0.8 Phy1050: units and dimensional analysis  (Page 6/11)

How many inches are in a mile?

At this point, we don't know how many inches are in a mile (but we could calculate it if we wanted to). However, we do know how many inches are in a foot and we know howmany feet are in a mile.

A conversion factor for inches to feet

For starters, lets calculate a factor that we can use to convert from inches to feet. We know that

12*inches = 1*foot

If we divide each side by 12*inches, we get

1 = 1*foot/12*inches

Let's refer to this as a conversion factor

Multiplication by 1

Since the expression on the right side is equal to 1, we can multiply any expression in an equation with the conversion factor

1*foot/12*inches

without changing the intrinsic value of that expression. (Once again, although we won't be changingthe intrinsic value of the expression, we may cause that intrinsic value to be expressed in different units.)

We can use this conversion factor to convert a distance value expressed in inches into the value for thesame distance expressed in feet.

A conversion factor for feet to inches

We could also use the reciprocal of the expression to convert a distance value expressed in feet into the value for thesame distance expressed in inches. I will have more to say on this later.

A conversion factor for feet to miles

Similarly, we could use the same procedure to show that

1 = 1*mile/5280*foot

Again, since the expression on the right side is equal to 1, we can multiply any expression in an equation with the expression on the right without changingthe intrinsic value of the original expression. This conversion expression can be used to convert from feet to miles. Its reciprocal could be used to convert from milesto feet.

Back to the problem at hand

We determined earlier that the distance expressed in inches from home to school is

distance = 63360 * inch

We can probably agree that multiplying the expression on the right as follows would not change the intrinsic value for the distance.

distance = 63360 * inch * 1 * 1

In fact, we could probably agree that it wouldn't change anything at all.

Substitute the two conversion factors

Having agreed on that, we can substitute the two conversion factors derived earlier, each of which has a value of 1, for the last two factors in our distanceequation.

distance = 63360 * inch * (1*foot/12*inch) * (1*mile/5280*foot)

Rearranging the terms

We can rearrange the terms and rewrite this equation as

distance = (63360 * inch * 1*foot * 1 * mile)/(12*inch*5280*foot)

Canceling like terms

Canceling out like terms in the numerator and denominator leaves us with

distance = (63360*mile)/(12*5280)

Doing the arithmetic, we get

distance = 1*mile

(Obviously, I started out with a set of numbers that were designed to cause the final answer to be one mile, but that was simply for convenience.)

There you have it

You have seen a detailed procedure for converting from a distance expressed in paces (for a specific individual) to the same distanceexpressed in miles. Obviously, once you understand the overall procedure, you could omit and/or combine some of the steps to simplify the process.