Video: Using Dimensional Analysis to Find the Dimensions of Constants in a Fifth Degree Polynomial

Consider the equation π  = π β + π£βπ‘ + πβπ‘Β²/2 + πβπ‘Β³/6 + πβπ‘β΄/24 + ππ‘β΅/120, where π  is a length and π‘ is a time. What is the dimension of π β? What is the dimension of π£β? What is the dimension of πβ? What is the dimension of πβ? What is the dimension of πβ? And what is the dimension of π?

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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 βdimβ and then in parentheses that variable. For example, weβre told the dimension of π  is a length, which weβll abbreviate capital πΏ, and that the dimension of π‘ is time, which weβll 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 itβs 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.