Calculating the uncertainty on an expression

In one of my labs I had to calculate the an expression for a constant $m$, where $m$ where $m$ is defined as $m = \frac{T}{L^2v^2}$, with $T$ being tension (constant), $L$ being length and $v$ being frequency. Now in order to do this accurately, I found 6 values for $L$ and $v$ - the tension was constant for all of these values. Then I used excel to plot a graph of $T$ against $L^2v^2$ and the gradient of the line of best fit gave me my value for $m$. Now the thing that I'm having problems with is finding out if it's possible for me to find the uncertainty on $m$. I know that I can find the uncertainty for $m$ if I know the uncertainties for $T$, $L^2$ and $v^2$ using the formula below:

$$\left. \begin{array}{l} Z = A \times B \\ or \\ Z = \frac{A}{B}\\ \end{array} \right\} \Rightarrow \biggl (\frac{\Delta Z}{Z} \biggr )^2 = \biggl (\frac{\Delta A}{A} \biggr )^2 + \biggl (\frac{\Delta B}{B} \biggr )^2$$

I'm able to calculate the uncertainty on $T$ but I'm not sure if I can calculate the uncertainties for $v^2$ and $L^2$. I know that the uncertainties for each value of $v$ and $L$ were $\pm 0.1$ Hz and $\pm 0.05$ Hz. However, in order to calculate the uncertainties for $v^2$ and $L^2$ I need to use the formula:

$$Z = A^n \Rightarrow \frac{\Delta Z}{Z} = n \biggr ( \frac{\Delta A}{A} \biggl )$$

which requires values for $v$ and $L$, which I do not have since I used the gradient of the line of best fit to find $m$ and not specific values of $L$ and $v$. So I'm wondering if it would be sensible for me to perhaps use the average of my values for $L$ and $v$ in order to calculate the uncertainties for $L^2$ and $v^2$ or if it is sensible at all to even calculate an uncertainty for $m$ in this situation.

• You can estimate the uncertainty in the slope from the deviations of the measured points to the fitted line. – Pieter Mar 5 '18 at 17:14
• In general, the uncertainty of $Z$ (let it be $\Delta z$) is defined as $\Delta z=\Sigma_{i}\big|\frac{\partial Z}{\partial x_i}\big|\Delta x_i$ if $Z=Z(\{x_i\})$ – MatMorPau22 Mar 5 '18 at 17:21
• @Pieter How exactly would I do that? Is there a reference for that sort of technique? – Hai Mar 5 '18 at 17:22
• Method of least squares, minimizing the sum of the squares of deviations. For a straight line it is called linear regression. These are terms you can google. An alternative is to make a graphic estimate by eye, see what slope is roughly consistent with data points. – Pieter Mar 5 '18 at 17:26