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I've recently been exposed to gyroscopes in my physics class. We discussed flywheels in relation to the gyroscopic effect.

I was wondering how the stationary flywheel scenario worked. Why is it that the gyroscopic falls along a "circular path?" I understand a torque is exerted by Fg, is it consequence of extended mass resulting in circular motion?

Also, how is angular momentum conserved in this case, considering Fg is a conservative force? The Fg increases the angular momentum of the flywheel, but how is the system's angular momentum conserved?

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I was wondering how the stationary flywheel scenario worked.

As Adrian said if the flywheel were not to spin, it would fall over (i.e. you would not observe what is called precessional motion). Why? Because there's no angular momentum due to the spin of the flywheel and thus no torque.

Why is it that the gyroscopic falls along a "circular path?"

This is basically because of the spinning of the flywheel. Let's say that it is spinning clockwise. Then by convention $\vec L$ points outwards along the symmetry axis.

The key is this: the weight of the flywheel exerts a torque on the gyroscope about the pivot. Such a torque changes the direction of the angular momentum, triggering precessional motion.

By the right hand rule you can guess that the direction of the torque is perpendicular to the plane in which the weight and O-CM distance vectors lie

how is angular momentum conserved in this case, considering Fg is a conservative force?

The angular momentum of the gyroscope system is not conserved. This is due to the presence of an external torque:

$$\vec \tau = (m \vec g)\vec r$$

Where $m$ is the mass of the flywheel and $\vec r$ is the O-CM distance.


Let me give you an example in which the angular momentum of the system is conserved.

Imagine a rocket containing a gyroscope that is initially not rotating (and it is in the space). Then, the total angular momentum of the system is zero at this point. Say you want the rocket to spin left. Then, if you make the gyroscope rotate clockwise (by remote control) you will generate an angular momentum pointing outwards. As the gyroscope-rocket system is isolated, angular momentum must be conserved. Thus, the rocket must rotate counterclockwise and generate an angular momentum pointing inwards so that both angular momentum vectors cancel each other out.

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The diagram is a little confusing, but when the flywheel is not spinning, it has no gyroscopic effect, it will fall just as anything else. The dashed arrow that says "path of free end" is meant to show that the end is pulled down, but in a curve, because the other end of the rod is held up by the pivot point. Say you put one end of a horizontal pencil on the pivot point, when you let go one end would stay on the pivot point, and the other end would fall towards the table in a radius around the pivot point. Hope this helps.

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  • $\begingroup$ So what role does the torque play in this case since it's pointing into the page (in y-direction)? $\endgroup$ – student Nov 12 at 7:03
  • $\begingroup$ Gravity is the only force acting on a stationary flywheel $\endgroup$ – Adrian Howard Nov 12 at 17:56

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