According to the POH, ground control is achieved by a spring loaded steering bungee.
But how does it work and why the nose wheel does not move when the aircraft is stationary?
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Because the nose wheel is only attached to the rudder pedals indirectly, through some springs (or maybe a bungee cord), that transfer the displacement of the pedals into a force that is applied to the position of the wheel. In a typical cross-wind landing scenario, where the aircraft would touch down with the nose wheel off center, that eliminates the possibility that the aircraft would turn hard to one side when the nose is lowered because the nose wheel is cocked to one side (due to the rudder pedals being displaced due to the crosswind). Because the force is only applied through the tension on the springs, the aircraft inertia (tendency to continue moving in the same direction), overcomes the springs and aligns the nose wheel with the ground track when the rubber touches the ground even though the rudder pedals are displaced.
The nose wheel does not turn when the aircraft is stationary, because there is no inertial force at work, and the friction of the wheel on the ground keeps the wheel from turning (the spring force is not strong enough to overcome the friction)
I'm not a pilot, nor have I ever been in a C172, but I'm going to go out on a limb here and answer from a logical physics point of view and say the answer is that it's because the nose-wheel steering is operated by a bungee cord.
When you step on the pedal, it pulls the bungee. Bungee cords, being by definition stretchy, will stretch. When the plane is stationary, the cord stretches, but there's more resistance between the wheel and the ground than the cord can overcome, so the wheel doesn't move.
When the plane is rolling, the bungee still stretches, but it wants to shrink back up. Since the wheel is constantly moving, it has less resistance against the pavement so as the bungee tries to shrink, it can actually pull the nose gear in the direction you want to go, thus you end up steering.
When you turn the steering wheel in your car, the wheels move whether you're rolling or not because there are steel links between the steering rack and the toe link on the wheel hub. Your movement of the steering wheel is directly fed (with very little rubber in between) to the wheel of the car. If you were to ever get into a car without power steering, you'd notice that it's much harder to turn the wheel when the car is stationary than when it's rolling because the stationary tire against the ground has much more resistance than when it's rolling.
The simple answer is that a slowly rolling wheel has less resistance to sideways motion.
The spring is there to prevent imperfections in the rolling surface from jarring or stressing the control linkages to the rudder (especially with unpaved airstrips). It provides some side force so the front wheel, as it rolls on its path, will change direction.
with infinite ruts, the odds off the wheel being pushed left or right become equal, allowing the spring to turn the wheel in the desired direction
Rubber, being an elastic polymer, will "sink in" to the little cracks and imperfections in the surface as several hundred pounds of aircraft weight sits upon it, increasing resistance to movement when the wheel is not turning.
Of course, at higher rotation speeds, gyroscopic effects will work against the pull of the spring, but the wheel will still turn.
Every once in a while, especially with insufficient yoke back pressure, the wheel can start resonating, causing a very unpleasant shudder while taxiing or on a take-off run. Increasing backpressure or changing speeds can help, and sometimes it's best to abort and make sure it's nothing worse.