deletions | additions
diff --git a/untitled.tex b/untitled.tex
index 2850d09..7e3a45c 100644
--- a/untitled.tex
+++ b/untitled.tex
...
\phantom{1}\\
\textbf{Practical Set-up Notes and Lab-Specific Advice}
\phantom{1}\\
It is recommend With the introduction to
the lab taking a maximum of 15 minutes, it is recommended that demonstrators divide up the 3 hours so that the first two parts take
45 ~40 minutes, and the third takes
90 ~90 minutes. Mark each section as you go to avoid a scramble at the end of the lab, and also to correct students who are not calculating uncertainties correctly,
etc. etc.\\
\phantom{1}\\
Each semester on the Intro lab we have a miscommunication issue between the demonstrators. Please, come at least 15 minutes before the lab and discuss (with your colleagues) what the answers are for the practical questions and how they are solved, especially if it is your first time or you have forgotten the approach from the last time when you did demonstration.\\
\phantom{1}\\
Do \textbf{not} argue with your colleague in front of the students! Even if you think that your colleague is wrong. Please have a word with him/her when he/she walks away from the student. Many times I have seen how demonstrators correct each other in front of the students, and students always get confused as they don’t know who is right and who is wrong. Sometimes they ask the same question to different demonstrators and when they get different answers they don’t know what to think. If they say that another demonstrator told you something else and you think it is wrong, ask them to wait, have a word with your colleague and then explain the agreed-upon approach to the student
together! together!\\
\phantom{1}\\
Finally, make the distinction between random errors and measurement uncertainty. Just because a student places \textit{the same weight on the same scale} 10 times does not mean they can increase their precision using $\sigma_{\bar{X}} = \frac{\sigma_{X}}{\sqrt{N}}$ - if the base uncertainty of the scale is much larger than any gaussian fluctuations, it is difficult (or impossible) to justify applying normal-distribution tricks such as adding variances rather than absolute uncertainties.
\phantom{1}\\
\textbf{Practice Problem - Approach and Solutions}
\begin{enumerate}
...
\begin{itemize}
\item{Find mean radius by dividing mean diameter by 2:
\begin{align*}
\mu_D
&= \frac{\sum{D_i}}{N} =
\frac{\sum{D_i}}{N} &= \frac{24.75+24.82+24.87+24.79+24.95+24.71+24.81+24.65+24.69}{9}\\
&= \frac{223.04}{9} = 24.78\text{mm}\\
\text{Therefore, } \mu_r &= \frac{\mu_D}{2} = \frac{24.78}{2}\\
&= 12.39\text{mm}\\
...
Based on data from Reserve Bank Coinage Specifications[1], the densities for 10 cent, 20 cent, 50 cent, 1 dollar and 2 dollar coins should come out as 6328, 6901, 6113, 7027, 6715 kg/m3 respectively. Having said this, values should roughly fall within 6000-8000 $kgm^{-3}$. Diameters and masses are pretty consistent between testing one coin and five coins, but thicknesses are very dependent on exactly where you test the coins. Students will probably need to measure the thickness of the coin at the *rim* to get a density in the Specification range. By measuring thicknesses at the center of each coin, the calculated densities are closer to 7500$kgm^{-3}$.\\
\begin{table}[!h] \begin{table}
\begin{tabular}{c|c|c|c|c|c}
& 10c & 20c & 50c & \$1 & \$2 \\ \hline
Thickness($\pm0.01$mm) & 1.37 & 1.48 & 1.39 & 2.59 & 2.37 \\ \hline
...
\phantom{1}\\
It is assumed that all of the measurements lie within the scale uncertainty of the calipers - if this is not the case, then adding the relative errors in quadrature, rather than linearly, is an acceptable approach.
\phantom{1}\\
\begin{table}[!h] \begin{table}
\begin{tabular}{ c|c|c|c|c|c }
$\rho=\frac{M}{V}$ & 10c & 20c & 50c & \$1 & \$2 \\ \hline
V (m$^3$) & $4.51\times10^{-7}$ & $5.47\times10^{-7}$ & $6.69\times10^{-7}$ & $1.07\times10^{-6}$ & $1.312\times10^{-6}$ \\ \hline
$\frac{\delta V}{V}$ \% & 0.828 & 0.768 & 0.8002 & 0.473 & 0.7986 \\ \hline
$\rho$ ($\frac{kg}{m^3}$) & 7538.8 & 7495.4 & 7623.3 & 7476.6 & 7469.5 \\ \hline
$\frac{\delta\rho}{\rho}$ \% & 3.77 & 3.207 & 2.761 & 1.723 & 1.819 \\
\end{tabular}
\caption{Volumes and densities for single coins}
\end{table}
\phantom{1}\\
In short, you should expect the uncertainties for the volume to be around $0.5-1\%$ and the density to be around
$2-5\%$. $2-5\%$.\\
\phantom{1}\\
For the last question of part 1,
the key thing for students to realise is that if
10 coins were used, each relative uncertainty would decrease by a factor of 10, which would ultimately propagate to 10 times the precision. It is not necessary for students to derive this, but they should be able to explain their reasoning.\\
Basically, if $r_i$ we assume that it is the
relative uncertainty of one measurement, then with $n$ coins, this will become $\frac{r_i}{n}$. From there, a quick derivation:
\begin{align*} & 10=\dfrac {r_{old}} {r_{new}}\\ & =\dfrac {\sqrt {\sum r_{i}^{2}}} {\sqrt {\sum \left( \dfrac {r_{i}} {n}\right) ^{2}}}\\ & =n\end{align*} thus n=10 coins.\\
\phantom{1}\\
Here is a good example for this question:\\
\textbf{Example}\\
(a) If 10 is added to each number in a set, what is random (gaussian) variations between coins that causes the
change difference in
our measurements, rather than the
mean? What is scale uncertainty of the
change in calipers/micrometers, then the standard
deviation?\\
(b) If each number in a set is multiplied by 10, what is error of the
change in calculated mean (as compared with the
mean? What true mean), calculated from $N$ independent measurements, is
the change in the standard deviation?\\
\textbf{Solution}\\
(a) Since each value increases by 10, the mean increases given by
10. Adding 10 to each number does not change the distances between the numbers and the mean. Since the standard deviation \begin{equation*}
\sigma_{\bar{X}} = \frac{\sigma_X}{\sqrt{N}}
\end{equation*}
where $X$ is
dependent on those distances, the standard deviation does not change. If you are not sure about the
answer, you can plug in some numbers and try it. The mean of {1, 2, 3} is 2. The mean set of
{11, 12, 13} measurements, $\bar{X}$ is
12. The mean increased by 10.The distances between {1, 2, 3} and their mean are 1. The distances between {11, 12, 13} and their mean
are also 1. The standard deviation did not change.\\
(b) Since each value value, $\sigma_{X}$ is
multiplied by 10, the
mean is multiplied by 10. Multiplying each number by 10 changes standard deviation of the
distances between measurements from the
numbers calculated mean, and
the mean by a factor of 10. Since the standard deviation $\sigma_{\bar{X}}$ is
dependent on those distances, the standard
deviation changes by a factor error of
10. If you are not sure about the
answer, you can Plug In some numbers and try it. The mean of {1, 2, 3} is 2. The calculated mean
of {10, 20, 30} is 20. The mean increased by a factor of 10. The distances between {1, 2, 3} and their mean are 1. The distances between {10, 20, 30} and their mean are 10. Since relative to the
distances changed true mean. In order to reduce the standard error by a factor of 10,
we need increase the number $N$ of measurements taken to $N= 10^2 = 100$. Similarly, by taking 10 measurements, we reduce the standard
deviation also changed error by a factor of
10.\\ $\sqrt{10} \approx 3.16$. \\
\phantom{1}\\
In part 3 (measuring reaction time), the free-fall distance associated with a reaction time of 0.18s is 16.2cm. This should be what people aim for, but it is difficult to test this in a reliable manner in the lab, so expect results to vary wildly - results will most likely be within the range of 10cm - 40cm.\\
\phantom{1}\\
\textbf{Key Discussion Points}
It Again, it is worth emphasising that students should `add errors in quadrature' when the errors are generated by random fluctuations. If the fundamental uncertainty of the measuring equipment is the limiting factor, the quadrature approach is no longer appropriate. This is covered in appendix A4.1 under the discussion of dependent and independent
quantities. quantities.\\
\phantom{1}\\
Section 1.3 - Archimedean method of determining volume by placing the coin in water and measuring the volume of displaced water would be good example. A reasonable way to do this is to use a graduated (marked) cylinder. Pour water into the graduated cylinder until it reaches a known level (seen by markings on the cylinder’s surface). Add the object to the water and record the new water level. The difference between the new water level and the original level will be the object's volume. This measurement is taken in milliliters, which are interchangeable with cubic centimeters. Once volume is determined, then weigh the object on a scale and do the same calculations as before. \\
\phantom{1}\\
Section 5.1 - Students should use $\frac{\mu_1 - \mu_2}{\sigma}$ formula\\
\phantom{1}\\
Section 5.2 - Ideally students should give a rough probability of difference occurring due to random error using the table from A.5.3\\
\phantom{1}\\
Section 5.3 - Firstly, students should distinguish between measuring their own reaction, and the average reaction time. Secondly, students should realise that their experiment cannot prove or disprove the reference value, only give evidence in support of or against it. Thirdly, students should realise that taking everyone’s value in the class would introduce sampling bias.\\
\phantom{1}\\
\textbf{References}
