**Names**

Hamish Huggard

Neelam Hari

Alex Smith

Anna Radionova

**Practical Set-up Notes and Lab-Specific Advice**

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 40 minutes, and the third takes 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.

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.

Do **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!

Finally, make the distinction between random errors and measurement uncertainty. Just because a student places *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.

**Practice Problem - Approach and Solutions**

Errors are differences between the measured value and the real value of a property due to systematic (miscalibration/bias) issues or random (presumably gaussian) fluctuations. Uncertainties come from lack of resolution in the scale of the equipment being used, or our own ability to measure. Don’t discuss human errors, as it’s too general to be useful, and if you’re following the correct experimental procedure then other (more important!) sources of uncertainty should dominate.

Under the assumption that the errors are normally distributed, so that the given measurements are both independent and random, we can do better than straight addition. We can add the

*variances*, rather than the errors themselves, and square-root the result to find the new standard error (this is ’adding in quadrature’): \[\begin{aligned} \text{Sum} &= \sum_{i=1}^N{x_i} \pm \sqrt{\sum_{i=1}^N{\delta x_i^2}}\\ &= (102.3+44.8+5.67) \pm \sqrt{0.2^2+0.2^2+0.08^2}\\ &= 152.77 \pm 0.29\\ &= 152.8 \pm 0.3 \text{cm}\end{aligned}\]Multiply the velocity and time values to find distance. Again, by assuming that errors are distributed normally, we will take the square-root of the sum of the squared relative uncertainties. We then multiply this new relative uncertainty by the distance to get the absolute uncertainty. \[\begin{aligned} \text{Distance} &= v\times t = 50\times1.2 = 60\\ \delta(\text{Distance}) &=D\sqrt{\frac{\delta v^2}{v^2}+\frac{\delta t^2}{t^2}} = D\sqrt{\frac{3^2}{50^2}+\frac{0.1^2}{1.2^2}}\\ &= 60\sqrt{\frac{12.96+25}{3600}}= 6.16\\ \text{Therefore, Distance} &= 60\pm6 \text{km}\end{aligned}\]

There are a few steps to this.

Find mean radius by dividing mean diameter by 2: \[\begin{aligned} \mu_D = \frac{\sum{D_i}}{N} &= \frac{24.75+24.82+24.87+24.79+24.95+24.71+24.81+24.65+24.69}{9}\\ &= \frac{223.04}{9} = 24.78\text{mm}\\ \text{Therefore, } \mu_r &= \frac{\mu_D}{2} = \frac{24.78}{2}\\ &= 12.39\text{mm}\\\end{aligned}\]

Here, we use \[\begin{aligned} \sigma &= \sqrt{\frac{\sum{(r_i - \mu_r)^2}}{N-1}}\\ &= 0.0469\text{mm}\end{aligned}\] Make sure that they use the radii here, and not the raw diameter values.

\[\sigma_{error} = \frac{\sigma}{\sqrt{N}} = 0.0156\text{mm}\]

Note that the relative error for the volume is three times larger than the error of \(\sigma_{error}\) as a fraction of the mean. \[\begin{aligned} V &= \frac{4\pi \mu_r^3}{3} = \frac{4\pi(12.39)^3}{3} = 7967\text{mm}\\ \delta V &= V(\frac{3\sigma_{error}}{\mu_r}) = 7967(\frac{3\times0.0156}{12.39})=30.1\\ \text{Therefore, } V &= 7970\pm30\text{mm}^3\end{aligned}\]

A couple ways students could approach measuring reaction time are:

if two students start stopwatches at the same time, then students 1 stops hers unexpectedly, student 2 has to stop his as soon afterwards as he can. Student 2’s reaction time is then given by student 1’s time - student 2’s time

If students have access to some counting down mechanism (watch, app, etc), then student 1 can set this to a random amount of predetermined time (which is unknown to student 2). Student 2 then starts his stopwatch at the same time as student 1 starts her countdown. Student 2 then has to stop his stopwatch as soon as the countdown reaches zero and some alarm is given off. Reaction time is then stopwatch time - countdown time

“Additional (if they’re super interested) but without a stopwatch: Catching the ruler experiment. See link:

http://www.hometrainingtools.com/a/measure-reaction-time-science-project

http://www.hometrainingtools.com/media/reference/ReactionTimeTable.pdf” - Neelam

**Sample Data and Expected Results**

The student will learn how to use the Vernier calipers and scale, and will then record their data. They might get stuck with the uncertainties, so we can direct them by asking questions:

What do the uncertainties on the instruments mean?

How do we use these uncertainties for our measurements?

Would the results differ if we used a ruler?

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\(kgm^{-3}\).

10c | 20c | 50c | $1 | $2 | |
---|---|---|---|---|---|

Thickness(\(\pm0.01\)mm) | 1.37 | 1.48 | 1.39 | 2.59 | 2.37 |

Diameter(\(\pm0.01\)mm) | 20.48 | 21.70 | 24.76 | 22.94 | 26.55\(\pm\)0.05 |

Mass(\(\pm0.1\)g) | 3.4 | 4.1 | 5.1 | 8.0 | 9.8 |

The volume is calculated using \[\begin{aligned}
V &= \pi r^2h\\
&= \pi\times(\frac{\text{Diameter}^2}{4})\times\text{Thickness}\\
\delta V &= V(\frac{2\delta D}{D} + \frac{\delta Th}{Th})\end{aligned}\] and so the density \(\rho\) is given by \[\begin{aligned}
\rho &= \frac{M}{V}\\
\delta\rho &= \rho(\frac{\delta M}{M} + \frac{\delta V}{V})\end{aligned}\]

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.

\(\rho=\frac{M}{V}\) | 10c | 20c | 50c | $1 | $2 |
---|---|---|---|---|---|

V (m\(^3\)) | \(4.51\times10^{-7}\) | \(5.47\times10^{-7}\) | \(6.69\times10^{-7}\) | \(1.07\times10^{-6}\) | \(1.312\times10^{-6}\) |

\(\frac{\delta V}{V}\) % | 0.828 | 0.768 | 0.8002 | 0.473 | 0.7986 |

\(\rho\) (\(\frac{kg}{m^3}\)) | 7538.8 | 7495.4 | 7623.3 | 7476.6 | 7469.5 |

\(\frac{\delta\rho}{\rho}\) % | 3.77 | 3.207 | 2.761 | 1.723 | 1.819 |

In short, you should expect the uncertainties for the volume to be around \(0.5-1\%\) and the density to be around \(2-5\%\).

For the last question of part 1, if we assume that it is the random (gaussian) variations between coins that causes the difference in our measurements, rather than the scale uncertainty of the calipers/micrometers, then the standard error of the calculated mean (as compared with the true mean), calculated from \(N\) independent measurements, is given by \[\sigma_{\bar{X}} = \frac{\sigma_X}{\sqrt{N}}\] where \(X\) is the set of measurements, \(\bar{X}\) is their mean value, \(\sigma_{X}\) is the standard deviation of the measurements from the calculated mean, and \(\sigma_{\bar{X}}\) is the standard error of the calculated mean relative to the 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 error by a factor of \(\sqrt{10} \approx 3.16\).

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.

**Key Discussion Points**

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.

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.

Section 5.1 - Students should use \(\frac{\mu_1 - \mu_2}{\sigma}\) formula

Section 5.2 - Ideally students should give a rough probability of difference occurring due to random error using the table from A.5.3

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.

**References**

[1] Retreived from http://www.rbnz.govt.nz/notes-and-coins/coins/new-zealand-coinage-specifications on 25th February 2016

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