1.3.5 Dimensional Analysis

Question 1

Dimensional Analysis

Answer 1

• A conversion factor is always a factor of one constructed from a known equality.
• Multiple conversion factors are sometimes necessary when converting from one unit to another.
• Temperature conversions are simpler when the proper units are placed on the conversion factor.

Question 2

conversion factor

Answer 2

• A conversion factor is always a factor of one constructed from a known equality. For example, 1 dollar = 106.823 yen, so 1 dollar / 106.823 yen = 106.823 yen / 1 dollar = 1.
• Conversion factors are useful for converting one quantity to another. For example, an amount in dollars could be converted to yen by multiplying by 106.823 yen / 1 dollar. The dollars in the numerator cancel with the dollars in the denominator, leaving yen. This method of converting quantities by multiplying by conversion factors to cancel units is called dimensional analysis.

Question 3

multiple conversions

Answer 3

• Multiple conversion factors are sometimes necessary when converting from one unit to another.
• For example, miles per hour (mi/hr) can be converted to meters per second (m/s) by using multiple conversion factors to convert from miles to feet, feet to inches, inches to centimeters, centimeters to meters, hours to minutes, and minutes to seconds. It is always good to check that the units cancel properly. If not, a conversion factor might have the numerator and denominator reversed.

Question 4

temperature

Answer 4

• Temperature conversions are more correct when the proper units are place on the conversion factor.
• For example, it is easier to make sure that the proper
  conversion formula is used if units are placed on the
  conversion factor from temperature in degrees Celsius(T(˚C)) to temperature in degrees Fahrenheit (T(˚F)). Because the size of a degree in Celsius is the same as a kelvin, temperature in degrees Celsius (T(˚C)) can be converted to temperature on the Kelvin scale (T(K)) by adding 273.15 K.
• Units should always be carried along when making
  conversions.

Question 5

In order to install new carpeting, it is important to first know the dimensions of a room. You should know the area (length ⋅ width) of the floor. Carpeting in the United States is typically sold in units of area, such as square feet (or ft 2 ). Suppose you want to buy carpeting for a room. You have only a meter stick to measure the room’s dimensions. The length is 5.50 m and the width is 3.70 m. If the conversion factor for meters to feet is approximately 1 meter = 3.25 ft, about how much carpeting should you buy? (Note: it is usually a good idea to buy a little extra.)

Answer 5

• about 225 ft^ 2

Question 6

One kilogram is equal to about 2.20 pounds. Both kilograms and pounds are units of mass. What is the equivalent weight in pounds of a 37.5 kilogram object?

Answer 6

• 82.5 lb

Question 7

In the United States, people sometimes find it necessary to convert between kilometers and miles. The standard conversion is 1 km = 0.62 miles. If a typical marathon race is 42.3 km long, about how long is the race in miles? (1 km = 0.62 miles.)

Answer 7

• 26.2 miles

Question 8

To convert inches to centimeters, you multiply by the number of inches by

Why does this work?

Answer 8

• The conversion factor is always equal to 1.

Question 9

What is 25°C in Kelvin?

Answer 9

• 298 K

Question 10

What is 315 K in degrees C?

Answer 10

• 42°C

Question 11

Suppose a cable factory creates a spool of wire that is 1.5 miles long. Which of the following conversion factors should 1.5 miles be multiplied by to convert the measurement to inches?

Answer 11

• (1,760 yd/1 mile) × (3 ft/1 yd) × (12 in./1 ft)

Question 12

There are always two conversion factors to choose from for every conversion that you want to do (e.g. 12 in./1 ft or 1 ft/12 in.). Which statement best describes why you should multiply by one conversion factor and not the other?

Answer 12

• When choosing a conversion factor, you have to know what units your data is presented in (i.e. your original piece of data) and what units you want to convert to. You want to choose the conversion factor that will cancel the original units, leaving the new units.

Question 13

Suppose you run a total of 4.35 miles in 58.75 minutes. If you want to determine your average running speed in units of km / hr, how many significant digits will your final answer have? The following conversion equivalents are given to you in order to do the conversions: 1 mile = 5,280 ft, 1 ft = 12 in., 1 in. = 2.54 cm, 100 cm = 1 m, 60 minutes = 1 hour.

Answer 13

• 3
• The only data that matters, as far as significant figures is concerned, is the original data. The lecture told us that conversions are thought to be exact and that, if it helps you to understand better, you can think of each conversion value to have many significant figures (e.g. 1 mile = 1.00000 mile). So the only two values that affect the number of significant figures are the original values of 4.35 (3 significant figures) and 58.75 (4 significant figures). Since 4.35 has less, then your final answer should have three significant figures.

Question 14

A baseball is thrown at 42.5 m / s (that’s meters per second). What is the speed of the baseball in miles / hour? (1 mi = 5280 ft, 1 in = 2.54 cm)

Answer 14

• 95.1 mi / hr