They say you never forget how to ride a bicycle, and as far as I can tell, it's true. I remember my nervous exhilaration the day we removed the training wheels from my first bicycle and I struggled frantically to balance. It took a day or two to get the hang of it. Or, as physicists call it, to achieve dynamic equilibrium.

But apart from a couple of spills and crashes over the years, I've never had to think twice about staying upright. There's something special about two wheels. Three wheels are for little kids, and one wheel is for the circus. The magic number is two.

At first, two wheels sounds like a bad idea. After all, a table or stool with only two legs won't stand up because a stationary object needs at least three legs to balance in our three-dimensional world.

But what works well at rest doesn't always work well in motion. The tricycle is a case in point. Picture a young tricyclist rolling recklessly downhill and making a sudden left turn. Instant disaster.

By steering left, the child creates a conflict between inertia and friction -- inertia tends to keep the combined mass of child and cycle moving straight ahead, but friction between the ground and the front wheel (way below the child-cycle center of gravity) tends to turn the tricycle toward the left.

If the turn is sharp enough, the tricycle will drive out from under the child and tip over. The problem isn't with the child; it's with the tricycle. The tricycle has one too many wheels.

Paradoxically, a bicycle's instability when stationary is its greatest asset when moving. You can lean a bicycle. And when you make a turn, you don't simply steer the handlebars; you also lean in the direction of your turn. Leaning compensates for your inertia and is so essential to biking that you don't even notice it.

If you steered the handlebars left without leaning left, you'd be no better off than a tricycle rider and would fall off the right side of the bicycle. Conversely, if you leaned left without steering the handlebars left, you'd fall off the left side of the bicycle. So when you lean left and steer left simultaneously, the two effects cancel.

However, leaning into turns requires a deliberate effort on your part because a moving bicycle naturally resists tipping. When accidentally tipped, the bicycle automatically steers you to safety. Its handlebars pivot spontaneously so that it drives underneath your center of gravity and stops you from falling.

The automatic steering results from two general principles. One is the fact that, whenever objects can reduce their stored or potential energies by moving or bending, forces push them in those directions. For the same reason, a bike falls over when it isn't moving.

A tipped bicycle pivots its front wheel to lower both its center of gravity and its gravitational potential energy. Try it. Stand by your bike and tip it to the left. It will automatically steer left to lower its center of gravity.

This phenomenon occurs because bicycles are designed so that the front tire touches the ground behind the steering axis [see illustration]. If it didn't, the wheel wouldn't naturally steer into a lean.

The second principle involved in automatic steering is gyroscopic precession. When you apply a twisting force to a rapidly spinning object, such as a gyroscope or bicycle wheel, the object starts to "precess," or gradually shift its axis of rotation.

In the case of the bicycle, when you lean to the left, the road pushes the tire up and to the right. This twisting force, or torque, causes the wheel to precess. It does so in the same direction as the steering, so they work together to keep you upright.

Automatic steering gives the bicycle its remarkable moving or dynamic stability and explains why bicycles can be ridden without hands. That classic "Look, ma!" stunt loses some of its glory when you realize that even a riderless bicycle tends to stay upright as it rolls forward.

In fact, moving bicycles are so stable that, to initiate the leftward lean needed for a sharp leftward turn, you must actually turn the handlebars briefly to the right and throw the bicycle off balance.

For once, steering without leaning is a good thing. You immediately begin to fall off the left side of the bicycle, the way a tricyclist would, and soon develop the leftward lean you need to complete the leftward turn. It's part of learning to ride a bike.

FLEXIBLE ENERGY

But while a bicycle may be able to stay upright on its own, it still needs you to supply it with energy. You transfer energy to the bicycle by doing work on its pedals. The crank then conveys this energy to the wheels. However, the simplest and most obvious scheme, using pedals to crank the front wheel directly, faded from general use about a century ago.

So merely passing along your energy to the wheel can't be the whole story. Why do modern bicycles use an elaborate system of levers, gears and pulleys to transfer your energy to the wheels?

The answer lies at the interface of physics and physiology. Your body is more than just a source of energy, and some ways of doing work on a bicycle are more comfortable than others. The usual pedaling motion is similar to that of jogging, and you find it relatively natural to push pedals around in a circle about 80 times a minute with forces approaching your weight.

Pedaling much faster than this is uncomfortable, as is having to push much harder on the pedals. Thus, the goal of all those gears and gadgets is to make sure that you never leave this comfort zone, even when the bicycle's wheels are doing vastly different things.

Whether you're traveling fast on level pavement or slowly on a steep incline, the work you do each second to maintain your pace is about the same. On level pavement, the rear wheel needs your energy in the form of gentle forces exerted over long distances of motion. But on the incline, the wheel needs energy in the form of strong forces exerted over modest distances.

This need for flexibility, along with the relatively small forces a person can exert on front-wheel mounted pedals, is what relegated direct-drive bicycles to the antique shop. Like all simple machines, the gears and levers of a modern bicycle provide a mechanical advantage by allowing force to be exchanged for distance or distance for force while leaving unchanged the amount of work being done and the energy transferred.

WHAT A DRAG

However, energy leaks out everywhere, and you must keep pedaling. Air resistance or drag is the biggest energy thief, and bicyclists are forever trying to reduce its effects.

The dominant form of drag here is pressure drag, which occurs when increasing air pressure in front of the moving bicycle isn't matched by an equal rise in pressure behind it. And if there's higher pressure in front of you than behind, the inequality will push you backward as the two air masses attempt to achieve equilibrium.

In an aerodynamic structure, the difference is minimal because the amount of turbulence -- chaotically moving air behind the structure -- is as small as possible. For a bicyclist, as for an airplane, bird or fish, the amount of pressure drag is roughly proportional to the cross-sectional area of the turbulent fluid pocket trailing behind.

The maximum increase in pressure in front of an ordinary rider traveling at 20 mph is only about .01 pounds per square inch (psi) above a normal sea-level pressure of 14.7 psi. But that translates into a backward force of several pounds, assuming that the air pocket behind the bike is about four square feet.

Speed enthusiasts use drop handlebars, aerodynamically tapered frames and seat posts and even reclining arrangements to minimize the size of their trailing air pockets. Known as recumbent bicycles, the reclining models have a small but loyal following, in part because they experience about 30 percent less air drag than ordinary bicycles.

Of course, when Chris Huber set the world speed record of 68.73 mph in 1992, his recumbent bicycle was equipped with a smooth, aerodynamic shell known as a fairing. This airfoil shape left almost no air pocket. But even then, it was helpful to ride in the thin Colorado air near 8,000 feet of elevation.

TIRES AND FRAMES

Air resistance isn't the only villain. Friction also can slow you. Although ball bearings between hubs and axles nearly eliminate sliding friction at the middle of a bicycle's wheel, the road takes a toll.

As a tire rolls, its surface repeatedly dents and recovers, and this process leads to sliding friction between tire and ground and to internal friction within the rubber itself. Keeping tires properly inflated reduces the denting and lightens your burden.

But firm tires have a drawback: You feel every bump. On smooth pavement, that's rarely a problem; on rough terrain, it can be torture. Whereas a springy bicycle frame might compensate for hard tires and cushion your ride, a frame that flexed a lot as you pedaled would waste energy.

You do work on the frame each time you push a pedal down, and the more the frame bends, the more work you do on it. If the frame merely converted this work to thermal energy through internal friction, that might be tolerable.

Instead, it returns some of the energy to your body as you withdraw your foot. Your body can't make much use of returned mechanical energy. It's almost as exhausting, for example, to lower a weight as to raise it. So you don't want a flexible frame that extracts and returns lots of energy when you pedal.

Thus, most energy-efficient frames are made of hard aluminum alloy tubes, arranged in triangle shapes that minimize weight while maximizing stiffness. So something else must cushion the ride. The latest breed of off-road bicycles includes full suspension, complete with springs and shock absorbers.

These suspended bicycles have rigid frames that barely budge as you pedal. But the frame floats above the wheels, supported by springs of various sorts, including coiled steel, compressed air and elastic solids.

HITTING THE BRAKES

In addition to improving comfort, a suspension keeps your tires in contact with the ground so static friction can allow you to control your bicycle. When you're airborne, you can't steer, pedal or brake. The same is true when your tires are skidding and, to make matters worse, you lose the automatic steering effect that helps to keep you upright.

Also, since antilock brakes are not made for bikes, you must have a delicate touch with the brake handles, even during an emergency.

To make it easier to feel when the wheels are beginning to lock during braking, brakes such as the Shimano V-Brake use longer levers to press the brake pads against the wheels. The increased leverage reduces the force you need to apply to the brake handles to stop the bike and leaves you with more tactile sense of what's happening to the wheel.

Remarkably, the wheel that skids most easily during a fast stop is the rear wheel. That's because the inertia of your body throws you forward when friction at the wheels slows the bicycle beneath you. In principle, this effect can toss you over the handlebars and onto the pavement, but skidding usually presents the bigger danger.

As the stopping bicycle pushes you backward to extract your forward momentum, you push forward on the upper half of the bicycle so it experiences a severe twist or torque about its center of mass. As the result of this torque, its front wheel is pushed hard into the pavement while its rear wheel is virtually lifted into the air.

Since the frictional forces between two surfaces are roughly proportional to how hard those surfaces are pressed together, the front wheel develops enormous traction during a stop while the rear wheel loses most of its grip on the road. That's why the front brake is so much more effective than the rear brake. You can easily begin to skid and fishtail if you squeeze the rear brake too tightly.

The brakes themselves exhibit interesting frictional effects. When you activate them, rubber pads press against the wheel rims, and sliding friction begins to transform your motional or kinetic energy into thermal energy.

More energy than you may think. A 150-pound rider on a 25-pound bike who slows gently from 20 mph to 10 mph generates 568 calories, sufficient heat to raise the temperature of a cup of water by four or five degrees. Riding the brakes while slowly descending a 1,000-foot hill produces 57,000 calories, enough thermal energy to turn a pint of water from freezing to boiling.

(One calorie represents the amount of heat needed to raise the temperature of 1 gram of water by 1 degree Celsius. Food Calories, usually spelled with a capital C, are actually kilocalories; that is, they signify 1,000 times as much energy.)

Once again, the proportionality between the forces pressing two surfaces together and the friction those surfaces experience gives you control over how quickly the brakes slow the wheels.

But occasionally, the brake and wheel begin to vibrate loudly as the brake pads alternately stick and slip along the wheel rim. That stick/slip phenomenon is the basis for all bowed instruments and is responsible for the resonant, bell-like tones you can produce by rubbing a damp finger around the rim of a crystal wineglass.

But the tone you get from a bicycle rim isn't bell-like. It's more of an irritating honk. Maybe if bicycles were made of lead crystal instead of aluminum, they would sound as good as they look.

Louis A. Bloomfield, a professor of physics at the University of Virginia, is the author of How Things Work: The Physics of Everyday Life (John Wiley & Sons, 1997).

CAPTION: Top: The smaller front wheel of Terry bicycles allows a shorter top tube for smaller riders. Above: The recumbent Cheetah design minimizes air resistance. Right: The poster for a Scottish tricycle about 1880.

CAPTION: Bicycling in Boston during the 1880s