#7 | Infusing the Soul of a 10-Yen Coin into Excel! Implementation Techniques for Equations of Motion
This article is part of the series found here👇.
If you prefer to listen, please play the audio commentary here👇😊
The goal of this series is to theoretically animate the unnatural movement of a 10-yen coin in Kokkuri-san using high school level mathematics and physics.
Last time, we created the template for the Excel model.
Starting now, we will input specific numerical values and formulas into that template!
The template file is available for free download in [ the previous article ].
The Excel file reflecting the work in this article is also available as an answer example in [ the download area of the previous article ].
✔︎ Inputting setting values
The fixed values are as follows:
Mass of the 10-yen coin: $${m=0.0045\ (kg)}$$
*The mass of a 10-yen coin is approximately 4.5g.Gravitational acceleration: $${g=9.80665\ (m/s^2)}$$
The values to be adjusted later are as follows:
Time step: $${\Delta t=0.0001\ (s)}$$
*We will start by checking with $${0.1\ (ms)}$$.
The constants to be entered later are as follows:
Static friction coefficient: $${\mu_s}$$
Kinetic friction coefficient: $${\mu_k}$$
We will discuss how to use these friction coefficients in the next installment and beyond.
When you enter these values into the Excel model template, it will look like this.
✔︎ Enter the calculation fields
Next, we will enter the numerical values and formulas into the calculation fields.
① Enter time in column G
First, we will create the flow of time.
Start time: Enter '0' in cell G4.
Next time: Enter a formula in cell G5 so that it becomes 'previous time (cell G4) + time step ($${\Delta t}$$)'.
For the time step ($${\Delta t}$$), refer to the setting value from earlier.
Copy this formula downwards to complete it.
② Enter the force applied by the fingertip in columns H to I
For the time created in ① above, we will enter how much force the 10-yen coin is being pushed with by the fingertip.
For now, we will enter '0' as the force applied by the fingertip for all time points.
Force applied by fingertip (horizontal direction): Enter '0' in column H for all time points.
Force applied by fingertip (downward direction): Enter '0' in column I for all time points.
This represents the state where the fingertip is not touching the 10-yen coin.
③ Enter gravity in column J
For the time created in (1), we will calculate the gravity acting on the 10-yen coin.
Enter a formula in cell J4 so that it becomes '= mass (m) × gravitational acceleration (g)'.
For mass and gravitational acceleration, refer to the set values.
Copy this formula downwards and enter it for all time steps.
🔎 Confirming the equation of motion for the 10-yen coin
Here, let's confirm the equation of motion for the 10-yen coin derived in the article below👇.
That equation of motion is as follows.
$$
\footnotesize
\begin{align*}
\\[0.1pt]
(Horizontal acceleration) = & \tfrac{(Force pushed horizontally from fingertip) - (Frictional force)}{Mass of 10-yen coin} \\[10pt]
(Frictional force) = & (Coefficient of friction) × (Normal force received from paper) \\[5pt]
& ※Coefficient of friction: To be considered at a later date \\[10pt]
(Normal force received from paper) = & (Gravity) + (Force pushed downward from fingertip)
\end{align*}
$$
④ Enter the normal force received from the paper into column K
From the formula above,
Enter a formula in cell K4 so that it becomes '= J4 (Gravity) + I4 (Force pushed from fingertip (downward))'.
Copy this formula downwards and enter it for all time steps.
⑤ Enter the coefficient of friction (horizontal direction) into column L
Since the coefficient of friction will be considered at a later date, we will enter a tentative value here.
Enter '0' in column L for all time steps.
This represents a state where the 10-yen coin receives no frictional force.
※This is almost the same state as an air hockey disk.
⑥ Enter the frictional force (horizontal direction) in column M
In cell M4, enter the formula: '= L4 (coefficient of friction (horizontal)) * K4 (normal force received from the paper)'.
Copy this formula down to apply it to all time steps.
⑦ Enter the sum of forces (horizontal direction) acting on the 10-yen coin in column N
In cell N4, enter the formula: '= H4 (pushing force from fingertip (horizontal)) - M4 (frictional force (horizontal))'.
Copy this formula down to apply it to all time steps.
⑧ Enter the acceleration (horizontal direction) in column O
In cell O4, enter the formula: '= N4 (sum of forces (horizontal) acting on the 10-yen coin) / mass (m)'.
For mass, reference the set value.
Copy this formula down to apply it to all time steps.
🔎 Confirming how to determine displacement from acceleration
To determine velocity from acceleration, and displacement from velocity, time integration is required.
Here, we will use the method for calculating integral values in Excel that we considered in the article below 👇.
⑨ Enter the velocity (horizontal direction) in column P
We will enter the formula to determine velocity from the changes in time and acceleration.
Since the 10-yen coin is initially at rest, enter '0' in cell P4.
Enter a formula in cell P5 so that it becomes '= P4 (previous velocity) + O4 (previous acceleration) * {G5 (current time) - G4 (previous time)}'.
Copy this formula downwards and enter it for all time steps.
In the formula you enter, do not reference the set value of 'time step ($${\Delta t}$$)', but instead use '(current time) - (previous time)'. By doing this, you can calculate correctly even if you change the time step in the middle of column G.
⑩ Enter displacement (horizontal direction) in column Q
We will now enter the formula to find displacement from the changes in time and velocity.
As a starting point, enter '0' in cell Q4.
Enter a formula in cell Q5 so that it becomes '= Q4 (previous displacement) + P4 (previous velocity) * {G5 (current time) - G4 (previous time)}'.
Copy this formula downwards and enter it for all time steps.
In the formula you enter, do not reference the set value of 'time step ($${\Delta t}$$)', but instead use '(current time) - (previous time)'. By doing this, you can calculate correctly even if you change the time step in the middle of column G.
✔︎ Conclusion
This time, we entered specific numerical values and formulas into the Excel model template created previously, and built a calculation model to derive the displacement of the 10-yen coin.
An Excel file reflecting the work in this article is available as an example solution. You can get it in the download area below.👇
#6 | Get the Calculation Model Here! Excel Models & Python Tools You Can Learn While Playing Are Now Available
Next time, we will perform an operation check (debugging) under simple conditions to see if the model we created this time works correctly. And if we have the energy, we plan to try making a real-time animation with Python!
How the movement of the 10-yen coin, which we have thought about using mathematical formulas up to this point, will be reproduced on the screen...
Please look forward to it!
🔗 Continues to the next article
If you found this even slightly helpful or
are curious about what comes next,
please consider liking or following,
as it encourages me to create the next article!
Subject: #Kokkuri-san
Theme: #Implementation #Excel
Use Case: #IndependentResearch #Relearning #Relearning #ClassroomIdeas #Physics
Topic: #ILovePhysics #IMadeThis
いいなと思ったら応援しよう！
もし具体的に何かのお役に立てたなら、チップで応援していただけると励みになります！
いただいたチップは、今後の活動に使わせていただきます😊