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I'm asked to write an analytical expression for the uncertainty in the ratio of the resonance frequency of some "string" to the mode of vibration, which works out to always be equivalent to the resonance frequency for $m = 1$. I'll clarify precisely what I mean by this:

The resonance frequency for some mode $m$ is given by: $$f = m f_1$$

Where $f_1$ is the resonance frequency at mode $1$, given by:

$$f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$$

Where $L$ is the length of the string, $T$ is the tension, and $\mu$ is the mass per unit length of the string.

We want to determine the uncertainty in $\frac{f}{m}$, which is equal to $f_1$.


I'm not sure how to proceed with this. I'm supposed to end up with an expression consisting of "three fractional uncertainties", and I then ignore all but $1$ which will be "much larger than the rest". My interperetation is that the "three fractional uncertainties" refered to are that of the length ($L$), the tension $T$, and the mass per unit length, $\mu$. So, am I simply being asked to write:

$$\frac{\delta f_1}{f_1} = \frac{\delta L}{L} + \frac{\delta T}{T} + \frac{\delta \mu}{\mu}\;\;\text{?}$$

However, we're also given the simplified "error propagation" rules that follow:

$$\delta (x+y+z) = \delta x + \delta y + \delta z$$ $$\frac{\delta x/y}{x/y} = \frac{\delta x}{x} + \frac{\delta y}{y}$$ $$\frac{\delta x^n}{x^n} = n\frac{\delta x}{x}$$

Would I not, for example, obtain a more complex expression for the error in $\sqrt{\frac{T}{\mu}}$ due to the division and square root, according to the given "error propagation" rules? Further, we're "measuring" the tension by computing it as $m_wg$, where $m_w$ is the mass of a weight attached to the spring, providing tension. This should further complicate the expression for error.


EDIT:

For instance, using the given error propagation rules, I obtain the following...

Starting with the square root, we have:

$$\sqrt{\frac{m_wg}{\mu}}$$

Where $m_w$ is the mass of the weight. Note that $\mu$ is given, and we know nothing about the uncertainty in the measurement. Surely, then, it is one of the three we're "supposed to ignore". So, first we have the error propagation rule for division:

$$\delta \frac{m_wg}{\mu} = \frac{m_wg}{\mu}\cdot\bigg[\frac{\delta m_w}{m_w} + \frac{\delta \mu}{\mu}\bigg]$$

Then we have the error propagation rule for powers/roots:

$$\delta \sqrt{\frac{m_wg}{\mu}} = \sqrt{\frac{m_wg}{\mu}} \cdot \bigg[\frac{1}{2}\bigg(\delta \frac{m_wg}{\mu}\bigg) \bigg]$$

... and things start getting (algebraically) very nasty...

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  • $\begingroup$ Note the square root is a power, which cuts relative uncertainty in half according to your third rule. Also, are you sure your error propagation rules read linearly and not like $\delta(x+y+z)=\sqrt{(\delta x)^2+(\delta y)^2+(\delta z)^2}$? $\endgroup$
    – DanDan0101
    Mar 4, 2021 at 17:32
  • $\begingroup$ @DanDan0101 Thanks for pointing this out. Yes, the rules that were given read linearly (which is a bit odd for me, as I'm somewhat more used to using the $\delta(x+y+z) = \sqrt{\ldots}$ rule you give, etc). $\endgroup$
    – 10GeV
    Mar 4, 2021 at 17:36
  • $\begingroup$ Real vibrating "strings" also have inharmonicity because of the non-zero stiffness of the material, and the vibration frequencies depend on the exact method of fixing the ends (i.e. "$x= 0$" is only a toy-physics-101 approximation to reality). I have no idea what you are really trying to find here. $\endgroup$
    – alephzero
    Mar 4, 2021 at 17:37
  • $\begingroup$ There are general formulas to handle error propagation, physics.stackexchange.com/questions/587041/… Please apply the general formula and forget the "simplified ones". If the simplified ones are applicable, you obtain them from the general formula in a single step. $\endgroup$
    – Semoi
    Mar 6, 2021 at 9:25

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