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The Art Of Dimensional Analysis

Dimensional analysis is a tool that can be applied in both science and mathematics, and one that I believe is heavily undervalued. It has come to my aid many times, in both exams and research.

Dimensional analysis is a simple and easy way of checking that an expression is consistent. But it can also be turned on its head and be used as a helping hand in deriving expressions that were badly revised or poorly understood. An expression, which is dimensionally consistent, is not necessarily correct but we can say that it could be correct or is probably a good approximation. Often it is enough to give you that little nudge to help you in the right direction.

An example:

The moon (with mass 7.3×10^22 kg) is 380,000 km away from the Earth (of mass 6.0×10^24 kg). What is the gravitational force that the Earth exerts on the moon? G = 6.67 x 10^-11 m^3 kg-1 s^-2

You’ve forgotten the expression for the gravitational force between two masses. However, you have remembered Newton’s Second Law; F=ma. And you also know that the units on the left hand side of the equals sign MUST equal the units on the right hand side.

This implies that the force has units kg m s^-2. You want to work out the gravitational force and the question gives you 4 quantities that you must somehow combine to have the same units as force.

[G] = m^3 kg^-1 s^-2

[m_E] = [m_M] = kg

[d] = m

So you can see that [F] = [G]*[m_E]*[m_M]/[d]^2 and hence you guess that the expression you have forgotten was;

F=Gm_Em_M/d^2

Which is in fact the correct answer. You then plug in the values given in the question and get full marks even though you hadn’t directly remembered the equation for gravitational force.

 

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