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Dimensional Analysis Part 3

So far we have concentrated on cancelling units when using dimesional analysis. Units can also be combined according to the rules of algebra. For example, 12x multiplied by 9/4x equals 27 because the x cancels out and 12/4 × 9 is 27. If the problem had been 2x multiplied by 5x then the answer is 10x^{2}. The x variables combine to be written together as x^{2} instead of x · x. Here are some examples with actual units:

AreaSpeed5 cm × 2 cm = 10 cm^{2}42 miVolume------ = 42 mi/hr 1 m × 2 m × 3 m = 6 m^{3}1 hr

The general idea is that if a unit does not cancel out during the calculation, it is still there at the end. Combine like terms and pay attention to whether the unit is in the numerator (on top) or in the denominator (on the bottom).

Here are two examples of problems solved by using combined units.

Convert 42 ft^{2}to m^{2}42 ft^{2}42 ft·ft 0.3048 m 0.3048 m ------ = --------- x ---------- x ---------- = 3.9 m^{2}1 1 1 ft 1 ft Enter this into your calculator as 42 × 0.3048 × 0.3048 = 3.90192768

Find the time to travel 50 miles at 35 mi/hr 50 mi 1 hr ------ x --------- = 1.43 hr or 1 hr 25.7 min 1 35 mi Enter this into your calculator as 50 ÷ 35 = 1.428571429

Areas and volumes are examples of combined units, for the most part. An acre is an example of a unit of area which is not a combined unit. A gallon is similarly a unit of volume which is not a combined unit. Square feet (ft^{2}) are really (feet) times (feet) because that’s how area is calculated. Similarly, a cubic meter (m^{3}) can be imagined as a cubic space one meter on a side. For area units you must convert with **two** conversion factors. For volume, use **three** conversion factors.

- Convert 0.25 m
^{3}to cm^{3} - Convert 5 ft
^{2}to in^{2} - Convert 42 mi
^{3}to km^{3} - Convert 1,000 mL (or 1 L) to in
^{3}

- Convert 15 mi
^{2}to km^{2} - Convert 10 gal to cm
^{3} - 1 acre is 43,560 ft
^{2}; convert 1 acre to m^{2} - Convert 42,000,000 gal to m
^{3}

Common units of speed are miles per hour (mi/hr), kilometers per hour (km/hr), meters per second (m/s), centimeters per second (cm/s), kilometers per second (km/s), and miles per second (mi/s). Show work as always. There is an example of this type of calculation in the introduction.

- Highway speed limit: 65 mi/hr

Convert to km/hr - Speed of Light 3.00 ×
10
^{8}m/s

Convert to mi/hr - Speed of Voyager 1 Spacecraft: 17.0 km/s

Convert to mi/hr - Speed of Voyager 1 Spacecraft: 17.0
km/s

How many miles does it travel 10 minutes? - Speed of Voyager 1 Spacecraft: 17.0
km/s

How long does it take to travel 1,000,000 mi?

- Speed of a fast snail: 0.28 cm/s

Convert to mi/hr - Speed of Sound in Air: 343 m/s

Convert to mi/hr - Speed of Sound in Water: 3,310 mi/hr

Convert to m/s - Speed of the Space Shuttle in orbit:
17580 mi/hr

Convert to km/s - How long does it take the Space Shuttle to travel 3,000 mi? (This is the approximate width of N. America)