Dimensional Analysis

1
Dimensional Analysis

• a problem solving process
• and
• unit conversion process

2
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!!!

3
The Steps in the Process

• Identify the Known/Given, including units

• Identify the Unknown, including units

• Write the Given over 1

• Write the conversion factor(s) as fractions

• Flip the fraction so that the units will cancel

• Multiply all the numerator values multiply all
  the denominator values and then divide the
  numerator answer by the denominator answer

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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 aConversion factor

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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

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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

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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
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Complete the problem

• Cancel the unit miles in the numerator and
  denominator.

9
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

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Multiple Steps (Conversion factors)

• You may need to use more than one conversion
  factor to arrive at the answer

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Multiple Steps (Conversion factors)

• You may need to use more than one conversion
  factor to arrive at the answer
• ExampleTo convert 4.6 meters to inches

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Multiple Steps (Conversion factors)

• You may need to use more than one conversion
  factor to arrive at the answer
• ExampleTo convert 4.6 meters to inches,you
  will need to convert meters to __________

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Multiple Steps (Conversion factors)

• You may need to use more than one conversion
  factor to arrive at the answer
• ExampleTo convert 4.6 meters to inches,you
  will need to convert meters to centimeters
  to ______

14
Multiple Steps (Conversion factors)

• You may need to use more than one conversion
  factor to arrive at the answer
• ExampleTo convert 4.6 meters to inches,you
  will need to convert meters to centimeters
  to inches

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Set up Multiple Steps

• Write the given over 1

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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.

17
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.

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Next Conversion factor

• Now convert the centimeters to inches

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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.

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Completing the problem

• Cancel the units meters in the numerator
  cancels meters in the denominator.

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Completing the problem

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

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Complete the problem

• Multiply all the numerator values multiply all
  the denominator values and then divide the
  numerator answer by the denominator answer

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Measurement-with a complex unit

• A complex unit is made up of multiple measurement
  units.

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Measurement-with a complex unit

• A complex unit is made up of multiple measurement
  units.
• Example Velocity 344 m/s

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Measurement-with a complex unit

• A complex unit is made up of multiple measurement
  units.
• Example Velocity 344 m/s
• Example Area 2.45 m2

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Measurement-with a complex unit

• A complex unit is made up of multiple measurement
  units.
• Example Velocity 344 m/s
• Example Area 2.45 m2
• Example Volume 138 cm3

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Set-up with complex units

• Each unit must be handled separately

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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

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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 (m2) the unit m2 is
  the same as m x m and each m must be handled
  separately

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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.

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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

344 m/s
Meters goes in the numerator seconds goes in
the denominator
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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

344 m/s
Meters goes in the numerator seconds goes in
the denominator
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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

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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.