Gyrating Gyroscopes

So, I was recently asked to describe how gyroscopes work, and all I could offer was a mumbly, handwavy, explanation – mumble conservation  mumble angular mumble momentum mumble – so I thought I should really just do my usual thing of writing it all down in one place, and hopefully, by the end of it, we’ll all understand a little better! I’ve also promised myself to try not to use any equations…. so at the end, you’ll find a link to a paper on gyroscopic motion that I found – you’re welcome

Let’s start with the word gyrate. Please don’t allow your mind to drift to a land full of embarrassing uncles, dads, bosses, random strangers murdering the dancefloor on a Friday night…..Definition 1: Dance in a wild or suggestive manner.

Instead, try to think of this…..Definition 2: To revolve around a fixed point or axis, or to move in a spiral-like course. You’ll be relieved to know that it is this second definition that we’ll be working with

The word ‘gyrate’ is, of course, related to the word ‘gyroscope’. And, at its most basic level, a gyroscope is just a spinning wheel or disk whose orientation will remain almost fixed, regardless of any motion of the surface underneath it – this is why it’s possible to set a gyroscope on your finger or on a suspended string without it falling off.

Because of this, gyroscopes are used to measure or maintain orientation, and have found applications from yo-yos and bicycles to planes and spacecraft. So, how do they work?

We need to talk about two things first

- Torque: In basic terms the torque is a measure of the turning force on an object – it describes is the tendency of a force to rotate an object about an axis. Think about a lever (a rod fixed at one end). In this case, it’s force can be measured as the magnitude of a force applied at a right angle to the lever multiplied by its distance from the centre of rotation (the fulcrum). So the torque on a spinning wheel can be calculated  by measuring the force applied, and multiplying it by the radius of the wheel (distance from the centre to the edge)

- Conservation of Angular Momentum:  Angular momentum is a measure of the momentum of an object that is spinning. And when we say it is conserved, it means that the amount of momentum remains constant, unchanging. Angular momentum is conserved in any system where there is no net external torque, or turning force, acting on it. We live in a universe of unbalanced forces. Newton tells us that things will remain as they are unless another force acts on it. Its not to say that a stationary tennis ball on a table has NO forces acting on it, it’s that all the forces that are acting on it BALANCE OUT. It will only move if we add another force, by pushing it, or by tilting the table or whatever.

So, what have these two things got to do with a gyroscope? Well, together they describe how gyroscopes work! A gyroscope’s behaviour is based on the principles of conservation of angular momentum, meaning that somehow, even with the rings moving, there is no overall torque acting on the central disk, and so it is very very stable.

So let’s get to it. I tried to draw a gyroscope, but my efforts were pants. So I lazily settled for the above image from Wikipedia. I’ll try to refer to their labelling below (apologies if I fail)!

- The  gyroscope frame is mounted so that it pivots about an axis in its own plane (determined by the surface it is sitting on). So the frame can only rotate around one axis.

- The gimbal is a ring mounted in the gyroscope frame in such a way that it can pivot freely, but only about an axis perpendicular to the axis of the frame.

- The rotor is a heavy-rimmed disk at the centre of the assembly – this is so the mass is kept a far away from the centre as possible (think about torque!). The axis that the rotor spins about is called the spin axis and is always perpendicular to the axis of the gimbal.

Are you still with me? Ok, so you’ve got the rotor wheel, which is nested inside the gimbal, which in turn is nested in the frame….  Both the frame and inner gimbal have very low-friction bearings, and together, these rings help to isolate the rotor from any outside ‘influence’, or torque, when it is set spinning.

Now, and this is important, if I tried to balance the gyroscope on a point, or even standing upright on a table without setting it spinning (via a string wrapped around the edge of the rotor for e.g) the gyroscope would simply fall over – think about it, balancing a heavy disk on a tiny point is rather difficult. BUT, set it spinning and Newton’s Laws, and the related conservation of momentum come into place. It is the spinning that gives the gyroscope its stability - once a heavy object (like the rotor disk in a gyroscope) has started moving, it’s very difficult to slow it down

Now, onto the weird stuff.  The precession of a gyroscope can be seen in this gif (also stolen from Wikipedia).

So, I’ve said that the gimbals and conservation of angular momentum ensure that there is no overall torque acting on the rotor in the centre. BUT, there is torque acting on the spinning gyroscope as a whole – of course there is, otherwise it wouldn’t be spinning! This torque produces a change in angular momentum.

But where does this torque on the gyroscope actually come from? Well, it is supplied by a couple of forces: gravity acting downward on the centre of mass of the gyro, and an equal force that acts upward to support one end of the device. The overall force that results from the combination of this torque and the spinning of the gyroscope, acts perpendicularly to both the gravitational torque (horizontal) and the axis of rotation (also horizontal)… meaning it acts about a vertical axis, and this causes the gyroscope to rotate slowly about the supporting point. THIS is called precession.

If you can’t get your head around that last bit, take your right hand and do this:

Your fingers are actually very good at vector maths. My index finger is the gravitational torque, my second finger is the axis of rotation – without actually using ANY vector equations, you can see that the resultant force is pointed in the direction of my thumb, vertically (give or take, I have a bendy thumb) – and this is the axis the gyro rotates slowly about.

The increase in precession is directly related to the spin of the rotor, so, once that slows down, the circular path ‘drawn’ by a precessing gyroscope gets so large that the device eventually falls under the influence of gravity.

Together, these effects say that as long as the rotor of the gyroscope is being spun, the angular momentum of the disk ensures that the gyroscope will keep pointing in the same direction.

This also explains why your bike is much more stable when it’s moving than when it’s stationary – your wheel is spinning which conserves angular momentum.

Right. Well, I’m done for the night – I will, no doubt revisit this as I come to terms with how horribly unclear I’ve left it….. until then, ciao

PS: If you don’t own a gyroscope, I highly recommend you buy one. I love mine, and spend ages just playing with it!

Maths: http://www.gyroscopes.org/how.asp

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