Conversions

There are so many units that you can use to express results that you need to becomeproficient at converting from one to another. Fortunately, there is an easy way to do thisand it works every time with no exceptions! You simply multiply by one. (The only questionis: which one?)

Since 12 in = 1 ft, you can get a one by dividing both sides of the equation by 1 ft, giving

12 in

1 ft= 1.

(Notice that this doesn’t say that 12 = 1.) You use this or any similar factor by noticingthat if you multiply anything by one, you don’t change it. It’s easy to see with an example;each factor in parentheses equals one.

1 yr = 1 yr ·(

365.24 days

1 yr

)·(

24 hr

1 day

)·(

60 min

1 hr

)·(

60 sec

1 min

)= 1 /yr ·

(365.24 /day

1 /yr

)·(

24 /hr

1 /day

)·(

60 /min

1 /hr

)·(

60 sec

1 /min

)= 365.24 · 24 · 60 · 60 · sec = 31556736 sec = 3.1557× 107 sec,

where there are only 5 significant figures in the result, because that’s the limit on the accuracyof the input data for the number of days in a year. (This is about π × 107 seconds — easierto remember.) Another example:

1 mi

1 hr=

1 mi

1 hr

(5280 ft

1 mi

) (1 hr

60 min

) (1 min

60 sec

)=

1 /mi

1 /hr

(5280 ft

1 /mi

) (1 /hr

60 /min

) (1 /min

60 sec

)=

5280

3600

ft

sec. = 1.47 ft/sec

For an example of what can go wrong, what is one mile in inches?

1 mi = 1 mi ·(

5280 ft

1 mi

) (1 ft

12 in

)= 1 /mi ·

(5280 ft

1 /mi

) (1 ft

12 in

)=

5280

12

ft2

in.

This isn’t wrong , but it’s pretty silly, and isn’t responsive to the original question of con-verting miles to inches. The key point to notice is that I kept track of the units honestly and Ididn’t assume that I’ll never make a mistake. A common error (the common error) in doingthese conversions is to write down the conversion factors and then to assume that they dowhat you want them to. If you skip the steps shown here, where you explicitly put in theunits and cancel them one pair at a time, you have about a 50-50 chance of messing it up.After all, if 12 in/1 ft = 1, then so does 1 ft/12 in = 1, and you can multiply anything by onewithout changing it.

1

2 Conversions

To avoid this trap, try writing the units first and fill in the numbers next, as in

1 mi ·(

ft

mi

)·(

yard

ft

)THEN 1 mi ·

(5280 ft

1 mi

)·(

1 yard

3 ft

)= 1760 yards

Don’t forget that each factor in a conversion must be done; if a unit is squared, itsconversion factor enters twice. Convert one square mile to square yards:

1 mi2(

5280 ft

1 mi

)2 (1 yard

3 ft

)2

= 3 097 600 yard2.

(I like the metric system more and more.)

Convert 10 cm1/2 using meters instead of centimeters.

10 cm1/2 ·( m

cm

)1/2then 10 cm1/2 ·

(1 m

100 cm

)1/2

=10√100

m1/2 = 1 m1/2.

For a particularly odd unit that I once encountered, convert 10 cm2/3 gm−1/3 sec2 tometers and kilograms. First, just get the units in place:

10 cm2/3 gm−1/3 sec2 ·( m

cm

)2/3·(

kg

gm

)−1/3

THEN

10 cm2/3 gm−1/3 sec2 ·(

1 m

100 cm

)2/3

·(

1 kg

103 gm

)−1/3

=10

1002/3 · 10−1m2/3 kg−1/3 sec2

=100

10 · 101/3m2/3 kg−1/3 sec2

=10

101/3m2/3 kg−1/3 sec2

Conversions 3

Practice Set 1

[1] 2.54 cm = 1 in. What is 1 meter in inches?

[2] g = 9.80 m/s2. Convert this to ft/s2.

[3] The conversion between Canadian and US dollarsis currently C$1.55=US$1.00. Gasoline in Canadacosts C$0.80 per liter; what is the cost per gallon inUS$? (1 liter is 0.264 gallons.)

[4] The light-year is the distance that light travels (invacuum) in one year. The speed of light is 3.00 ×108 m/s. What is one light-year in meters?

[5] The standard class hour is 50minutes. What is thisin microcenturies?

4 Conversions

Solutions for Practice Set 1

[1] 2.54 cm = 1 in. What is 1 meter in inches? 1 m(

100 cm1 m

) (1 in

2.54cm

)=

1002.54

cm = 39.37 in

[2] g = 9.80 m/s2. Convert this to ft/s2. 9.8ms2

(100 cm

1 m

) (1 in

2.54cm

) (1 ft12 in

)= 32.2

fts2

[3] The conversion between Canadian and US dollarsis currently C$1.55=US$1.00. Gasoline in Canadacosts C$0.60 per liter; what is the cost per gallonin US$? (1 liter is 0.264 gallons.)

0.80C$liter

(1 liter

0.264 gallons

) (1 US$C$1.55

)=

US$1.95 per gallon

[4] The light-year is the distance that light travels(in vacuum) in one year. The speed of light is3.00× 108 m/s. What is one light-year in meters?

3.00× 108 ms· 1 yr

(365.24 days

1 yr

)·(

24 hours1 day

) (3600 s1 hr

)= 9.47× 1015 m

[5] The standard class hour is 50 minutes. What isthis in microcenturies?

50 min(

1 hr60 min

) (1 day24hr

) (1 yr

365.24 days

)·(

1 century100 yr

) (106 microcenturies

1 century

)= 0.95 microcenturies

Conversions 5

Practice Set 2

[1] What is 60 mi/hr in ft/sec?

[2] Given 2.54 cm = 1 in exactly, what is 1 mile in kilo-meters?

[3] Water flows down a river at a rate of 500 cubic me-ters per second. What is this in gallons per hour?(1 liter = 10−3m3, 1 liter = 0.264 gallons.)

[4] g = 9.8 m/s2 is the standard gravitational field atthe surface of the Earth.Convert this to light-years/year2.

[5] Astronomers most often use the unit called a “par-sec,” which is about 3.26 light-years.The farthest objects ever observed are measured tobe about 1010 light-years away. What is this in mega-parsecs?

6 Conversions

Practice Set 3

[1] The density of water is 1 gm/cm3. Convert this tokg/m3?

[2] Start with 10 m3/2 and put the meters into centime-ters.

[3] If you go on a diet, and lose at a rate of 1 kg/week,what is this in milligrams per second?

[4] The speed of light is 3.00 × 108 m/s. What is it infeet per nanosecond?

[5] A certain physical constant is 1.57 gm1/2cm−3/2.Convert the grams to kilograms and the centimetersto meters.