====== Introduction ======
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So far, you have performed a number of experiments with an emphasis on **interpretation of data** – both for the purposes of informing your experimental technique and for the evaluation of your final results. This is a key component of experimental science. Many of the most important experiments in science are ones which resulted in the measurement of something which – up to that point – was unknown and for which there was no theory or other means of determining whether or not the experimental results were correct.

So how do you know if your experiment was a "success" when there is no known "right" answer to compare your results to? The answer is that you evaluate your data //throughout// your experiment and use your data to refine your methods until you are confident that you fully understand each part of the experiment. When you understand each part, you can in turn trust the results and then allow //only// conclusions which are supported by the data. Other scientists will come along and build on the results of your experiment by developing theoretical models or performing additional experiments – maybe to verify what you found, to poke holes in the interpretation by finding conflicting information, or to use the model to make predications about new experimental results to look for – and over time, the community reaches a consensus.

The process of performing an experiment and reporting on the results of the experiment (as well as limitations on your results – i.e. experimental uncertainties) is of fundamental importance to the advancement of scientific knowledge.

====== Procedure ======


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===== Objective =====

In today's lab, you will perform an experiment to determine how much energy is required to rupture a piece of tissue paper. Since there is no "known value" for this process, you will have to operate under the same conditions as many research experiments.

You are given an apparatus which is suitable for making this measurement. It is up to you to use the physics you have been studying this quarter to determine what measurements need to be made, and how they will be used to accomplish the goal. Your goal is to report the **amount of energy required to rupture a piece of tissue paper**, along with an estimate of how well you know that value. In the process, you will need to look at your data to understand how well your experimental technique is performing and where you can make improvements to your experiment.

===== Apparatus =====

For this lab, you will use the apparatus setup as shown in figure 1. A steel ball bearing released from point (a) will travel down the tube gaining kinetic energy until it emerges at point (b) with some velocity $v$. The ball bearing will then travel some distance $d$ and fall a distance $h$, before striking the floor at point %%(c)%%.

If a piece of tissue paper is placed at point (b) and the ball bearing is again released from the same point (a), one would expect the distance travelled in this case will be less than in the case when there is no tissue paper. The difference in distance travelled will be related to the energy lost in punching through the piece of tissue paper.

{{ phylabs:lab_courses:phys-120_130-wiki-home:fall-experiments:tissue-rupture:apparatus.png?direct&500 |}}
;#;
**Figure 1:** The apparatus
;#;


Using what you have learned in lecture about kinematics and energy, it is up to you – with guidance from your TA – to decide what measurements to make and what calculations to perform. The energy lost to rupturing the tissue paper will be small, so you will need to make repeated measurements of the distances travelled to obtain averages with small enough uncertainties to produce a meaningful answer. You can use a magnet to ensure that you release the ball bearing from the same starting location (a) for each trial.

You will need to propagate your measured quantities and their associated uncertainties through some calculations in order to arrive at a final result with its uncertainty.

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<details>
<summary>  How do I give the height an uncertainty? Won't it be the same all the time? </summary>

When measuring some static quantity such as the height of the ramp or the mass of the ball, it doesn't make sense to measure quantities multiple times and use the standard deviation, because it is entirely possible that your measurements could be identical each time.  In this case, the uncertainty in the measurement is determined by the //resolution// of the measurement device.  For instance, see the photo below:

{{ phylabs:lab_courses:phys-120_130-wiki-home:fall-experiments:tissue-rupture:measurement.png?direct&500 |}}

In this instance, we can probably agree that the stapler is at least $18.5\text{ cm}$ long.  We might estimate that it is $18.51$ or $18.52\text{ cm}$ even, but we can't make very good estimates here because we're trying to measure something smaller than our ruler's resolution (i.e. the $1\text{ mm}$ scale).

As a rule of thumb, we typically will use half the resolution of a measurement scale ($0.5\text{ mm}$ in this case) when estimating the uncertainty due to a measurement device.

For this lab, we encourage you to **//not//** try and use this for your distance measurements ($d$). This is because it will almost certainly be overshadowed by the statistical fluctuations (standard deviation) of the ball's travel distance.

A complete treatment of uncertainties takes into account both the limits due to statistical fluctuation and due to device resolution, but in these labs it is usually clear which will be more important.
</details>
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===== Propagation of Uncertainties =====

In many of the previous experiments, the main source of experimental uncertainty was random statistical fluctuations in measured quantities. In order to estimate such uncertainties, you repeated measurements and took the average as your "measured value" and the standard deviation as the "uncertainty" in that value. In this experiment, the quantity we are interested in – the energy required to rupture the tissue paper – is not a directly measured quantity, but instead is quantity which will be calculated from other measured quantities.

The uncertainty on a calculated quantity comes from propagating uncertainties through a calculation. That is to say, if your calculated quantity $f$ is a function of variables $A$  and $B$  – i.e. $f = f(A,B)$ – then the uncertainty in your calculated quantity $\Delta f$ will be some combination of the variables and their uncertainties: $\Delta f = \Delta f(A, B, \Delta A, \Delta B)$.

The general rules for propagating uncertainties can be found <wrap em>[[phylabs:lab_courses:supplemental-material:reports-presentations-and-notebooks:uncertainty_analysis_introductory_labs:start| here]]</wrap>, but for this lab we will highlight the rules for the most common cases.

==== Sums and differences ====

If your calculated quantity includes sums or differences – $f(x,y,z) = x - y + \dots +z$ – then the individual uncertainties are squared, summed, and square-rooted:
\begin{equation*}
\Delta f = \sqrt{(\Delta x)^2 + (\Delta y)^2 + \dots + (\Delta z)^2}.
\end{equation*}

Notice that the uncertainty terms //add//, even if you are performing a //subtraction// in your calculation. 

==== Products and quotients ====

If your calculated quantity includes multiplication or division – $f(x,y,z) = \dfrac{xy}{z}$ – the fractional uncertainties are squared, summed, and square-rooted:
\begin{equation*}
\frac{\Delta f}{f} = \sqrt{\left(\frac{\Delta x}{x}\right)^2 + \left(\frac{\Delta y}{y}\right)^2 + \left( \frac{\Delta z}{z}\right)^2}.
\end{equation*}
==== Powers  ====

If your calculated quantity includes a variable raised to a power – $f(x) = x^n$ – then the fractional uncertainty is multiplied by the exponent:
\begin{equation*}
\frac{\Delta f}{f} = n \frac{\Delta x}{x}.
\end{equation*}

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Note that for a square root $n = 1/2$.
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==== An example ====

As an example, consider the function for the period of a pendulum from an earlier lab this quarter: $T = 2\pi\sqrt{\frac{L}{g}}$. If we measure $L \pm \Delta L$ and $T \pm \Delta T$, and wish to calculate $g \pm \Delta g$, we can rearrange the function as
\begin{equation*}
g = 4\pi^2LT^{-2}
\end{equation*}
and calculate the association uncertainty as
\begin{equation*}
\frac{\Delta g}{g} = \sqrt{\left(\frac{\Delta L}{L}\right)^2 + \left(-2\frac{\Delta T}{T}\right)^2}.
\end{equation*}
====== Grading ======


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The grading this week is identical to last week. The rubrics are repeated below for convenience.

===== In-lab rubric (4 points) =====

Your TA will come around at some point in the lab to speak with your group and award the in-lab points.

<div participation_notebook_rubric>
|  Participation (2 points)  ||  Lab Notebook (2 points)  ||
|  Acceptable (2)  |  Unacceptable (0)  |  Acceptable (2)  |  Unacceptable (0)  |
| Participates in a meaningful way in group discussions and data taking/record keeping.  | Arrives late or leaves early. \\ \\ Allows partners to do most of the work. \\ \\ Takes a superficial data set with no attempt to analyze data and improve measurements in order to leave the lab early.\\ \\ Is disruptive or otherwise disrespectful to the group.  | Keeps notebook up to date as the experiment progresses.\\ \\ Notebook is neat, legible and organized.\\ \\ All major elements of the experimental process are documented, including setup, procedure, notes on decisions made over course of the experiment, etc.  | Students fill out the notebook at the end of the period.\\ \\ Notebook entries are unintelligible.\\ \\ Significant aspects of how the experiment was performed and what happened over the course of the experiment are omitted.  |
</div>
===== Report rubric (6 points) =====

Your TA will grade your group's report and return it with a grade before your next lab period.

<div rubric three_section>
| |  Acceptable (2)  |  Needs Improvement (1)  |  Not Acceptable (0)  |
|  Experimental Procedure  | Experimental procedure adequately described, including diagrams as needed.\\ \\ Reasons given for decisions made regarding procedure.\\ \\ Sources of uncertainty are clearly described.\\ \\ Decisions regarding how uncertainties will be estimated are presented clearly.  | Some elements of the experimental procedure are omitted or unclear.\\ \\ No justification given for choices made during the experiment.\\ \\ Important sources of uncertainty were missed.\\ \\ No reasons presented for how uncertainties were estimated.  | It is unclear how the experiment was performed.\\ \\ No uncertainties considered.                                                                                              |
|  Data and analysis  | Data are clearly presented in tables and graphs.\\ \\ All tables and graphs are appropriately labeled and include units.\\ \\ Uncertainties in data are propagated through calculations as appropriate.                                                                       | Some data not recorded.\\ \\ Tables or graphs inadequately or incorrectly labeled.\\ \\ No units assigned to measured values.\\ \\ Uncertainties not propagated through calculations.                                                                       | No data present in report.\\ \\ Data presentation is confusing and does not relate to description of experimental procedure.\\ \\ No treatment of experimental uncertainties.  |
|  Conclusions | Conclusions are clearly supported by the data.\\ \\ Comparison of data to models or predictions include assessment of uncertainties. | Conclusions are overstated based on the data.  |Conclusions are contradicted by the data.  |
</div>
====== Notebook & Conclusions ======
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Remember to update you lab notebook and the group report as you work though the lab. A lot of this experiment and the subsequent analysis is of your own design, so make clear notes for yourself and your TA. You should write down details of what you are doing, how you are doing it, and what results/conclusions you arrive at. Make sketches of the apparatus you use and how you set it up. Record important parameters. Make notes about difficulties you encounter while taking data and changes you make to your procedure. Do all of this in real time as you are doing the experiment, not at the end when you have forgotten most of the details.

At the end of your notebook you should write up a brief summary of your final conclusion. Demonstrate to the reader that they should trust your measurement of this new unknown quantity!

<div template>
[[https://docs.google.com/document/d/1YKPXHqFDUwz1BDYmQRJqhb3mbDdT2QDen1pmdWXD-mk/copy|Lab 4 Template]]</div>

<WRAP download>
{{ phylabs:lab_courses:phys-120_130-wiki-home:fall-experiments:tissue-rupture:lab_four_notebook.ipynb | Lab Four Notebook.ipynb}}</WRAP>

<div submit>
[[https://forms.gle/CUK1fNyBL5f5DvvC7|Use this link to submit your report]]
</div>

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**Remember to log out of your Google account after you submit!**
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