Is there a difference in the types of handles of a bike? On some bikes, there is the (upright) handlebar and on some the (drop) handlebar. Is there a purpose?

The shape of the handlebar determines your riding position. The upright position is generally more comfortable but, by sitting you upright, it increases the pressure drag you experience. Drop handlebars lower your body and make you more aerodynamic, but that position isn't as comfortable.

How is it that the torques cancel when you turn a bicycle?

During a turn, you lean the bicycle into the turn. For example, when you turn left, you lean the top of the bicycle toward the left. The result is that you (and the bicycle) experience two torques. First, the support force from the ground tries to rotate you one direction-it tries to make your head go left and your feet go right. Second, friction from the ground, which is making you and the bicycle accelerate toward the left as part of the turn, tries to rotate you in the opposite direction-it tries to make your head go right and your feet go left. These two torques will cancel one another if you are leaning just the right amount. As a result, the bicycle doesn't undergo angular acceleration and you don't tip over.

Why is it easier for you to make sharp turns more quickly when your center of gravity is over the handle bar?

The force that causes you to turn is friction between the front wheel and the ground. When you turn left, friction pushes the front wheel left and you turn left. By putting all of your weight over the front wheel, you accomplish two things. First, you increase the maximum static frictional force between the ground and the front wheel. You push them together harder so that they are less likely to slide (skid). Second, you make it easier for that sideways friction force to accelerate you; the force acts closer to your body and more directly on you. There are fewer torques on the bicycle that might cause it to skid about on either the front or rear wheel.

Please explain the advantage of air tires with less mass.

The rim of the wheel travels at a different speed from the rest of the bicycle. The top of the wheel heads forward faster than the bicycle, while the top of the wheel heads forward more slowly than the bicycle. But because kinetic energy is proportional to the square of speed, the increase in the top of the wheel's energy caused by its increased speed more than makes up for the decrease in the bottom of the wheel's energy caused by its reduced speed. The overall result is that the wheel rim has twice as much kinetic energy as it would have if it were simply sliding forward without turning. This fact is important because it means that you want as little mass in the wheel rim as possible. Every kilogram there counts double when you are trying to start up from rest. By putting air inside the tire, rather than rubber, you reduce the mass at the wheel rim and make the bicycle easier to start.