Editing Uncertainties

To understand just how precisely we must make our measurements, we must look at the propagation of uncertainty which can be described by the relationship:

: <math> \delta f(x_n)^2 = \sum\limits_{i=1}^n \left( \frac{\partial f(x_n)}{\partial x_i} \delta x_i \right)^2 </math>

== 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.

== Uncertainty in Fit ==

Instead of trying to precisely match the SMU and SMD periods we can plot a curve of the period vs the small mass position in the SMU orientation and superimpose this graph on a graph in the SMD orientation. Then by finding the crossing point of the fit functions we can extrapolate the crossing point. This crossing point is the value to which we must know the uncertainty to <math>\pm 0.00002 s</math>. So, once again an analysis of the uncertainties of this fit must be examined.

If we are able to take data close enough to the crossing point, we can effectively "zoom" in our graph to such an extent that a linear fit should be possible. If this is possible, the equation for the fit will simply be:

: <math>T = m L + b</math>

Where <math>L</math> is the position of the small mass. We will have two such equations. One for the SMU orientation and one for the SMD orientation. Since we are searching for the point where <math>L</math> and <math>T</math> are the same, we can set the equations equal to one another and solve for <math>L</math>:

: <math>L = \frac{b_{smd} - b_{smu}}{m_{smu} - m_{smd}}</math>

Substituting this back into either line equation (SMU shown here) results in our equation for the period in terms of :

: <math> T = \frac{m_{smu}b_{smd} - m_{smd}b_{smu}} {m_{smu} - m_{smd}}</math>

This gives us a relationship which we can use to determine how our uncertainties make up the crossing point uncertainty we are aiming for. Performinf the derivatives results in:

: <math> \left(\delta T\right)^2 = \left(\frac{m_{smd} (b_{smu}-b_{smd})} {(m_{smu}-m_{smd})^2}\right)^2 (\delta m_{smu})^2 + \left(\frac{m_{smu} (b_{smd}-b_{smu})} {(m_{smu}-m_{smd})^2}\right)^2 (\delta m_{smd})^2 + \left(\frac{m_{smd}} {m_{smu}-m_{smd}}\right)^2  (\delta b_{smu})^2+ \left(\frac{m_{smu}} {m_{smu}-m_{smd}}\right)^2  (\delta b_{smd})^2</math>