Why does a bicycle (without any support of stand) falls down being at rest, but not under motion? I have always seen a bicycle not standing without any support, it either falls down to the right or to the left, may even to some other direction. But,the same two wheeler when under motion,moves balanced, without falling to either side,even it not falls, when the two wheeler is bent to left or right by the driver, like the one which we could see in the bike races. So, what makes the bicycle at rest to fall? And what makes it not to fall when under motion? 
MY VIEW ON THE CONCEPT
If the bicycle is at rest, it will be acted upon by downward force ($mg$,where $m$ is mass and $g$ is acceleration due to gravity). If it is affected just by this force, it should have stood without falling, because there is no force which pulls it  to either side to make it fall. I think, it might fall, either due to the direction of wind, or due to the nonuniform distribution of mass, or due to other external forces. If we consider the same bicycle to be under motion, I don't know what makes it not to fall,even affected by nonuniform distribution of force,or direction of wind. I think there might be something related with vectors, angular momentum,and forces like centripetal and centrifugal. (What ever I said, is just my opinion about the concept, I don't claim it to be true. Correct me,if I am wrong anywhere.) 
 A: Forget two wheels. Look at one wheel. Just take a wheel of any kind and roll it down a hill. It stays up until it stops rolling.
The reason is gyroscopic precession.
If it starts to fall to the right, that is no different than if you strike it with your hand at the top on the left side.
The top of the wheel is moving forward, so when you strike it on the left side, you are deflecting the material on the top into a path that is angled to the right of its original direction.
This has the effect of turning the wheel to the right, which (if the tilt was caused by falling to the right) brings its point of support back under its center of gravity.
That's how it balances itself.
When it stops turning, this effect stops working.
A bicycle uses this effect, plus the rider also steers in the direction of fall.
But a bicycle will stay up all by itself as long as it has any speed.
A: Physics has conservation laws which have been imposed because of observations and are incorporated in all mathematical theories describing physical systems. 
One studies physics at an elementary level in order that everyone does not have  to rediscover the wheel.
Angular momentum conservation is one of these strict conservations laws. Once a wheel gets an angular momentum then it will keep it until external forces ( frictions slowing the wheel or something hitting it or a displacement of the weight of the driver) change its value. Angular momentum is a vector is space.
A: If the steering would be locked, it would fall over. The reason it doesn't is because the rider is constantly adjusting the steering to provide centrifugal force that will act against the centripetal force of gravity.
For example, when the vehicle is turning, it is tilted to one side. Gravity is pulling one way, causing the vehicle to fall toward the ground. Centifugal force caused by turning is causing the vehicle to return upright. If the forces are in balance, the vehicle will continue to be tilted and be making a turn. If the rider changes one of the forces, the vehicle will either fall to the ground or return upright.
If the vehicle is standing still, the rider will not be able to generate any centrifugal force by steering, and the vehicle will fall.
If the rider is unable to correct any imbalance of forces (such as wind) even while moving, he will fall. Observe any person learning to ride a bicycle, as empirical evidence.
Another thing to note is that the wheels become gyroscopes when you are moving. The faster you are moving, the greater the angular momentum. This slows the rate of falling, due to the vehicle trying to precess, held in place by friction on the road. It also tends to precess the steering wheel. This is usually the front wheel, which means that as the vehicle tilts, the steering tends to turn in a way that automatically makes a balanced turn. This is why with sufficient speed, you can ride a bicycle without steering. If you're not careful though, there may be enough force imbalance to completely fall to the ground. Without a rider to correct for these imbalances, the vehicle will again fall.
