Editing Uncertainties (section)

== Uncertainty in g ==

For Kater's pendulum, our function is the relationship:

: <math>g = \frac{4\pi^2l}{T^2}</math>

So, our propagation of error is:

: <math>\delta g^2 = \left( \frac{\partial}{\partial l}\frac{4\pi^2l}{T^2} \delta l \right)^2 + \left( \frac{\partial}{\partial T}\frac{4\pi^2l}{T^2} \delta T \right)^2</math>

or

: <math>\delta g^2 = \frac{16\pi^4}{T^4}\left(\left[\delta l \right]^2 + \left[ \frac{2l}{T}\delta T \right]^2\right)</math>

By looking at the relationship of the uncertainty of <math>g</math> to <math>g</math> we get the relationship: 

: <math>\left(\frac{\delta g}{g}\right)^2 = \left(\frac{\delta l}{l} \right)^2 + \left( 2\frac{\delta T}{T} \right)^2</math>

To know what precision we must maintain in our measurements we can use estimated values for <math>g</math>, <math>l</math>, and <math>T</math> If we estimate a value of <math>g</math> as <math>9.810 \tfrac{m}{s^2}</math>, the length as <math>0.9990 \pm 0.0001 m</math> as reported by one of our Kater's Pendulum, and a period of <math>\approx 2 s</math> and we hope to achieve an uncertainty of <math>\pm 0.001 \tfrac{m}{s^2}</math>, we can then calculate what our acceptable uncertainty in the period measurement can be.

: <math>\delta T = \sqrt{\frac{T^2}{4}\left[\left(\frac{\delta g}{g}\right)^2 - \left(\frac{\delta l}{l} \right)^2\right]}</math>

: <math>\delta T = \sqrt{\frac{\left(2s\right)^2}{4}\left[\left(\frac{0.001}{9.810}\right)^2 - \left(\frac{0.0001}{0.9990} \right)^2\right]}</math>

: <math>\delta T = \pm 0.00002 s</math>

This indicates that a single period measurement simply is not accurate enough to determine the period. If the uncertainty of the photogate is <math>\pm 0.0001 s</math> then we need a number of measurements to reduce the uncertainty. This can be calculated by using the relationship:

: <math>\delta T = \frac{\sigma}{\sqrt{N}}</math>

where:

: <math>\sigma = \sqrt{\frac{1}{N-1}\sum\limits_{i=1}^N \left( T_i - \bar T\right)^2}</math>

If <math>\sigma</math> is taken as <math>\approx 0.0001 s</math>, a quick calculation indicates we will need at a minimum of 27 measurements to even get our period uncertainty close to the necessary value.

However, this is assuming that we can precisely match the period in the SMU and SMD orientations.