Dimensional Analysis

[gdmiccc]

Hi,
In the Dimensions section of Chapter 5, one of the examples states that since "exp(x) is dimensionless, we know that lambda(t) must be dimensionless, in other words t has the same dimension as 1/lambda"

Also, in the solution explanation of Question 5.7 is states that "Since the argument of exp(x) is dimensionless, we know that x has the same dimension as 1/lambda"

If anyone could elaborate or explain how "we know" these statements, I would greatly appreciate it. Thanks.

 

[John Lee]

Hi,
In the Dimensions section of Chapter 5, one of the examples states that since "exp(x) is dimensionless, we know that lambda(t) must be dimensionless, in other words t has the same dimension as 1/lambda"

Also, in the solution explanation of Question 5.7 is states that "Since the argument of exp(x) is dimensionless, we know that x has the same dimension as 1/lambda"

If anyone could elaborate or explain how "we know" these statements, I would greatly appreciate it. Thanks.

Yeah - it's not particularly clear. I guess the way I think about it is that if it wasn't then the power series expansion would cause problems:

\( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... \)

So if x is measured in cm then:

1 is dimensionless, x is cm, x² is cm² and so on...