calculus-based physics for scientists and engineers


Introduction to Units

There are three basic units associated with each of the three fundamental dimensions of mechanics. The three units of the SI system are defined in cooperation by national laboratories throughout the world. The basic units of other systems (e.g. British imperial System) are defined relative to the SI units through established conversions (next topic).

  •   Dimension  
  • Length
  • Mass
  • Time
  •   SI Unit Name  
  • meter
  • kilogram
  • second
  •    Symbol  
  • m
  • kg
  • s

Don't confuse a unit symbol with a symbol for a physical quantity. In this text, physical quantities are in italics, while unit symbols are in regular type.

example: A distance \(d\) will have units of meters, m, while a mass \(m\) will have units of kilograms, kg.
This can make equations confusing. Also, undoubtedly, neither you nor I distinguish between italic and non-italic in our handwriting. You write symbols on the page, but when you read the equations in your head, read the full words the symbols represent.

Units follow the same algebraic rules as dimensions. If you don't remember these, check out the help button in the More Info section for this module.

Unit Conversions

quotation: You will be astounded by the number of math problems you solve in physics by the multiplication of a judicious choice of the number one. -me

Converting between units:

  1. Find a conversion equation between the unit you have and the unit you need (or a unit that will get you closer to the unit you need).
  2. Convert those equations into ratios that equal the number one.
  3. Multiply.

This is one of those topics that is best shown through example.

Unit Conversions with Powers

In the previous topic, we only looked at expressions with linear powers of units. Working with other powers of units make things more complicated but the same rules are obeyed. Units follow algebraic rules like dimensions.


Do Now!   Do these exercises immediately.

Not Now!   Do these after you start to forget the topic, say in a week.

More!   More exercises if you want. Maybe review before a test.

Required Problems


1. A furlong is 660 ft and a fortnight is 14 days. What is 85 mph is furlongs/fortnight?


2. A cord of wood is a volume that is 8 ft by 4 ft by 4 ft. What is this in m3?

Optional Problems


1. Density is a mass/volume. If aluminum has a density of 2.7 g/cm3, what is the density in kg/m3? What is the density in Pg/nautical mile3? (Pg is petagram. Interestingly, the symbol for nautical mile is nm, which is identical to nanometers. Other than ridiculously contrived physics textbook problems, I do not know of an example where this creates confusion.)