Dimensional Analysis

What is Dimensional Analysis? Dimensional analysis is a problem-solving method that uses the idea that any number or expression can be multiplied by one without changing its value. It is used to go from one unit to another.

Organized method of problem-solving Used in chemistry, physics, engineering, and medicine Communicates the path to scientists that follow your work Records your own path for your future use

How Does Dimensional Analysis Work? A conversion factor, or a fraction that is equal to one, is used, along with what you’re given, to determine what the new unit will be.

You are not changing the size, just the name. CONVERSIONS You are not changing the size, just the name.

Examples of Conversions 60 s = 1 min 60 min = 1 h 24 h = 1 day

Fill in the Missing Numbers 12 1 foot = _____ inches 1 meter = _____ centimeters 1 pound = _______ ounces 1 minute = ______ seconds 1 hour = ________ minutes 1 day = __________ seconds 100 16 60 60 What is the purpose of knowing these facts? Give me some examples. 86,400

Unit conversion factor A unit conversion factor is a fraction whose numerator and denominator are equivalent measures. Some common unit conversion factors are given below. You can also use the reciprocal of these. 1 ft ? in. 1 yd ? ft 1 mi 1lb ? oz 1 pt ? c 1 qt ? pt 1 gal ? qt 1 hr ? min 1 min ? s 1m ? cm 1km ? m Fill in the blanks for as many as you know. Compare with your neighbor.

Unit conversion factor A unit conversion factor is a fraction whose numerator and denominator are equivalent measures. Some common unit conversion factors are given below. You can also use the reciprocal of these. Ask for some reciprocals so you know they understand.

Choose a unit conversion factor that… Introduces the unit you want in the answer Cancels out the original unit so that the one you want is all that is left.

“Canceling” out Words

Examples of Conversions You can write any conversion as a fraction. Be careful how you write that fraction. For example, you can write 60 s = 1 min as 60s or 1 min 1 min 60 s

Examples of Conversions Again, just be careful how you write the fraction. The fraction must be written so that like units cancel.

Practice: Choose the appropriate conversion factor. Inches to feet Minutes to hours Meters to centimeters

Use the one that will cancel out the units you want to change Convert 8 yards to feet… Make a decision: What conversion factor will you use? Set up the problem: Multiply the measurement by the conversion factor. Hint! Use the one that will cancel out the units you want to change Solve the problem: Perform the multiplication

A bucket holds 16 quarts. How many gallons of water will fill the bucket? Use a unit conversion factor to convert the units. What are the two conversion factors comparing quarts and gallons? Which one will “cancel” quarts? 16 qt 

You Try it! One bag of apples weighs 64 ounces. How many pounds does it weigh? Darren drank 2 liters of water. How many milliliters of water did he drink?

Steps Start with the given value. Don’t forget the units!!!! Draw the chart. Write the multiplication symbol. Choose the appropriate conversion factor. Multiply the numbers & cancel the units. Remember, cancel like units. Reduce the fraction (if needed). Count significant figures and round to the appropriate number.

Let’s try some examples together… Suppose there are 12 slices of pizza in one pizza. How many slices are in 7 pizzas? Given: 7 pizzas Want: # of slices in 7 pizzas Conversion: 12 slices = one pizza

conversion 7 pizzas 12 slices = 84 slices 1 pizza Suppose there are 12 slices of pizza in one pizza. How many slices are in 7 pizzas? conversion Starting amount End Amount 7 pizzas 12 slices = 84 slices 1 pizza

Let’s try some examples together… 2. How old are you in days? Given: 15 years Want: # of days old Conversion: 365 days = one year

conversion 15 years 365 days = 5475 days 1 year How old are you in days? conversion Starting amount End Amount 15 years 365 days = 5475 days 1 year

Let’s try some examples together… 3. There are 2.54 cm in one inch. How many inches are in 17.3 cm? Given: 17.3 cm Want: # of inches Conversion: 2.54 cm = one inch

There are 2.54 cm in one inch. How many inches are in 17.3 cm? conversion Starting amount End Amount 17.3 inches 2.54 cm = 43.9 cm 1 inch

Now, you try… Determine the number of eggs in 23 dozen eggs. If one package of gum has 10 pieces, how many pieces are in 0.023 packages of gum?

Multiple-Step Problems Most problems are not simple one-step solutions. Sometimes, you will have to perform multiple conversions. Example: How old are you in hours? Given: 17 years Want: # of days Conversion #1: 365 days = one year Conversion #2: 24 hours = one day

How old are you in hours? = 131,400 hours 15 years year 365 days Starting amount conversion conversion End Amount 15 years 365 days 24 hours = 131,400 hours year 1 day

Combination Units Dimensional Analysis can also be used for combination units. Like converting km/h into cm/s. Write the fraction in a “clean” manner: km/h becomes km h

Combination Units Example: Convert 0.083 km/h into m/s. Given: 0.083 km/h Want: # m/s Conversion #1: 1000 m = 1 km Conversion #2: 1 hour = 60 minutes Conversion #3: 1 minute = 60 seconds

Convert 0.083 km/h into m/s 0.083 km hour = 0.023 m s 1000 m 1 km reduce Starting amount conversion conversion multiply End Amount 0.083 km hour 1000 m 1 km 1 hour 3600 s 83 km 3600 s = 0.023 m s