Use of Dimensions to Derive Equations

If we have some idea or can make an educated guess as to how one physical quantity relates to another we can use dimensions to derive the form of the equation. As an example, consider the equation for the period of pendulum bob. We might suppose that the period depends on the mass of the bob, the length of the pendulum and the acceleration due to gravity

We can express this as T=mxlygz. Where x, y and z are as yet undetermined indices.

To find the values of x, y and z we convert the formula into its dimensions. On the left-hand side the dimension of the period is [T], the dimension of mass is [M]x, the dimension of the length of the pendulum is [L]y and the dimension of g is [LT-2].

[T]=[M]x[L]y[LT-2]z.

Equating left-hand indices with matching dimensions on the right-hand side.

[M]: 0=x

[L]: 0=y+z

[T]: 1=-2z

From this we can deduce that z=-1/2, while y=1/2 and x=1/2

Substituting these values into the original equation we obtain. T=m0l1/2g-1/2= k(l/g)1/2. Where k is a constant of proportionality. Compare this with the equation for the period of a pendulum T=2π(l/g)1/2. The form of the equation is correct, but it cannot determine the constant of proportionality.

Prefixes and Magnitudes

To make sense of the vast range over which physical quantities are measured, prefixes are used as a short-cut to writting the magnitude using scientific notation

Other prefixes which are commonly used but are not strictly part of the SI system.

Scientific Notation

Measuring physical phenomena in the real world we enevitably encounter numbers of large magnitude. The estimated (insert example here) is 1,000000000 * n zeros while the time taken for a photon of light to cross the radius of an atom is 0.0000000000000000000000000000000000000001.

Clearly, performing calculations with such unweildy numbers is entirely impractical. Therefore, we need ‘short-hand’ method for writing these numbers. For number with upto 18 significant figures (18 zeros to the right). We can use the prefixes. For anything that falls between the gaps, we write the number of zeros as the index of a power of 10.

For example, 1,000,000 is 1 x 106 as there are 6 zeros after the 1. and for 0.1 is 1 x 10-1. 100 = 1.