Is Precession Influenced by Conventional Vector Directions?

[FrankJ777]

I'm curious about how precession works. From what I thought I understood about angular momentum and torque vectors, the direction that they point are by arbitrary convention using the right hand rule. They are just a convenient expression of the normal or the plane that the torque acts on the moment arm and the magnitude, or in the case of angular momentum, the vector describes the magnitude and the normal of the plane the mass is rotating in. The way I understand precession is that a force acts on the angular momentum vector, which produces a torque with a vector pointing from the normal of the plane that the force vector and angular momentum vector produce.

What's odd to me though, is that I thought the angular momentum vector and torque vector directions were an arbitrary convention, but they seem to act like ordinary force vectors and moment arms, in that the force F acts on L as though it were a moment arm and produces the torque vector, $\tau$ which seems to act like a conventional force vector, causing L to have an angular acceleration. I don't understand how it works if the direction of L and $\tau$ are just arbitrary conventions. It seems like there is something more I'm missing. could someone please clear it up for me. Thanks

 

[Andrew Mason]

I'm curious about how precession works. From what I thought I understood about angular momentum and torque vectors, the direction that they point are by arbitrary convention using the right hand rule. They are just a convenient expression of the normal or the plane that the torque acts on the moment arm and the magnitude, or in the case of angular momentum, the vector describes the magnitude and the normal of the plane the mass is rotating in. The way I understand precession is that a force acts on the angular momentum vector, which produces a torque with a vector pointing from the normal of the plane that the force vector and angular momentum vector produce.

What's odd to me though, is that I thought the angular momentum vector and torque vector directions were an arbitrary convention, but they seem to act like ordinary force vectors and moment arms, in that the force F acts on L as though it were a moment arm and produces the torque vector, $\tau$ which seems to act like a conventional force vector, causing L to have an angular acceleration. I don't understand how it works if the direction of L and $\tau$ are just arbitrary conventions. It seems like there is something more I'm missing. could someone please clear it up for me. Thanks

Precession is a difficult aspect of what is a very difficult subject - classical mechanics.

First of all, the force (of gravity) acts on the body; it does not act on the angular momentum vector of the body. The torque of gravity adds to the angular momentum of the top.

1. If the top is not spinning, the torque resulting from gravity acting on the centre of mass about the tip will cause it to fall over immediately. The torque is represented mathematically by a vector perpendicular to the plane of the tip and the centre of mass as the top falls over. Since torque = dL/dt, the change in angular momentum (ΔL) as the top falls over is represented by a vector in the same direction as the torque. The value of ΔL is the integral of the torque x the time through which it acts. That is all easy enough.

2. Now let's see what happens when the top is already spinning and is perfectly vertical. It already has angular momentum that is represented by a vector $\vec{L}$ in the direction of the spinning axis of the top - i.e. it is perpendicular to the torque vector that made the top fall over in 1. Suppose I push the top over a little by applying a lateral force F on the upper portion of the top at a height h above the tip for time t.

This produces a momentary torque about the tip $\tau = \vec{F}\times\vec{h}$ for a time t so it imparts a change in angular momentum of $\Delta \vec{L} = \vec{F}\times\vec{h}t$. The total angular momentum is now $\vec{L}+\Delta \vec{L} = \vec{L} + \vec{F}\times\vec{h}t$. The change in the angular momentum vector is perpendicular to L, the angular momentum from the axis spin, so the resultant vector is a vector that is not quite vertical. In order to achieve that, the axis has to precess. When it does that, gravity starts exerting a torque. And the angular momentum vector keeps changing - by increasing the degree and rate of precession. All of this is quite complicated. Hyperphysics site has quite a good introductory explanation: http://hyperphysics.phy-astr.gsu.edu/hbase/top.html

AM