Date modified: Mon 2023-09-18 . 09:01 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Pendulum swing. Let us analyze a **pendulum** ! A pendulum is just a weight hung from a pivot point (by a rope or an arm) so that it can swing freely. In an **idealized** situation, the rope is massless and taut (like a massless rod), the weight is a single point with no size, and the pivot has no friction. Of course real world pendulums are not ideal, but this is a good approximation as a start. Here is a crude pendulum made with some twine and battery, wrapped in blue tape, and the pivot is just a piece of chopstick where we wrap the excess twine. This will do for us. ![[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-06 11.04.37.excalidraw.svg]] %%[[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-06 11.04.37.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-06 11.04.37.excalidraw.dark.svg|dark exported image]]%% Notice the two things we can adjust in an (idealized) pendulum is the pendulum **arm length** $L$ and the pendulum weight **mass** $M$. ## Begin by questioning. We will make this pendulum in a moment. But let us think about some questions we could have: - As the pendulum swings, what is its frequency? - How would the frequency $f$ depend on the variables we can adjust, namely the length $L$ of the arm and the mass $M$ of the weight? ![[---images/---assets/---icons/question-icon.svg]] What are some questions that you might have about a pendulum? ## Make some hypotheses. Let us start by making some **hypotheses** (guesses based on limited knowledge). It is a good practice to do this for any scientific or mathematical investigation, so we know what we are trying to test for later. ![[---images/---assets/---icons/question-icon.svg]] Make a guess: What do you think is the **general relationship** between the **frequency** $f$ of the pendulum (how often it swings back and forth in a fixed unit amount of time), and the variables we can adjust: **The length of the pendulum arm** $L$, and **the mass of the pendulum weight** $M$? **Your guesses: ** As $L$ increases, you expect $f$ to .......... (increase / decrease / stay the same / other?) As $M$ increases, you expect $f$ to .......... (increase / decrease / stay the same / other?) ![[---images/---assets/---icons/question-icon.svg]] More sophisticated guess: What do you think is the mathematical **functional relation** between $f$ and $L$, and the **functional relation** between $f$ and $M$? That is to say, how is $f$ as a function of $L$, and how is $f$ as a function of $M$? Your guess could be, where $X$ is $L$ or $M$: (1) $f$ is linear in $X$: $f\sim aX+b$ (2) $f$ is quadratic in $X$: $f\sim aX^{2}$ (3) $f$ is some power $p$ of $X$: $f\sim aX^{p}$, specify what $p$. (4) $f$ is exponential in $X$: $f\sim a b^{X}$, specify if $b > 0$ (exponential growth) or $b < 0$ (exponential decay) (5) $f$ is ???? **Your guesses:** The functional relation between $f$ and $L$ is .................... The functional relation between $f$ and $M$ is .................... ![[---images/---assets/---icons/question-icon.svg]] Any other guesses? ## Construct the experiment. Take a long enough piece of twine (an arm length or so), a dead battery (as weight), and some blue painters tape. Tape the twine to the side of the battery so that the twine is going out parallel to the long side of the battery. Then, use just enough blue tape to cover the battery so that the battery is all blue. We will track this blue color later. Take a piece of chopstick and wrap the other end of excess twin on it, this will help us adjust the length and server as a pivot and anchor. ![[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 01.53.21.excalidraw.svg]] %%[[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 01.53.21.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 01.53.21.excalidraw.dark.svg|dark exported image]]%% Great! Now on your **smart phone** (the most technologically advanced device in your pocket), we will use this [tracker at https://bonsoon.net/dev/tracker/](https://bonsoon.net/dev/tracker/) to track the horizontal position of this blue battery. Aim the **red cross** over the blue tape and hit **calibrate**. Calibrate on a different parts of the blue tape so the battery is highlighted as much as possible. ![[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 02.04.00.excalidraw.svg]] %%[[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 02.04.00.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 02.04.00.excalidraw.dark.svg|dark exported image]]%% Now, the **blue cursor** will be tracking the center point of the highlighted portion. You can hit **start track** and it will start recording the horizontal position made by the blue cross. You can hit **stop track** to stop collecting. The box window below will plot out the horizontal position as time goes on. Wiggle back and forth and see if it is working. It would look something like this: ![[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 02.08.30.excalidraw.svg]] %%[[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 02.08.30.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 02.08.30.excalidraw.dark.svg|dark exported image]]%% Now, **practice having the pendulum swinging, and track its motion of the pendulum** with the tracker program. Use the chopstick to help you adjust the length and anchor. You may have to calibrate a few times since lighting can cause the blue to shift to a different hue. The program will try to match similar colors shown in the five boxes that you calibrated on. Note, you want to do this facing a direction so there is no "blue" color other than the battery on your camera! Help each other in your group if you need help. But everyone should try to build one and have their own experiment on their own phone. ## A note on the tracker program. Once you start tracking, the program will make a measurement at each "tick", and remember the last 200 measurements it made. **A full 200 measurements will fill the box window drawn on the bottom part of the tracker program.** Each tick is set to be around 100ms, **however** due to difference in the phones you may have, it may take longer in reality. **So everyone's phone will be different**. In other words, we won't know how long it will take for your phone to make 200 measurements. To counteract this, as the position measurements are being made (in pixels), a **timestamp** will also be recorded per measurement made. The timestamp is in **Unix time**, which is the amount of time since **Unix Epoch**, a reference date and time set at Thursday 1 January 1970 00:00:00, **Universal Time**. For this program, it records the Unix time in milliseconds on your device. If you click **export csv**, it will save a .csv file with two rows, the first row are the Unix times it made the measurements, and the second row are the position measurements in pixels. You should read up Unix time, Unix epoch, and Universal time if you are interested. ## Conduct the experiment! **The effect of** $L$. Read through this part first and then conduct the steps described. First let us investigate the relationship between the **frequency** $f$ of the battery pendulum and the **length** $L$ of the pendulum. **As careful as you possibly can**, measure out the pendulum at 30 cm with a ruler, from the **pivot** to the **center of the battery** (this is roughly the center of mass of the pendulum, since in an idealized situation, the mass is just a single point.) This measurement is your $L$. Position your pendulum using your chopstick so that it can swing freely (as shown in the very first image of this worksheet), calibrate the tracker on the blue battery, and make sure the tracker can see the battery in a full swing. Now start tracking. You want to hold your phone as steady as possible. Wait until the tracker has drawn the position of the battery and **fills the drawing box window.** When it has filled the box window, you can stop tracking. It should look something like this (this is me shamelessly doing it in a cafe):![[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 02.43.44.excalidraw.svg]] %%[[1 teaching/summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 02.43.44.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/pendulum-swing 2023-09-07 02.43.44.excalidraw.dark.svg|dark exported image]]%% Since we don't know (yet) how long that was, the box window is some set unit amount of time. This is fine because so long as we are using the same device, this set time should be roughly the same each time. Call this time **box time**. Now count how many cycles are there per box time. This can be simply done by counting how many peaks on one side there are. The frequency is therefore number of peaks per box time! Repeat the experiment three times per length $L$, and do it for lengths $L=$ 30 cm, 25 cm, 20 cm, 15 cm, and 10 cm. ![[---images/---assets/---icons/exclaim-icon.svg]] **IMPORTANT! !** For each length, also export and save a csv file of the data, and **email it to yourself**. **Label** it in your email which **length** it was. We will use this data later in computer class!!! The more you can make the better. Record them below. Measurements of Frequency in **number of peaks/box time** (of the same device): $$ \begin{array}{c|c|c|c|c|c} \\ \text{length } L \text{ in cm}&\text{trial 1} &\text{trial 2} & \text{trial 3} & \text{average } f \\ \hline 10 & \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} &\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} &\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & \\\hline 15 & \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline 20 & \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline 25 &\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline 30&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline \end{array} $$ **Some analysis of the data.** Make a **table** of with one column $L$ and the other the average $f$, and plot it in [DESMOS https://www.desmos.com/calculator](https://www.desmos.com/calculator). (Refer to the **chromatic scale** worksheet if need help.) ![[---images/---assets/---icons/question-icon.svg]] What is the relation between $f$ and $L$? Try plotting the log plot $(L,\log f)$ and the log-log plot $(\log f, \log L)$. Which one appears to be a straight line? Deduce a reasonable relation between $f$ and $L$. If either of these give a straight line, what are the **slope** and **intercept** of that line? ![[---images/---assets/---icons/question-icon.svg]] Use the $f\sim ....$ command of Desmos to find a best fit model. (Like last time with the chromatic scale.) Write down all the parameters (like $a,b$ etc), as well as $R^{2}$. . . . ![[---images/---assets/---icons/question-icon.svg]] What is the **functional relation** between $f$ and $L$ as suggested by your experiment and Desmos? . . . . **The effect of $M$.** Now **fix a length**, say $L$ = 30 cm. Weigh the battery on your pendulum. And take another battery and tape it on with blue tape. You can weight the new pendulum. Effectively the mass has now doubled. Now use the tracker and measure what is the frequency, peaks per box time. ![[---images/---assets/---icons/question-icon.svg]] Do you see any noticeable difference between 1 battery vs 2 batteries at 30 cm? What conclusion can you draw? Try with three batteries if you can. . . ![[---images/---assets/---icons/question-icon.svg]] Now look up pendulum on google or wikipedia from the internet, and find what is the relationship between frequency $f$, length $L$ , and mass $M$. Compare it with what you deduced earlier. Isn't it cool?! Hint. Frequency $f$ is the **reciprocal of period** $T$. . . . ![[---images/---assets/---icons/question-icon.svg]] Using this theoretical relation between $f$, $L$, and $M$, what should the frequency be (in Hz) when the length $L$ is 30 cm? 25cm? 20cm? 15cm? and 10cm? . . . Great job! ## Some discussion. ![[---images/---assets/---icons/question-icon.svg]] What is an overall conclusion that you can make about pendulum frequency $f$, and its length $L$ and mass $M$ ? . . . . ![[---images/---assets/---icons/question-icon.svg]] Are there any factors that might have affected your experiment? Notice our pendulum is not an ideal one. . . . As your pendulum swung, it may have swung in a more complex manner than just back and forth. It spins on its own and rotatses about the pivot 2-dimensionally. This adds extra wiggles to the pendulum. So the motion is comprised of many oscillations together. Later in computer class, we will determine what the main frequency is, and see all other frequencies using **discrete Fourier transform (DFT)!**