#10 | Breaking Free from Zero Friction! Modeling Practices for Kinetic Friction

This article is part of the series 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 were able to reproduce the movement of the 10-yen coin like in Kokkuri-san by modeling the trembling of the fingertips using 'random numbers based on a normal distribution'.

But!

Since the frictional force (coefficient of friction) was still zero, the 10-yen coin moved easily with a small force, just like an air hockey puck.

So, this time, let's take a look at what happens when we incorporate kinetic friction!

The Excel model created in the steps of this article is available to the public.
You can get it from the download area below 👇.

✔︎ How to incorporate kinetic friction

The Excel model we have created so far is already set up so that the frictional force is reflected in the calculation if you input the 'force pushed from the fingertips (downward)' and the 'coefficient of friction'.

First, let's start by inputting the 'force pushed from the fingertips (downward)' (column I).

The same method used for the 'force pushed from the fingertips (horizontal)' (column H) that we entered in the previous article 👇 is fine.

Just like the 'force pushed from the fingertips (horizontal)' (column H) in the previous article ☝️, we will input 'random numbers based on a normal distribution' into the 'force pushed from the fingertips (downward)' (column I) as well.

This completes the input for the 'downward force pushed by the fingertip'.

All that remains is the coefficient of friction.

Here, let's look back at the equation of motion incorporated into the Excel model.

This equation of motion was formulated on the premise that when the 10-yen coin is moving to the right, as shown in the figure below, the frictional force acts to the left.

However, what happens when the 10-yen coin is moving in the opposite direction, to the left?

If the coefficient of friction is set to a constant such as 0.1, the frictional force can only act to the left, which would lead to the bizarre phenomenon of the 10-yen coin accelerating due to friction.

Therefore, when the 10-yen coin is moving to the left, we must ensure that the frictional force acts in the opposite direction, to the right.

To address this, as shown in the graph below, we switch the +/- of the coefficient of friction according to the velocity of the 10-yen coin. ($${\mu_k: Kinetic friction coefficient}$$)

With this, when the 10-yen coin is moving to the right, the coefficient of friction is 'positive (+)', so the frictional force acts to the left; when the 10-yen coin is moving to the left, the coefficient of friction becomes 'negative (-)', reversing the direction of the frictional force to act to the right.

However, there is still a problem with this.

If the direction of the frictional force is suddenly reversed at a velocity of 0 (zero), the change in frictional force becomes too drastic, causing the 10-yen coin to oscillate slightly instead of coming to a stop.

To prevent this, we solve it by adding a transition velocity $${v_s}$$ as shown in the figure below, so that it changes smoothly near a velocity of 0 (zero). ($${\mu_k: Kinetic friction coefficient}$$)

In short, you can just enter the coefficient of friction into Excel as a function that takes the velocity of the 10-yen coin as an argument, as shown in the graph above.

✔︎ Made it into a video

I have made the calculation results incorporating kinetic friction into a video.

The main calculation conditions are as follows.

The kinetic friction coefficient was set to a small value of $${\mu_k=0.001}$$.

Even so, since it barely moved with the same 'force received from the fingertip (lateral direction)' as in the previous article where friction was zero, the calculation was performed with that force increased tenfold.
*Please refer to the previous article for the specific calculation conditions used last time.

By incorporating kinetic friction in this way, the movement of the 10-yen coin changed as follows.

• Greater force is required to move it:Last time, it would stop with only a slight force. This is to be expected.
  *When measuring and inputting the force of the fingertip, you will need to adjust this friction coefficient to match the movement of the 10-yen coin.

• No major changes in how it moves: The irregular movement characteristic of Kokkuri-san did not change enough to be visually noticeable.

I hope you all enjoy simulating under various conditions, such as by changing the friction coefficient!😊

The Excel model created in the steps of this article is available to the public.
You can obtain it from the download area below👇.

Please try making a 10-yen coin dance using equations of motion on your own computer!

✔︎ Conclusion

This time, we incorporated kinetic friction.
With this, we have been able to reproduce the resistance when the 10-yen coin is in motion.

The modeling for 'when it is moving' is now complete.

Next time, we will incorporate static friction, which acts at the moment it starts to move.
Let's see together what kind of difference this creates.

Stay tuned!

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• Subject: #Kokkuri-san

• Theme: #KineticFriction #CoulombFriction

• Use Case: #IndependentResearch #Relearning #Refresher #ClassroomIdeas #Physics

• Topic: #ILovePhysics #MadeIt