Dimensional analysis can be a useful tool for, at this stage, deriving units
of measurement or checking formulas. Dimensional units, or units of measurement,
can be treated as algebraic expressions (i.e., they are like variables). Take
Given this formula: y=5gt2, where t=10 s and g=the acceleration due
to gravity=9.80 m/s2
Find the units of y.
Solution by dimensional analysis:
Seconds are a measure of time and meters are a measurement of length aren't
they? So t= Time, g= Length/Time2
This equation has the dimensions y=(L/T2)T2, which can be
algebraically simplified to y=L because the T2 and the 1/T2
cancel each other out; the dimension for y in this example was Length, so the
units were meters.
This notation is just another way to write numbers, sort of a scientist's
"short hand." Take the number 6 million: 6,000,000. Instead, you can
write it as 6 multiplied by a power of 10 (in this case, it is expressed as 6 x
Here's another example: Express 25 in scientific notation. 2.5 x 101.
Why can't you just write it like 25 x 100? Because scientific
notation is always in the form a x 10b, where a is a number is a
decimal less than 10. Examples of such: 4.5627, 3.00, 5.2, etc.
Significant Figures, or the slang, "sig figs" are significant in
determining the accuracy of measurements! If an engineer is a tenth of a degree
off in determining a projectile's trajectory (path of flight), it could land
quite far away from where the scientist had calculated, or in any calculation in
which precision is important. Sometimes this could be a grievous error. Just
think, if astronomers were "off" about the path of an asteroid
approaching Earth...(well how MANY movies about that have you seen?? =])
Here are some rules for preserving significant digits in figures:
- Addition & Subtraction: Ever heard the phrase "a chain is as
good as it's weakest link"? Well, when adding two numbers together, there
is a similar rule. The precision (farthest decimal place) is as good as its
least precise measurement (the least farthest decimal place) Take this
The first number has an accuracy to 3 decimal places, but the second number
has only 2. So, the answer is precise to only 2 decimal places.
68.391+2.55=70.94 (70.941 rounded to the nearest hundredth)
- Multiplication & Division: The rule for multiplication and
division is different from that for addition and subtraction. Try not to get
the two mixed up because the results (obviously) may not always be the same.
For multiplication and division, count the number of figures in the smallest
number. The answer must have no more number of accurate figures than this
number. If this sounds confusing, analyze these examples:
2.55 seems to have the least amount of digits (3), so the answer must have
a maximum of 3-digit accuracy.
68.391·2.55=174 (174.39705 rounded to the nearest one's place)
16.7 may seem to be the number with the least number of digits (3), HOWEVER,0.00000000035
has only 2-digit accuracy! How? What? Write it in scientific
notation: 3.5 x 10-10. Only 3.5 are the significant digits. (lots
of zeros in a long string don't count!
0.00000000035·16.7=(3.5 x 10-10)·16.7=5.8 x 10-9
(5.845 rounded to the nearest tenth)
- Let's compare the number 200. to 200 :
The first one has 3 sig. figs. whereas the second one only has one. Why?!?
(They're the same number aren't they?) The value (200) is the same, but
the decimal point at the end of the first one means that all three digits
count as sig. figs. (all are certain to be accurate). For the second one,
because there is no decimal point, the rule "lots of zeros in a
long string don't count!" applys.