Gyroscopes are weird. The weird stuff they do is difficult to describe properly, and even more difficult to explain, but it's extremely useful and easy to demonstrate. Toy gyroscopes are cheap and easily available, and it is highly recommended that anyone with even a passing interest in how the world works gets hold of one.
What Is a Gyroscope?
Anything which spins on an axis can be considered a gyroscope. A bicycle wheel, a basketball on the finger of a Harlem Globetrotter, a plate on top of one of those bendy poles that strange men set spinning on television magic shows for no readily apparent reason - all of these things can be considered gyroscopes.
While any object will do, for a really good demonstration of the stuff that makes gyroscopes weird, the ideal shape is a heavy disc with a thin axle. Various arrangements of rings can be added to protect the disc as it spins and allow the user to pick it up, but the most important factor is that as much as possible of the disc's mass should be in the rim. For that reason, the discs on toy gyroscopes usually have holes drilled in them to minimise the weight, and thickened rims to maximise the effect.
Take a football (or a soccer ball, if you're American1), or a basketball, and just try to balance it on your finger. It won't be there long. Now try spinning it on your finger. With a little practice, you can keep it there for several seconds. With a lot of practice, you can keep it there for several minutes, occasionally topping up the spin rate with a carefully applied chopping motion.
Get a bicycle wheel and hold it by the axle. Have someone else spin it as fast as they can. Now just try to turn around while holding the axle.
Buy a toy gyroscope, and before spinning it up, try to balance it on the small plastic model of the Eiffel tower which is usually supplied, or the pointed end of a pencil. Now, spin it up using the string or piece of toothed plastic provided, and try again. It will, at first, stand up straight. It will then start to wobble, with the free end of the axle moving in larger and larger circles, but won't fall off. Even though its centre of gravity is clearly no longer above its only point of contact with the ground, it doesn't fall over.
All of the above weird effects are results of a property of gyroscopes called 'precession'. The following explanation calls for a bit of visual imagination.
Imagine a large gyroscope, say about a metre high, perfectly balanced upright in front of you. Its axle is vertical, the bottom end is fixed to the ground, and it's not spinning. If you push the top of the axle to the north, it will fall over, to the north. So far, so exactly what you'd expect.
Now imagine the same gyroscope, but this time spinning at speed. It's standing in front of you, and again you push the top of the axle to the north. However, it doesn't move to the north, it moves to the east, at exactly 90 degrees to the applied force.
This is precession, and a lot of what's weird about gyroscopes, is down to this simple fact: when a force is applied at right angles to the spinning axis, this will give rise to movement in a direction at right angles to both the spinning axis and the applied force.
What Causes Precession?
Precession is due to the law of conservation of angular momentum. Momentum is what you get if you multiply the mass of an object by how fast it is moving, so a bullet has a lot because it's moving very fast, and a steamroller has a lot because it's very heavy. Angular momentum is momentum in a circle - an object spinning on the spot still has momentum, even though it isn't going anywhere. As with matter and energy, you can't create or destroy momentum, you can only move it about or change it into something else. For instance, some of the angular momentum of a toy gyroscope is converted into heat in the bearings. Some of the angular momentum of a spinning basketball is converted into heat on the end of your finger! This is why they eventually run down.
A precise explanation of exactly why conservation of angular momentum causes a gyroscope to precess is beyond the scope of this entry, but there are many easily available physics and applied mathematics textbooks which can give the details, with diagrams. The important point to realise is that it is not in any way magical, and it's most definitely not anti-gravity. It is perfectly within the rules of ordinary Newtonian physics. Many inventors in the past have claimed to have built working anti-gravity devices2 which use gyroscopes, but not a single one has ever flown, or even demonstrably reduced its own weight. However, research in this area continues, and significantly the US government continues to grant patents for reactionless drives - something it won't do for perpetual motion machines.
What's the Use?
Apart from looking cool on a basketball court, there are many uses of gyroscopes. Their tendency to resist movement makes them useful as stabilisers in ocean-going ships. If you set a gyroscope spinning and mount it on a gimbal3, the axis will remain pointing in exactly the same direction. This allows the gyroscope axis to operate as an inertial frame of reference. This allows it to be used as an artificial horizon in an aircraft, and as a navigational aid in all sorts of applications. If you spin such a gyroscope axis parallel to the Earth's axis, it will always point to the north pole - the true north pole, not the magnetic north pole which has the unfortunate habit of moving about several hundred miles from the true north pole.
Is the Earth a Gyroscope?
Since it spins on its axis - yes, and it is precessing. Its period of precession4 is about 25,800 years.