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Dimensional Homogeneity


A dimensionally homogeneous equation is an equation which has balanced units on each side.
For example, the very simple equation:
F=ma
Has Force in Newtons on one side and on the other has mass (in kg) and acceleration (in [math]m/s^2[/math]).
so we can examine the dimensions on each side of the equation:
[math]N = kg \times m/s^2[/math]
1 newton is 1[math]kgm/s^2[/math] so this equation is said to be dimensionally homogeneous.

Not Dimensionally Homogeneous example 1:
now, say for we are doing a question and we are given a mass in tonnes in a question, we check for dimensional homogeneity
[math]N = t \times m/s^2[/math]
one newton is not 1[math]tm/s^2[/math] so we need to do some unit conversion before we can apply this formula to our given information.

Not Dimensionally Homogeneous example 2:
Now, lets say for example we are now incorrectly given the mass of an object in m/s (velocity). Obviously mass is not measured in m/s but for some reason that is what it has been to be determined in this example. so lets see if our force equation will work and be dimensionally homogeneous:
[math]N = m/s \times m/s^2[/math] Therefore: [math]N = m^2/s^3[/math]
Because 1 Newton is not 1 [math]m^2/s^3[/math] this equation is invalid and not dimensionally homogeneous.