Math Review Topic

- Conversion - - Of Units -

Introduction

hysics is a science of measurement. A measurement is the process of comparing the size of an unknown quantity to the size of a recognized standard by means of P a secondary standard (one that’s a duplicate of the primary standard), device, or instrument. The purpose of making measurements is to quantify physical parameters in order to better describe and understand the physical world in which we live.

A measurement result always consists of two parts:

1. a magnitude, and 2. a unit

An example of a measurement is

15.8 meters

In the example above, the magnitude is 15.8, while its unit is meters. Without units, measurement results are incomplete and meaningless. A measurement’s magnitude tells how many times larger or smaller the quantity is with respect to a given standard, while its unit identifies the particular standard to which the quantity is being compared. We interpret the measurement above to mean that the quantity of interest is 15.8 times longer than a standard meter.

Usually two or more measurements have to be combined arithmetically, through some equation, to obtain a final result. When this happens, we have to deal with both the magnitude and the units in the calculation. The numerical portion is treated just like any calculation involving pure (or dimensionless) numbers. The units are treated like the unknowns in an algebraic equation. For example,

A car travels a distance of 200 miles in 4 hours. What’s its average speed?

Total Distance 200 miles mi Average Speed    50 Elapsed Time 4 hours hr

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In this example we first divide the value 200 by the value 4 to get the quotient 50. In this particular case, the unit mathematics doesn’t produce a single unit, so the units are kept in a fractional format --- namely miles / hour (read “miles per hour”).

Example 2

As a second example, suppose we ask the question “How far will a car travel in 4 hours if it’s traveling at an average speed of 50 miles per hour?”

mi Distance Traveled  Average Speed  Time  v  t  50  4 hr  200 miles hr

We carried out the arithmetic on the numerical part (the magnitudes) as usual. But now we’re able to perform some arithmetic on the units as well. The unit of time (hours) in the denominator divides into (cancels) the unit of time (hours) in the numerator leaving the single unit of miles in the product.

Example 3

As a third example, let’s see what happens when two similar units are multiplied. A rectangle has a length of 5 inches and a width of 3 inches. What’s its area?

Area  Length  Width  l w  5 in3 in  (53) (inin)  15 in2

In this example we use the exponential shorthand in2 to denote the multiplication of the unit “inch” by itself. In other words, (in x in) = in2.

If we were calculating the volume of a box, the process would be the same.

Volume  Length  Width Height  l w h

 20 cm30 cm10 cm  6,000 (cmcm cm)  6,000 cm3

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Conversion of Units

One of the fundamental rules of working with measurement is that all the units have to be consistent with a specific measurement system --- either the mks, cgs, or fps measurement systems. It would be meaningless, for example, to calculate the area of a rectangle whose base was expressed in inches and its height in centimeters.

Sometimes this mixing of units is unavoidable due to the difference in read-out units on various instruments. Moreover, values may have been obtained from other sources or reference tables. If this happens, it becomes necessary to convert from one set of units to another. There are four types of conversions that you’ll need to perform.

1. Direct Conversions

Direct conversions are the easiest of all. It’s a simple 1-step process that takes us directly from one set of units to another. Suppose we make a measurement using an instrument that reads out in a particular set of units, but we need to report the result in a different set of units. How do we go about changing from one set to the other?

Example 1

A steel rod is measured with a tape measure and found to be 3.5 inches. What’s its length in centimeters?

Solution:

We start by looking in a table of conversion factors (like the one in your textbook) to see if it lists a relationship between inches and centimeters. In this case we find that

1 in = 2.54 cm

Technically speaking, this expression is not a true equation, since the equality symbol links two dissimilar items. It’s the same as saying

1 apple = 3 bananas

A statement of this type is called an equivalency. But even though it’s not a true equation, we can still perform the same arithmetic operations on both sides without affecting the validity of the expression – just as if it were a true equation.

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If we divide both sides of this expression by 1 inch, we have

1 in 2.54 cm   1 1 in 1 in

In other words, we wind up with a special “1”. And since we know that multiplying any value by 1 leaves the value unchanged, we can multiply our measurement of 3.5 inches by this special “1” to get

 2.54 cm   3.5  2.54   in cm  3.5 in1  3.5 in        8.89 cm  1 in   1   in 

Notice that the magnitude changed in addition to the units. This is because we’re now comparing the quantity to a different standard. Even though the quantity itself hasn’t changed, it’s now a multiple of another standard.

2. Chain Conversions

It may not always be possible to convert directly from one set of units to another. This happens when no specific conversion factors exist. The process then requires several steps, as shown in the example below.

Example 2

Given the following information, how many seconds are there in 1 week?

1 week = 7 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds

Solution:

This is done by first converting the weeks to days. Next, the days are converted to hours. Then the hours are converted to minutes. And finally the minutes are converted to seconds. In equation form, this sequence of steps is written as

 7 da  24 hr   60 min  60 sec 5 1 wk  1 wk           6.048 x 10 seconds 1 wk   1 da   1 hr   1 min 

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3. Power Conversions

Units that have been raised to powers often appear in measurement values. When this happens, and the results need to be converted to a different set of units, it’s VERY IMPORTANT to remember that a separate conversion has to be performed for each occurrence of a unit.

Example 3

A rectangular card has a length of 3.0 in and a width of 2.0 in. Its area was calculated to be 6.0 in2 (area = length x width). What is its area in units of square centimeters?

Solution:

Since the unit of inches appears twice (raised to the second power) in the calculated area, we need to apply the inches-to-centimeters conversion factor two times.

2   2.54 cm   2.54 cm  6.0 in  2.0 in3.0 in  2.0 in  3.0 in    1 in   1 in 

 6.0  2.54 cm2.54 cm  38.7 cm2

4. Mixed Conversions

The last category of conversions involves changing two or more units at the same time. These units may appear in both the numerator and the denominator of a measurement. Derived quantities like speed (mi/hr), pressure (lb/in2), and heart rate (beats/min) are examples of quantities that require two or more units to completely describe them. When converting values having multiple units, each unit must be handled separately.

Example 4

A car is moving with a speed of 60 mi/hr. What is its speed in units of ft/sec?

Solution:

In this example, we have to convert miles to feet in the numerator and hours to seconds in the denominator.

mi mi ft 1 hr 1 min ft 60  60 5,280    88 hr hr mi 60 min 60 sec sec

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Exercises:

Perform the following conversions using only the conversion factors provided. Don’t look up any conversion factors on the Internet or use the “Convert” feature on your calculators!!! The purpose of these exercises is to learn a process --- not to get an answer.

Direct Conversions:

1. 2.9 kg = gm

2. 14.0 in = cm

3. 3.2 lbs = kg

Chain Conversions:

4. 2.7 kg = ozs

5. 9.75 hr = sec

6. 2.0 mile = yds

Power Conversions:

7. 4.6 m2 = cm2

8. 5 yd2 = ft2

9. 12 in3 = cm3

Mixed Conversions:

10. 30 mi/hr = ft/sec

11. 6 lbs/ft = ozs/in

12. 30 kg/hr = lb/min

Combinations:

13. 2 liters/sec = gal/min

14. 9.0 gm/cm3 = lbs/ft3

15. 8.0 lbs/in2 = oz/ft2

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Some helpful conversions:

1 inch = 2.54 cm 1 km = 1,000 meters 1 day = 24 hrs 1 foot = 12 inches 1 kg = 1,000 gm 1 min = 60 seconds 1 mile = 5,280 feet 1 kg = 2.20 lbs 1 hour = 60 min 1 yard = 3 feet 1 lb =16 ozs 1 liter = 1,000 cm3 1 meter = 100 cm 1 oz = 28.4 gm 1 gal = 3.78 liters

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Answers:

1. 2,900 gm

2. 35.6 cm

3. 1.45 kg

4. 95 oz

5. 35,100 sec

6. 3,520 yds

7. 46,000 cm2

8. 45 ft2

9. 197 cm3

10. 44 ft/sec

11. 8 oz/in

12. 1.1 lb/min

13. 31.8 gal/min

14. 560 lb/ft3

15. 72 oz/ft2

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