Dimensional analysis to determine unknown exponents

[Vasili]

Homework Statement

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.

Homework 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.

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?

 

[Vasili]

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

 

[thomate1]

I would say that you are on the right track with your dimensional analysis approach to solving for the unknown exponents in Equation (1). Your initial step of determining the dimensions for each term is correct, and your dimensional equation is also correct.

To solve for the unknown exponents, you can use the equations you derived for [L], [T], and [M] and substitute them into your original dimensional equation. This will give you a system of three equations with three unknowns (l, m, and n). You can then solve this system using algebraic manipulation and substitution to find the values of l, m, and n.

Alternatively, you can use the relationships between the dimensions (i.e. [L] is equal to [M]^l [T]^n) to determine the values of the exponents. For example, from equation (iii), you can see that l=-n, which can then be substituted into equations (i) and (ii) to solve for m and n. This approach may be simpler and more straightforward.

Overall, your approach using dimensional analysis is a valid and effective method for solving for the unknown exponents in Equation (1). Keep up the good work!

 

[baba89]

Great job breaking down the dimensions for each term in the equation and setting up the dimensional equation. From here, you can solve the three equations simultaneously to find the values of l, m, and n. To do this, you can use substitution or elimination methods.

Using substitution, you can rearrange equation (iii) to solve for l in terms of n: l=-n. Then, substitute this into equation (i): -1=-(-n)+n. Simplifying, you get n=1/2.

Next, substitute this value of n into equation (iii) to solve for l: l=-1/2.

Finally, substitute these values of l and n into equation (ii) to solve for m: 0=-1m-2(1/2). Simplifying, you get m=-1.

Therefore, the unknown exponents in equation (1) are l=-1/2, m=-1, and n=1/2. Great job using dimensional analysis to determine these values!