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

Consider the physical quantities 𝑚, 𝑠, 𝑣, 𝑎, and 𝑡 with dimensions. [𝑚] = 𝑀, [𝑠] = 𝐿, [𝑣] = 𝐿𝑇⁻¹, [𝑎] = 𝐿𝑇⁻², and [𝑡] = 𝑇. The equation 𝑇 = 𝑚𝑠/𝑎 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 𝑚, 𝑠, 𝑣, 𝑎, and 𝑡 with dimensions. Dimensions of 𝑚 equals capital 𝑀. Dimensions of 𝑠 equals capital 𝐿. Dimensions of 𝑣 equals 𝐿𝑇 to the negative one. Dimensions of 𝑎 equals 𝐿 times 𝑇 to the negative two. And dimensions of 𝑡 equals capital 𝑇. The equation 𝑇 equals 𝑚𝑠 over 𝑎 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 𝑚 times the dimension of 𝑠 divided by the dimension of 𝑎. If we enter these dimensions in. Where the dimension of 𝑚 is capital 𝑀. The dimension of 𝑠 is capital 𝐿. And the dimension of 𝑎 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.