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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 $m/s^2$).
so we can examine the dimensions on each side of the equation:
$N = kg \times m/s^2$
1 newton is 1$kgm/s^2$ 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
$N = t \times m/s^2$
one newton is not 1$tm/s^2$ 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:
$N = m/s \times m/s^2$ Therefore: $N = m^2/s^3$
Because 1 Newton is not 1 $m^2/s^3$ this equation is invalid and not dimensionally homogeneous.