *m* places right, multiply by 10^{−m}. For example:

and

### Video Illustration

Back in 1977, artists and filmmakers Charles and Ray Eames made a great video called Powers of Ten that nicely illustrates the very large and the very small. Watch it on YouTube here.

### Table of Prefixes

Units can be scaled up or down with prefixes. Prefixes that multiply are written with capital letters, prefixes that divide are written with lower-case letters. Exceptions to this rule: the first three “multiply” prefixes — deca, hecto, kilo – are written with lower-case letters. Really, the only one you need to remember is kilo (k); deca and hecto are not widely used.

The table below lists the standard prefixes. Notice that they (mostly) step up or down by a factor of 1000 (10^{3}).

yotta | Y | 10^{24} |
1 000 000 000 000 000 000 000 000 |

zetta | Z | 10^{21} |
1 000 000 000 000 000 000 000 |

exa | E | 10^{18} |
1 000 000 000 000 000 000 |

peta | P | 10^{15} |
1 000 000 000 000 000 |

tera | T | 10^{12} |
1 000 000 000 000 |

giga | G | 10^{9} |
1 000 000 000 |

mega | M | 10^{6} |
1 000 000 |

kilo | k | 10^{3} |
1 000 |

hecto | h | 10^{2} |
100 |

deca | da | 10^{1} |
10 |

10^{0} |
1 | ||

deci | d | 10^{-1} |
0.1 |

centi | c | 10^{-2} |
0.01 |

milli | m | 10^{-3} |
0.001 |

micro | μ | 10^{-6} |
0.000 001 |

nano | n | 10^{-9} |
0.000 000 001 |

pico | p | 10^{-12} |
0.000 000 000 001 |

femto | f | 10^{-15} |
0.000 000 000 000 001 |

atto | a | 10^{-18} |
0.000 000 000 000 000 001 |

zepto | z | 10^{-21} |
0.000 000 000 000 000 000 001 |

yocto | y | 10^{-24} |
0.000 000 000 000 000 000 000 001 |

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