# Dimensional analysis to determine unknown exponents

1. Oct 25, 2008

### Vasili

1. The problem statement, all variables and given/known data
1. Use the method of Dimensional Analysis to show that the unknown exponents in Equation (1) are l=-1/2, m=-1, and n=1/2.

2. Relevant equations
Equation (1) is $$\lambda = k \mu ^{l} f ^{m} T^{n}$$

Where:
$$\lambda$$ is the wavelength;
f is the frequency of the sound;
T is the tension in the string;
$$\mu$$ is the mass per unit length of the string.
k is a dimensionless constant.

3. The attempt at a solution
The dimensions for the above terms should be:
$$\lambda = [L]$$ (Simple enough)
$$f=[L] ^{-1}$$ (Since the frequency is the inverse of time. Is this correct?)
$$T=[M][L][T] ^{-2}$$ (Since the tension in the rope is just the force exerted on it, right?)
$$\mu = [M][L] ^{-1}$$ (Since it is the mass per unit length)

Which gives the dimensional equation as:
$$[L]=([M] \cdot [L]^{-1}) ^{l} \cdot ([T] ^{-1}) ^{m} \cdot ([M] \cdot [L] \cdot [T]^{-2})^{n}$$

Which can be used to make equations for [L], [T], and [M], respectively:

1=-1l + 1n ([L]) (i)
0=-1m - 2n ([T]) (ii)
0=1l + 1n ([M]) (iii)

And from here I don't know where to go. If I manipulate (ii) to state n in terms of m, I get n=-1/2m. But where do I go from here? I need to solve these three equations simultaneously?

Last edited: Oct 25, 2008
2. Oct 25, 2008

### Vasili

Oh, I got it. Sorry, I keep doing this with my posts here. XD