Dimensional Analysis a problem solving process and unit conversion process

The “Steps” in the Process We need to follow 6 steps in every problem - make sure you never leave out the units!!!!! Follow the Steps!!!

The “Steps” in the Process Identify the Known/Given, including units Multiply all the numerator values & multiply all the denominator values and then divide the numerator answer by the denominator answer Flip the fraction so that the units will cancel Write the conversion factor(s) as fractions Write the Given over 1 Identify the Unknown, including units

Sample Problem We want to convert miles to kilometers The “given” would be 25.6 miles, which we want to convert to kilometers (the “unknown”) To convert you will need a “Conversion factor”

What is a “Conversion factor”? A “conversion factor” is an equality of measurements Example: 1 foot = 12 inches Example: 1 meter = 100 centimeters Example: 2.54 centimeters = 1 inch Example:.621 miles = 1 kilometer

How a Conversion factor works Since a Conversion factor is an equality - it can be written as; 2.54 centimeters = 1 inch or 1 inch = 2.54 centimeters As a fraction it can be written as;

Setting up the problem Write the given over 1 The unit “miles” is in the numerator, so the miles part of the Conversion factor must be in the denominator. The horizontal line means to divide & a vertical line means to multiply

Complete the problem Cancel the unit miles in the numerator and denominator.

Complete the problem Cancel the unit miles in the numerator and denominator. Then multiply all the numerator values & all the denominator values and then divide the numerator answer by the denominator answer

Multiple Steps (Conversion factors) You may need to use more than one conversion factor to arrive at the answer

Multiple Steps (Conversion factors) You may need to use more than one conversion factor to arrive at the answer Example: To convert 4.6 meters to inches

Multiple Steps (Conversion factors) You may need to use more than one conversion factor to arrive at the answer Example: To convert 4.6 meters to inches, you will need to convert; meters to __________

Multiple Steps (Conversion factors) You may need to use more than one conversion factor to arrive at the answer Example: To convert 4.6 meters to inches, you will need to convert; meters to centimeters to ______

Multiple Steps (Conversion factors) You may need to use more than one conversion factor to arrive at the answer Example: To convert 4.6 meters to inches, you will need to convert; meters to centimeters to inches

Set up Multiple Steps Write the given over 1

Set up Multiple Steps Write the given over 1 The unit “meters” is in the numerator, so the meters part of the Conversion factor must be in the denominator.

Set up Multiple Steps Write the given over 1 The unit “meters” is in the numerator, so the meters part of the Conversion factor must be in the denominator.

Next Conversion factor Now convert the centimeters to inches

Next Conversion factor Now convert the centimeters to inches You can have multiple conversions, BUT in the final step you will still only multiply twice (the numerator and denominator) and divide once.

Completing the problem Cancel the units – meters in the numerator cancels meters in the denominator.

Completing the problem Cancel the units – meters in the numerator cancels meters in the denominator Now cancel the centimeters, leaving only the unit inches.

Complete the problem Multiply all the numerator values & multiply all the denominator values and then divide the numerator answer by the denominator answer

Measurement-with a complex unit A complex unit is made up of multiple measurement units.

Measurement-with a complex unit A complex unit is made up of multiple measurement units. Example: Velocity 344 m/s

Measurement-with a complex unit A complex unit is made up of multiple measurement units. Example: Velocity 344 m/s Example: Area2.45 m 2

Measurement-with a complex unit A complex unit is made up of multiple measurement units. Example: Velocity 344 m/s Example: Area2.45 m 2 Example: Volume138 cm 3

Set-up with complex units Each unit must be handled separately

Set-up with complex units Each unit must be handled separately Complex units like Velocity (m/s) – the “m” goes in the numerator and the “s” goes in the denominator and each must be handled separately

Set-up with complex units Each unit must be handled separately Complex units like Velociy (m/s) – the “m” goes in the numerator and the “s” goes in the denominator and each must be handled separately Complex units like Area (m 2 ) – the unit “m 2 ” is the same as “m x m” and each “m” must be handled separately

Velocity Example The speed of sound in air is 344 m/s, what is it in km/hr? Follow the steps that we have learned.

Velocity Example The speed of sound in air is 344 m/s, what is it in km/hr? Follow the steps that we have learned. Write the given over 1 Meters goes in the numerator & seconds goes in the denominator 344 m/s

Velocity Example The speed of sound in air is 344 m/s, what is it in km/hr? Follow the steps that we have learned. Write the given over 1 Meters goes in the numerator & seconds goes in the denominator 344 m/s

Velocity Example Now use your conversion factors to change “m” to “km” and “s” to “hr”, so our answer will be in “km/hr”? Convert the “m” to “km” first and then “s” to “hr” next. Multiply all the numerator values & multiply all the denominator values and then divide the numerator answer by the denominator answer

Remember/Reminder Your Given is always written over one. You must have units with all conversion factors. You multiply twice and divide once. You must cancel the units.