Boomerangs and Gyroscopes

The gyroscopic effect

One of the problems with a bicycle wheel is that if you try to hold it on the spindle you get your fingers stuck in the spokes. But if you go to a bike shop and get a stunt peg (preferably with little grooves) you can attach it to the bike wheel, giving you a handle, and making things much safer. Now spin the wheel. You'll feel really curious gyroscopic effects. It's rather hard to describe them, but I want to make it clear to you. So take a piece of string. If you get stunt pegs which have little grooves in them, the string can be attached very easily. Now spin the wheel, holding it up (vertically) on the piece of string. You should see clearly that something rather amazing is happening - the wheel spins round in the vertical plane. This is the gyroscopic effect, and it's called gyroscopic precession. In broad terms the reason why this is happening is because there is a spinning object, the wheel, and there is a couple or moment or torque being provided which is at right angles to the direction of spin, and I will explain this more in a minute.

This is nothing special - bike wheels aren't the only things that do this. I have my 2 year-old daughter's spinning top with me. I can attach a piece of string to the bottom, then put the string in my mouth so I have both hands free, then hold the top in one hand and spin it with the other. If you try this, then hold the string and watch the top, you will see it precess around the string. Even a 2 year-old's top shows these amazing gyroscopic effects! The question is, why?

Before we can really understand the gyroscopic effect properly we have to understand some important concepts. The first is the concept of the couple (also known as moment or torque). In the diagram you can see a couple defined as an angular force. Note the two pink forces marked f applied in opposite directions to a bar, this is called a couple - because there are two forces. I can use the right hand rule to define the direction of the couple. If my fingers curl in the actual turning direction, then my thumb will point upwards in the direction of the couple.

I can also demonstrate a couple if I sit on a rotating stool, and someone pushes my shoulders round, so I go round and round. What happens is that if one of my shoulders is pulled and the other pushed, I go round, because there are two forces in opposite directions moving me.

The next concept we have to understand is that of moment of inertia. Moment of inertia is the angular equivalent of mass. On the left of the picture to the left, I have a dumbbell shaped object which has two masses relatively close together on it, on the right of the slide there is the same dumbbell shaped object but the masses are further apart. The one on the right has a higher angular mass than the one on the left, as we shall see shortly. The angular mass we call moment of inertia.

The other thing we can talk about is called angular momentum. Angular momentum is the product of the moment of inertia and the angular velocity, 1#1 where J is moment of inertia, and 1#1 is angular velocity. This is just the same as defining linear momentum as the product of mass and linear velocity. A heavy car has more linear momentum when going at a certain speed than a light car travelling at the same speed - it takes more effort to stop it. Similarly a heavy pair of dumbbells has more angular momentum than a light pair of dumbbells, and it takes more effort to stop it rotating.

What is interesting about this, is that I can change the moment of inertia of the dumbbells by moving the weights in and out. I can demonstrate the concept of moment of inertia by holding two heavy weights (5kg) close to my chest. It is quite easy for someone to apply a couple to my shoulders and spin me around. If I hold the weights at arm's length, it is much harder for someone to apply a couple to my shoulders and spin me round. The reason is that I have a higher moment of inertia. It's like having a higher mass. A heavier car is harder to push than a lighter one. An object with high moment of inertia is harder to spin than one with a low moment of inertia.

I changed my moment of inertia in the middle of that experiment, simply by moving my arms in and out with the heavy weights. You can't change the mass of a car so easily. In changing my moment of inertia, angular momentum on this (almost) frictionless stool is conserved - this means it stays constant. So if my moment of inertia is reduced by bringing the weights in, or increased by bringing the weights out, then my angular speed must increase or decrease accordingly (since 1#1 ). If I start spinning with the weights in, I go quite fast. If I then stretch out my arms, so I hold the weights at arm's length, I slow down. If I bring them back, I speed up again.

Let's look at the circular motion formula again. Remember the mass going round in a circle, with the centripetal force acting on it. The force is at right angles to the direction in which the object is travelling at any point. According to Newton's Second Law of Motion, there is a change in momentum in the direction of the applied force, so the linear momentum is constantly changing. It is equal to mv, and v changes as the object changes direction (velocity incorporates the idea of direction since it is a vector, speed is just the size of the velocity, without the direction).

If we look at a spinning disc or spinning bicycle wheel, which has a certain angular moment, and then apply a couple [provided by the two orange forces] to the spin. The angular momentum changes in the direction of the applied couple, so the direction of the angular moment has been tilted over [see the small diagram]. If we apply Newton's Second Law of Motion to angular motion, it says that angular momentum changes in the direction of the applied couple.

We are now in a position to look at a bicycle wheel properly. Imagine a bicycle wheel spinning, with a couple applied. The string on the stunt peg holding the bicycle wheel creates a couple. The bicycle wheel's weight provides a downwards force at the wheel's centre, and there is also an upwards force through the string. This is a couple since there are two opposite forces which are not aligned with each other. If I spin the bicycle wheel this is at right angles to the couple and so at right angles to the angular momentum. This means that the angular momentum changes, creating motion in the direction of the couple. This is gyroscopic precession. This is not straightforward stuff, but it is important to realise all this can be analysed in this way.

I can demonstrate this to make it clearer to you. If I hold the bicycle wheel up with the piece of string through the stunt peg, then there is a couple acting on the wheel. If I then spin the wheel, it precesses round steadily [see the directions on the diagram]. If I point my right thumb upwards my fingers point in the direction of the precession.

To check the formula, I could spin the wheel backwards (changing 1#1 to a negative value), and I should then find the wheel precesses backwards ( 1#1 has to be negative also since C and J are the same). This turns out to be what happens. I can also spin the wheel slowly, and the precession should then be fast - and it is (the product 1#1 has to remain constant). A very fast spin causes quite slow precession. It's really quite wonderful stuff!

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