# Help understanding a result from Euler's laws of rotation

So I'm trying to learn how to apply Eulers laws of rotation. I'm currently looking at an example in the book Engineering Mechanics 3 by Dietmar Gross et. al that can be found on page 190. Seems like a great book btw.

A mill is constructed as in the picture below.

A key result is that the normal force $N$ increases with the rotational speed $\omega_0$, and the book states that this is due to a gyroscopic effect. I find this result absolutely absurd. Am I getting this right? A wheel that is rotated on a flat surface like this, will have a higher normal force the faster it is rotating? Really? If so, is there any intuitive explanation for this?

Edit: So solving this problem gave me a crash course in how gyroscopes work. This problem is closely related to something called gyroscopic precession. I will post links below that I used for gather the knowledge I needed. It turns out that this problem is much easier solved using the the laws directly related to the angular momentum around the center-point of the mill. I think they used Eulers equation for rotation in the book just to verify that these give the same result.

The approach I used was analyzing the change in angular momentum

$\dot{L} = M$

First I checked $M$ to see about where torque $M$ would act. There is a great answer to this question below where a user posted a picture of this too. Then I solved for the vector norm $||\dot{L} ||$to see how large the torque would be. This gave the same result as in the textbook. I also bought a toy gyroscope to really feel this effect, which is both weird, absurd and amazing to me. I have learned tons the past days, it has been great.

• DId you work out the equations before you did a blanket statement that is is absurd? The best way to explain this is to show the equations of motion and then try to explain where each part comes from. Then it becomes more or less obvious. – ja72 Jan 26 '17 at 21:10
• Useful links that I werent allowed to post in topic: Good quick intro to gyroscopes: youtube.com/watch?v=ty9QSiVC2g0 A more thorough walkthrough: youtube.com/watch?v=4t87kZ0pAUc A fast summary of the above video: scienceworld.wolfram.com/physics/GyroscopicPrecession.html A more detailed derivation, including an effect that was neglected without mentioning in the previous: web.mit.edu/8.01t/www/materials/modules/guide16.pdf Furthermore I found the chapters in the book mentioned in topic really, really useful. – Martin Jan 30 '17 at 9:51
• ja72: I won't discuss your personal feeling about gyroscopic effects, that would be like having a long discussion about what ice cream is the tastiest. But I myself find gyroscopic effects to be everything from absurd to counter-intuitive to amazing. – Martin Jan 30 '17 at 9:56

I find this result absolutely absurd. which shows that most of rotational dynamics is counter intuitive.

Let me try and explain by hand-waving that

A key result is that the normal force $N$ increases with the rotational speed

is correct.

Looking from the top the angular momentum of the wheel changes from $\vec L_{\rm old}$ to $\vec L_{\rm new}$.

The change in angular momentum is $\Delta \vec L$ as is shown in the vector diagram on the right.

Now change position and look at the wheel from the side.

The change in angular momentum is out of the screen and this must be the direction of the torque $\vec \tau$ which causes that change in angular momentum.

So about the left hand pivot point the torque has to try and rotate the wheel anticlockwise which must mean that $N >W$ remembering that when the wheel was not moving $N=W$
So the normal reaction force on the wheel $N$ is greater than the weight of the wheel $W$.

If the wheel is made to go round faster the magnitudes of $\vec L_{\rm old}$ and $\vec L_{\rm new}$ are larger and so the magnitude of $\Delta \vec L$ must be larger.
In turn the torque must have a larger magnitude and so $N-W$ must be larger with $W$ constant.
So $N$ does increase as the speed of the wheel increases.

• Thanks for a very detailed answer! This is the foundation for a solution that is much simpler than Eulers laws of rotation. I worked some on this and got the same analytical result as in the textbook by using L_dot = M and analyzing the direction of the torque M and the magnitude of it |M|. I'll edit my question. – Martin Jan 29 '17 at 20:25

Yes; nothing "absurd" about it. It's a consequence of angular momentum conservation. Rotating the axis of rotation of the wheel as in this mill will create a moment perpendicular to the wheel axle which has the effect you mention. This is explained in the book you mention. Did you read page 189?

• Absurd might not be the word, maybe amazing is a more appropriate description. Sure I can see it happening using mathematical analysis. I just could never have looked at the system and foreseen this effect using intuition and experience. But hmm. Thinking about it, the good old demonstrations using a spinning bicycle wheel should apply here actually. I'm getting my miniature gyro on mail order tomorrow, psyched to play around with it :) – Martin Jan 26 '17 at 20:34
• Yes, the bicycle wheel is a good object to demonstrate the effect. Or, if you have access to a power tool (electric drill or something of that sort, with a fast spinning rotor inside) you can also experience this moment. Just run the tool and then rotate it around an axis normal to the axis of rotation of the rotor. – Pirx Jan 26 '17 at 20:37