Dimensional Analysis

The correct scientific measurement for any physical quantity is represented by an accurate numerical value expressed with the desired SI unit.

Dimensional analysis, also known as the factor label method, is a mathematical approach that is based on the principle that the units of quantities must be subjected to the same mathematical operations as their associated numbers.

Often, a single physical quantity can be expressed in different, but equivalent, units. For instance, the length of an object can be expressed as meters or centimeters, where 1 meter represents the same amount of length as 100 centimeters.

Dimensional analysis facilitates the conversion between such equivalent units through the use of the unit conversion factor.

The unit conversion factor is a ratio between two different units that measure the same physical quantity.

For instance, length in meters or centimeters can be interconverted, using conversion factors: 1 meter per 100 centimeters, and 100 centimeters per 1 meter.

The ratio of choice depends on the desired unit in the result. Thus, to determine the length in meters, the correct unit conversion factor is the ratio that cancels the units of centimeters and leaves meters.

Consider a 500 centimeter tall giraffe. To express its height in meters, both numbers and units, should be multiplied by the appropriate conversion factor. The numbers yield the product quantity, 5, and the units cancel out each other, except meters. Thus, the height of the giraffe is equal to 5 meters.

Sometimes a physical quantity that cannot be measured directly is calculated from other directly measured properties using equations and arithmetic operations. For instance, the density of an object can be calculated from its mass and volume.

Consider a plastic ball that has a mass of 12 g and a volume of 6 cm³. Its density can be determined by dividing its mass by its volume. The numbers and units are divided to yield the product quantity as 2 g/cm³.

Units, like numbers, are carried through all the steps of a calculation. In the end, the product quantity should have the desired units, and if not, this indicates errors in the use of conversion factors.

For example, the kinetic energy of a dog with a mass of 45 kg running at a speed of 11 m/s can be calculated using the mathematical equation, mass times velocity squared, divided by 2.

The speed of the dog is squared to 121 m²/s² and multiplied by the mass of 45 kg.

Lastly, the total quantity is divided by 2 — yielding the kinetic energy of the dog as 2722.5 kg·m²/s². Taking into account significant figures rounds the kinetic energy to 2700, or 2.7 × 10³ kg·m²/s².

The SI unit of energy is the joule, which is equal to one kilogram meter squared per second squared, or 2.7 × 10³ joules.