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Vasili
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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?
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