### Video Transcript

Consider the equation π equals π zero plus π£ zero π‘ plus π zero π‘ squared divided by two plus π zero π‘ cubed divided by six plus capital π zero times π‘ to the fourth divided by 24 plus π times π‘ to the fifth divided by 120, where π is a length and π‘ is a time. What is the dimension of π zero? What is the dimension of π£ zero? What is the dimension of π zero? What is the dimension of π zero? What is the dimension of capital π zero? And what is the dimension of π?

This is a question in dimensional analysis, where we want to solve for the dimension of each unknown term in the equation for π . We can symbolise the dimension of a variable by writing βdimβ and then in parentheses that variable. For example, weβre told the dimension of π is a length, which weβll abbreviate capital πΏ, and that the dimension of π‘ is time, which weβll abbreviate capital π.

Considering this equation, we can now solve for the dimensions of each of the variables in it. Looking at π zero where it appears in the equation, we see that itβs combined with no other terms but stands on its own, so the dimension of π zero must be the same as the dimension of π . That means the dimension of π zero is length. Next, we look at π£ zero. π£ zero we see is multiplied by π‘, which has units of time, so the dimension of π£ zero multiplied by capital π must be equal to the dimension of π length. Therefore, π£ zero has dimensions of length per time.

One way to write that is length times inverse time. When we consider π zero, we see that that value is multiplied by π‘ squared, so the dimension of π zero when multiplied by π‘ squared must yield the dimensions of π , length, therefore π zero has dimensions of length per time squared.

Another way to write that is length times time to the negative two. Looking next at π zero, that value is multiplied by π‘ cubed since π zero times π‘ cubed must have units of length. In order for it to match with the units of π , the dimension of π zero is length per time cubed or length times time to the negative three. Next, we look at capital π sub zero.

Capital π sub zero multiplied by π‘ to the fourth has units of length, so the dimensions of π zero equals length per time to the fourth power or length times time to the negative fourth. Finally, we look at π. This value multiplied by π‘ to the fifth has the same units as π , the units of length, therefore the dimensions of π are length per π‘ to the fifth, which can be equivalently written length times time to the negative five.

These are the various dimensions of the variables written in this equation.