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A dimensional equation is a tool used in what is called dimensional analysis. It is a procedure to determine whether or not an equation is plausible. If you are doing theoretical research or just solving a physics problem all equations must be true in terms of the units used on both sides of the equation. Lets take the basic equation F=ma. In a dimentional equation we surround each variable by ="the units of":
[F]=[m][a] means the units of F equals the units of m times the units of a
In SI units this means that Newtons = kilograms x m/sec2. A Newton is in fact 1 kgm/sec2 so it checks out. If it didn't then there is something wrong with the equation. It would mean that either I did the derivation of the problem incorrectly.
A dimensional equation can also help us to determine the units of unknown quantities. When Newton wrote his law of universal gravitation F=GmM/d2 G is a constant of proportionality that makes both sides of the equation correct. We can do experiments to determine the numerical value of G but what are the dimensions of G? We will rewrite Newton's equation and them make a dimantional equation out of it.
G=Fd2/mM. So [G]=[F][d]2/[m][M].
In SI units this means the units of G are Newtons m2 / kg2. In English units the units of G are pound ft2 / stone 2.
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