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Wize University Physics Textbook (Master) > Foundations / Introduction / Measurement

Units

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SI Units

Some quantities have a lot of different units you can use. For example, for distance, we could use meters, yards, feet, miles, angstroms, astronomical units, parsecs, light-years...

For communication, it helps to have a standard set of units. In chemistry we use base SI units as follows (SI: systeme internationale):


Prefixes
To represent orders of magnitude, we can add prefixes to any unit to scale them to the settings of our measurements.


Exam Tip
Memorizing these units can help you solve questions quickly on exams!

Ex. if you see wavelength in nm instead of m, you should know that 1 nm = 1 x 10-9 m

Unit Analysis
  • Unit analysis is a method of checking the dimensional consistency of a equation.
  • Two sides of an equation must be equal numerically, but also in dimension.... units can cancel each other out!
  • We can often use units to help guide us in answering questions and making sure we are on the right track.

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Example: Unit Conversion


It's important to be able to convert between different types of units. Let's practice with a few examples.

a) How many seconds are in a year?


1 year =1 year⋅365 days1 year⋅24 hours1 day⋅60 min⁡1 hour⋅60 sec⁡1 min⁡=31,536,000 sec⁡1\ year\ =1\ year\cdot\frac{365\ days}{1\ year}\cdot\frac{24\ hours}{1\ day}\cdot\frac{60\ \min}{1\ hour}\cdot\frac{60\ \sec}{1\ \min}=31,536,000\ \sec



b) The speedometer in your car is in km/h. You go on a road trip to the US, where speed limits are posted in mph (miles per hour). You're on a highway with a 70 mph speed limit. If you are driving 110 km/h, are you speeding? (1 mile is 1.61 km.)

We need to convert our speed to mph.
110 kmh=110 kmh⋅1 mile1.61 km=68.2 mph110\ \frac{km}{h}=110\ \frac{km}{h}\cdot\frac{1\ mile}{1.61\ km}=68.2\ mph
This is less than the posted speed limit. We are not going to get a ticket!

Practice: Smartest Unit of Volume


I have a giant balloon, and it can hold up to 54100 human brains. A human brain is actually a measure of volume (1 human brain = 0.046 ft3). There are 3.28ft=1m3.28ft=1m , so what is the volume of the giant balloon in m3m^3?
Extra Practice
Dimensions and units of a constant
This is the Stefan-Boltzmann equation. σ\sigma is the Stefan-Boltzmann constant.
PA=σT4\frac{P}{A} = \sigma T^4
a) Knowing that P=[ML2T3]P = [\frac{ML^2}{T^3}] is power, A=[L2]A = [L^2] an area, what are the dimensions of the constant?
Finding the exponent from dimensional analysis
X=YZ32WnX = \frac{Y Z}{32 W^n}
If X=[Mθ]X = [M\theta] , Y=[MT2]Y = [MT^2] , Z=[θ/L2]Z = [\theta / L^2] , and W=[T/L]W = [T/L] , what is the exponent n?
Dimensions and units of a constant
This is the Stefan-Boltzmann equation. σ\sigma is the Stefan-Boltzmann constant.
PA=σT4\frac{P}{A} = \sigma T^4
a) Knowing that P=[ML2T3]P = [\frac{ML^2}{T^3}] is power, A=[L2]A = [L^2] an area, what are the dimensions of the constant?
Counting heartbeats
How many times does your heart beat in a year?
Fermi question: paperclips
How many paperclips can you fit in an average cup (1 m3=1000 L)?
Filling your tub
How much water (in m3) can you keep in your bathtub without it spilling?
Let's say you are a really fast runner. I mean, like 50km/h fast. Convert that into the following units.
Symbols
In dimensional analysis, the symbol for time is
T
, for mass is
M
, for length is
L
.
In the SI unit system, the symbol for nanogram is
ng
, for millikelvin is
mK
, and centinewton is
cN
.
Power of a variable from dimensions
If the dimensions of the variables
W=[L3] , X =[MT] , Y=[M6L], Z =[M2T]W=[L^3]\ ,\ X\ =\left[\frac{M}{T}\right]\ ,\ Y=\left[\frac{M^6}{L}\right],\ Z\ =\left[M^2T\right]
What is the exponent n?