General Review | Review
Significant Figures
Example: 37.12 meters has 4 significant figures if measured at centimeter scale. For our class, the general rule is to keep 4 significant figures throughout your calculations, or 5 for exponentials and logarithms. Your answer should use 3 significant figures.
Scientific Notation
Example: Speed of light in a vacuum $c = 299,792,458 m/s$ (not scientific notation), or approximately $3.00*10^8 m/s$ (scientific notation). Use the commutative property of multiplication to carry out calculations in scientific notation, for example:
Find the distance light travels in 1.2 pico seconds.
$c$ is the speed of light in a vacuum, and let $\Delta t$ represent the change in time. |$\Delta \overrightarrow{r}$| will represent the distance travelled, which you will see more of in Kinematics. Start out by representing the situation symbolically:
$$|\Delta \overrightarrow{r}| = |\vec{v}|\Delta t = (3.00*10^8 m/s)(1.2*10^{-12}s)$$
Now use the commutative property to switch the exponents:
$$= 3.00*1.2*10^8*10^{-12}m$$
$$=3.6*10^{-4}m$$.
Dimensions and Dimensional Analysis
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 add or subtract 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 $[L]$
Units
In this class, we will default to SI (International System of Units), which are derived from dimensions. SI units are sometimes calles MKS (meters, kilograms, seconds). Here's a nice tabulation of SI units:
| Symbol | Name | Dimension |
|---|
| s | second | time |
| m | meter | length |
| kg | kilogram | mass |
| A | ampere | electric current |
| K | kelvin | temperature |
| mol | mole | amount of substance |
| cd | candela | luminous intensity |
Here's an example that uses Newton's second law $\Sigma \overrightarrow{F} = m\overrightarrow{a}$, which you'll see in Forces. Let's determine the units of $\overrightarrow{F}$ by looking at dimensions:
$$\Sigma \overrightarrow{F} = m\overrightarrow{a} \to [M]\frac{[L]}{[T]^2}$$ So $N$ has units of $\frac{kg*m}{s^2}$.
Conversions
Easy (No Powers)
Let's convert 100 feet per second to ? kilometers per day:
| 100 ft | 0.3048 m | 60 sec | 60 min | 24 hrs | 1 km |
| 1 sec | 1 ft | 1 min | 1 hr | 1 day | 1000 m |
= 2633 km/day
Less Easy (Powers)
Now, what if we go from $10 m^3 \to cm^3$? There are 100 cm per 1 meter, so use that to "cancel out" the $m^3$:
| 10 $m^3$ | 100 cm | 100 cm | 100 cm |
| | 1 m | 1 m | 1 m |
=
| 10 $m^3$ | [100 cm]$^3$ |
| | [1 m]$^3$ |
= $10*10^6 cm^3$
Orders of Magnitude
Orders of Magnitude basically corresponds to counting zeros. More concretely, the powers of 10. For example, how many orders of magnitude larger is 1000 than 1? Well, count the zeros (or powers of 10). 1000 has 3 more zeros, so it is 3 orders of magnitude larger.
We have focused so much on creating introductions to the physics content we haven't had time to create text that reviews algebra and other prerequisite knowledge. We do however have good videos on what you'll need to know starting this class. Check them out below in the drop down called BoxSand Videos.
