# Video: Using Dimensional Analysis to Determine the Dimenstions of a Quantity

Consider the physical quantities π, π , π£, π, and π‘ with dimensions. [π] = π, [π ] = πΏ, [π£] = πΏπβ»ΒΉ, [π] = πΏπβ»Β², and [π‘] = π. The equation π = ππ /π is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

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### Video Transcript

Consider the physical quantities π, π , π£, π, and π‘ with dimensions. Dimensions of π equals capital π. Dimensions of π  equals capital πΏ. Dimensions of π£ equals πΏπ to the negative one. Dimensions of π equals πΏ times π to the negative two. And dimensions of π‘ equals capital π. The equation π equals ππ  over π is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

We want to solve for the dimension of the term capital π that appears on the left-hand side of the given equation. Since weβre given an equation that weβre told is dimensionally consistent, we can write that the dimensions of π is equal to the dimension of π times the dimension of π  divided by the dimension of π. If we enter these dimensions in. Where the dimension of π is capital π. The dimension of π  is capital πΏ. And the dimension of π is πΏ times π to the negative two. We see that, in this fraction, the factors of length πΏ cancel out. And we can write it in a simplified form as capital π times π squared. This is the net dimension of the right-hand side of the equation. And since the equation is dimensionally consistent, itβs also the dimension of capital π. So our result is capital π times capital π squared.