Units and Dimensional Analysis
Units are part of every measured number in science. Think about the following
scenario. You turn to your coworker and ask them for the length of a box for storing
tools. He comes back and says it’s 2.5 . You know he’s from Europe, so there’s a
question: 2.5 ft., 2.5 m., 2.5 dm. ? Without a unit, you don’t know what a number
means.
In calculating, treat units the same way you treat variables in an equation. Below are
a few examples of handling units in calculations.
1.5 ft. x 12 in.
= 18 in.
10 mi. x 4 hr. = 40 mi.
1 ft.
hr
(this is 10 mph)
Another way of writing this would be
1.5 ft. | 12 in.
= 18 in.
10 mi. | 4 hr. = 40 mi.
| 1 ft.
hr. |
Labels can be strung along:
1 mi. | 5280 ft. | 12 in. |2.5 cm. |
|
1 mi. |
1 ft. |
1 in.
1 m.
|
1 km.
| 100 cm. | 1000 m.
Notice that all the units, except km. cancel.
= 1.584 km.
Units within functions
With waves and damped waves, we now have sine, cosine, exponential and log
functions. Thinking about units, we know how to interpret meters per second or
square meters, but what does sin(meter) mean? Or exp(meter), log(seconds), etc.??
These are nonsensical units. What this requires is that the arguments for these and
other functions must be dimensionless; the dimensions must all cancel inside the
function.
Checking our formulae so far:
sin(ωt), ω has units of radians/sec and t is in seconds : seconds cancel leaving
radians, a dimensionless quantity.
exp(-t/τ), t is in seconds, so τ must also be in seconds to cancel.
cos[2π(x/λ-ft)], x and λ must be in the same length units and f is in cycles/sec, t is in
seconds, leaving seconds. A further note: λ is actually length/cycle, so both x/λ and
ft result in units of cycles. The 2π actually is radians/cycle converting the other
results to radians for the trig functions.
cos[2π(x/λ-ft)]:
cos[2π rad/cycle(xm/{λm/cycle}-fcycles/sec tsec)]
= cos[2π rad/cycle(x/λcycles -ftcycles)]
= cos[2π(x/λ -ft)rad]