Summary: Learning Objectives

Students will be able to…

1. Identify the dimensions of physical quantities.
2. Perform dimensional analysis on derived expressions.

Don't compare apples and oranges! That is, don't add, subtract, or equate objects with different dimensions. But you can (and will!) multiply and divide them.

Dimensions ([D]) are fundamental quantities like time, length, and mass. Dimensions are written with brackets, with time [T], length [L], and mass [M]. Other quantities in the study of the motion can be decomposed into time, length and mass, for example:

$$Speed = \frac{Length}{Time} = \frac{[L]}{[T]}$$

Apples $\neq$ Oranges $\to$ [A] $\neq$ [B]

To Reiterate, DO NOT and quantities with different dimensions!

Example:

$$\Delta x = v_{ix} + \frac{1}{2}a_{ix}\Delta t^2$$ (You'll see a lot more of this equation in kinematics. For now, just worry about the dimensions involved.)

$$[L] = \frac{[L]}{[T]}[T] + [?][T]^2$$

So, the dimensions of $[?]$ must be $\frac{[L]}{[T]^2}$ to even out. Overall $\frac{1}{2}a_{ix}\Delta t^2$ has dimensions of $[L]$

Dimensional Analysis (11 min)       

Dimensional Analysis (11 min)

Introduction to dimensional analysis. This one is a little dense, so don't worry too much about the symbols and ignore the final page.

Introduction to dimensional analysis.

https://www.youtube.com/watch?v=s49jC0w3EOc

More on dimensional analysis.

https://www.youtube.com/watch?v=FUC-KoQfg60

simulations here

Recommended example practice problems 

Dimensions should always "check out". Dimensions are like the more fundamental versions of units. For example, centimeters and inches are in different unit systems, but they both have dimensions of "length." Likewise, acceleration has units of "meters per second-squared" (m/s^2) but it has dimensions of "length per time-squared" ([L]/[t]^2).