Measuring g Using the Pendulum Data

Brokk Toggerson

Measuring g Using the Pendulum Data

Brokk Toggerson

You should now have figured out that the power law for a pendulum

$T = 2 \pi \left( \frac{L}{g} \right)^p$

can be linearized using logarithms to

$\ln \left( \frac{T}{2\pi} \right) = p \ln ( L ) - p \ln (g)$

How can we separate $g$ from the power $p$ ?

Rearrange the equation!

From Determining the Uncertainty on the Intercept of a Fit, we know that the intercept $p \ln (g )$ in this case, can be determine from the average x and y values (as all fit lines must pass through this point!):

$\overline{\ln \left( \frac{T}{2\pi} \right)} = p \, \overline{\ln ( L )} - p \ln (g)$

Divide through by the power (i.e. the slope of our line) $p$

$\frac{1}{p} \overline{\ln \left( \frac{T}{2\pi} \right)} = \overline{\ln ( L )} - \ln (g)$

and rearrange to isolate $\ln (g)$

$\ln (g) = \overline{\ln ( L )} - \frac{1}{p} \overline{\ln \left( \frac{T}{2\pi} \right)}$ .

Thus, you can use the power $p$ , as measured by the slope, and the average values $\overline{\ln ( L )}$ and $\overline{\ln \left( \frac{T}{2\pi} \right)}$ to get $\ln (g)$ . Since we assume that the average values do not have uncertainty, we can use the uncertainty on the power $\sigma_p$ and a Monte Carlo error propegation to determine the uncertainty $\sigma_{\ln (g)}$ .

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Physics 132 Lab Manual by Brokk Toggerson is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

Rearrange the equation!

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