In Chapter 5 "Planck Invents a Better Yardstick" he describes how Planck came up with a much more canonical system of units than the metric system. In this system the 3 big constants all have value 1. These are: c the speed of light, ℏ Planck's reduced constant, and G Newton's gravitational constant.
It is an interesting exercise both algebraically and in unit conversion to figure out what the basic units of length, time and mass should be in order to make this happen. Start with the following definitions of the basic constants:
- c = 299 792 458 m / s
- G = 6.67300 × 10-11 m3 kg-1 s-2
- ℏ = 1.05457148 × 10-34 m2
Then your goal is to convert units so that these three constants all end up equaling 1. You'll get 3 non-linear equations in three conversion factors. A very capable 9th grader should be able to do it.
I won't show you how to do it, but I'll give you the answer so you can check your work:
Letting l, t, and m are respectively the Planck units of length, time and mass and 1 meter = L l, 1 second = T t, and 1 kg = M m, and c, h, and G are respectively the speed of light, Planck's constant and Newton's gravitational constants as expressed in SI, you'll get the following results:
- L = √ ( c3 / Gℏ ) = 6.19 × 1034 Planck lengths per meter
- T = √ (c5 / G ) = 1.86 × 1043 Planck time intervals per second
- M = √ ( G / ℏ c ) = 4.59 × 107 Planck mass units per kilogram
