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Consider the four quantities π, π», π, and π, where the dimensions of π are mass times length squared times time, the dimensions of π» are mass to the negative one times length to the negative three, and the dimensions of π are mass squared times time to the negative one.
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The compound quantity π times π times π» over π is dimensionless.
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What are the dimensions of π?
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Of the four quantities named here, weβre given the dimensions of three of them: π, π», and π.
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Weβre also told that the quantity π times π times π» divided by π is dimensionless.
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Symbolically, we can write that as the dimensions of π times π times π» divided by π equals one.
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Knowing all this, we want to solve for the dimensions of π.
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Another way to write this fraction is as the dimensions of π times the dimensions of π times the dimensions of π» all divided by the dimensions of π.
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If we substitute in all the known dimensions of these quantities, we get this result.
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What weβll do is simplify this fraction as far as possible.
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And that will help us understand the dimensions of π.
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Notice that in our numerator we have mass times mass to the negative one.
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When multiplied together, these values equal one.
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Similarly, we have a length squared multiplied by a length to the negative three.
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The overall result of this is length to the negative one.
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This gives us the dimensions of π times πΏ to the negative one times π all divided by π squared times π to the negative one.
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Note that if we multiply numerator and denominator by the time π, then in the denominator, π cancels with one over π.
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Our result can be written this way: the dimensions of π times πΏ to the negative one times π squared over π squared.
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Letβs now recall that this product on the right is equal to one.
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Therefore, the dimensions of π, whatever they are, must effectively cancel out these dimensions.
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This means that the dimensions of π are equal to the inverse of these dimensions.
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Therefore, πβs dimensions are π squared divided by πΏ to the negative one times π squared.
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This is equal to π squared times πΏ all divided by π squared.
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So, in our dimensionless expression, π times π times π» divided by π, the dimensions of π are π squared times πΏ divided by π squared.