Uncertainty

Variable Definitions

$ l\rightarrow $ length of pendulum

$ g\rightarrow $ gravity

$ T\rightarrow $ period of the pendulum

$ \delta T\rightarrow $ uncertainty in the period

$ \delta g\rightarrow $ uncertainty in the gravity

$ \delta l\rightarrow $ uncertainty in the length

Uncertainty in the Measurement of g

The theory section shows that $ g $ is found to be:

$ g={\frac {4\pi ^{2}l}{T^{2}}} $

From this, the error can be found by:

$ \left({\frac {\delta g}{g}}\right)^{2}=\left({\frac {\delta l}{l}}\right)^{2}+\left(2{\frac {\delta T}{T}}\right)^{2} $

This formulation is then used for calculations of the uncertainty and the accuracy necessary for keeping the error withing 1/10000. The error on the period is found from:

$ \delta T={\frac {\sigma }{\sqrt {N}}} $

where:

$ \sigma ={\sqrt {{\frac {1}{N-1}}\sum \limits _{i=1}^{N}\left(T_{i}-{\bar {T}}\right)^{2}}} $