1.4.3 Dimensional Analysis

Often while calculating, there is a need to convert units from one system to the other. The method used to accomplish this is called factor label method or unit factor method or dimensional analysis.

This is illustrated below.

Example

A piece of metal is 3 inch (represented by in) long. What is its length in cm?

Solution

We know that 1 in = 2.54 cm From this equivalence, we can write

1 in / 2.54 cm = 1 = 2.54 cm / 1 in

Thus, (1 in / 2.54 cm) and (2.54 cm / 1 in)  equals 1.Both of these are called unit factors. If some number is multiplied by these unit factors (i.e., 1), it will not be affected otherwise.

Say, the 3 in given above is multiplied by the unit factor. So,

3 in = 3 in × (2.54 cm / 1 in) = 3 × 2.54 cm = 7.62 cm

Now, the unit factor by which multiplication is to be done is that unit factor which gives the desired units i.e., the numerator should have that part which is required in the desired result. It should also be noted in the above example that units can be handled just like other numerical part. It can be cancelled, divided, multiplied, squared, etc. Let us study one more example.

Example

A jug contains 2L of milk. Calculate the volume of the milk in m3.

Solution

Since 1 L = 1000 cm3 and 1m = 100 cm, which gives

(1m / 100 cm) = 1 = (100cm / 1m)

To get m3 from the above unit factors, the first unit factor is taken, and it is cubed.

(1m / 100 cm)3= 1 m3/ 106cm3= (1)3 = 1

Now 2 L = 2×1000 cm3

The above is multiplied by the unit factor

2×1000 cm3x 1 m3/ 106cm3 = 2 m3 / 103 = 2 X 10-3 m3

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