Problem Solving, Solving for the Unknown and Converting Units (Dimensional Analysis)

Solving for the Unknown

• An equation is given in a form to find a particular unknown, such as Density.

D = _m_ V• However, in many problems, you

may know the Density and need to find the Mass or the Volume.

Solving for the Unknown

• To find what you are looking for you may need to modify the equation to solve for the unknown.

• Suppose you were looking for Volume instead of Density, you would need to modify the equation as shown below.

D = _m_ VD = _mV_ V = _m_ V V D

Problem Solving

• 1. List the data and unknown.• 2. Convert the units as required.• 3. Select the equation.• 4. Modify the equation to solve for

the unknown.• 5. Substitute the data into the

equation.• 6. Solve for the answer units.

Problem Solving

• Example 1: If A = 2, B = 3, C = 3, what is D?

• Equation:A x B = C x D

Problem Solving

• 1. List the data and unknown:– A = 2– B = 3– C = 3– D = ?

Problem Solving

• 1. List the data and unknown:– A = 2– B = 3– C = 3– D = ?

• 2. Convert the units as required:– No units so no conversions required.

Problem Solving

• 3. Select the EquationA x B = C x D

Problem Solving

• 3. Select the Equation:A x B = C x D

• 4. Modify the Equation to Solve for the Unknown:

D = A x B C

Problem Solving

• 5. Substitute the data into the equation:

D = 2 x 3 3

Problem Solving

• 5. Substitute the data into the equation:

D = 2 x 3 3

• 6. Solve for the unknown.D = 2

Converting Units

• Units give numbers an identity. If I say a person is 6, you may assume I am talking about their age. However, I could mean that they are six feet tall, finished in sixth place, is six years from retiring, etc..

• You have to have units to understand what a number means.

Using Units

• Units work just like numbers. If you multiply 2 x 2, you get the square of 2 which is 4.

• If you multiply meters x meters you get meters square which is the unit of area.

Using Units

• How you handle units depends on their location and the type of mathematical operation you are performing.

• If the same units are both multiplied on the top or the bottom of the equation, then you will get the unit squared.

meters x meters = meters2

Using Units

• If the units appear on the top and the bottom of the equation, they cancel out.

_meters_ = 1

meters

• If the units are added or subtracted, they must be the same (meters and meters not meters and yards).

Converting Units

• In all mathematical operations, units must be the same or you cannot perform the mathematical operation.

• When the units are the same, we refer to this as using “common units.”

Converting Units

• If your units are not the same, you must convert one or both to perform the calculation.

• Converting units is also referred to as “Dimensional Analysis.”

Converting Units

• I know that you have been taught various “short-cut” methods in math. You need to forget these. They only work in limited situations and their use will result in loss of points.

• The one method that works in all types of unit conversion, except temperature, is “Dimensional Analysis.”

Converting Units

• Dimensional Analysis is a process. It is best illustrated by a simple example. I measure the length of a piece of wood and find that it is 3 feet. However, I need to know the length in inches so I can see how many 9 inch pieces I can produce.

Converting Units

• The calculation would be as follows;3 feet x _12 inches_ = 36 inches

1 foot

_36 inches_ = 4 pieces 9 inches/piece

• In order to do this calculation, I had to convert the units.

Constants

• Constants (unchanging values) occur in many equations. They usually come with units. An example is the Universal Gravitational Constant (G).

G = 6.67 x 10-11 N-m2/kg2

Constants

• In order to do a calculation using this value (G), you must match the units in the constant. This means that;– All forces must be in Newtons– All distances must be in meters– All masses must be in kilograms

Note: If your units do not match you constant, you must convert your units to the constant’s units. The constant controls the units.