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)}.


Icon for the Creative Commons Attribution-ShareAlike 4.0 International License

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

Share This Book