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

Consider the physical quantities π‘š, 𝑠, 𝑣, π‘Ž, and 𝑑 with dimensions. [π‘š] = 𝑀, [𝑠] = 𝐿, [𝑣] = 𝐿𝑇⁻¹, [π‘Ž] = 𝐿𝑇⁻², and [𝑑] = 𝑇. The equation 𝑇 = π‘šπ‘ /π‘Ž 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‘š, 𝑠, 𝑣, π‘Ž, and 𝑑 with dimensions. Dimensions of π‘š equals capital 𝑀. Dimensions of 𝑠 equals capital 𝐿. Dimensions of 𝑣 equals 𝐿𝑇 to the negative one. Dimensions of π‘Ž equals 𝐿 times 𝑇 to the negative two. And dimensions of 𝑑 equals capital 𝑇. The equation 𝑇 equals π‘šπ‘  over π‘Ž 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 𝑇 that appears on the left-hand side of the given equation. Since we’re given an equation that we’re told is dimensionally consistent, we can write that the dimensions of 𝑇 is equal to the dimension of π‘š times the dimension of 𝑠 divided by the dimension of π‘Ž. If we enter these dimensions in. Where the dimension of π‘š is capital 𝑀. The dimension of 𝑠 is capital 𝐿. And the dimension of π‘Ž is 𝐿 times 𝑇 to the negative two. We see that, in this fraction, the factors of length 𝐿 cancel out. And we can write it in a simplified form as capital 𝑀 times 𝑇 squared. This is the net dimension of the right-hand side of the equation. And since the equation is dimensionally consistent, it’s also the dimension of capital 𝑇. So our result is capital 𝑀 times capital 𝑇 squared.

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