Sorry circumference.
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The only reason why the inner portion covers the same distance is because it is the part of the larger wheel. If you had a wheel that was only the smaller portion in total size, it would cover a smaller distance.
The distance traveled is essentially the amount of wheel that touches the road/surface that the wheel is traveling on (which is the outermost part of the wheel which has the largest circumference/radius)
The inner part of the wheel is not rotating along the ground, it is rotating a couple of inches off the ground
Only the outermost surface determines the distance travelled. The math is very simple. Diameter of the wheel x Pi = circumference and thus distance travelled for one revolution. All the portions of the wheel are making exactly one revolution too, but they are not covering the same distance. Think about you standing on the north pole for 24 hours. You’d cover no distance, your body would just spin. But if you were on the equator you’d cover 25000 miles in the same time.
If you had a wheel with a removable inner ring and you removed that ring and rolled it next to the wheel, one rotation of that removable ring would not cover the same amount of distance as the wheel it came out of because it is smaller.
Imagine two ants that want to race by getting on the wheel of a bike. One is on the tire and one is near the hub.
At the beginning, the tire ant is near the ground. As the wheel starts to turn, the hub ant moves forward while the tire ant just kind of moves upward. Hub ant thinks he’s gonna win for sure.
As the wheel keeps turning, tire ant picks up speed upward and forward. When the tire ant is at the top of the rotation, he’s going twice as fast as the hub ant! Tire ant thinks he’s got the race in the bag and hub ant is shocked.
As the wheel keeps turning, the tire ant slows down again and starts moving downward. At the bottom of the rotation, the tire ant comes to a complete stop. The hub ant takes the lead again.
Ultimately, both ants travel the same average speed, with the tire ant going twice as fast and then coming to a stop over and over again while the hub ant maintains a more constant speed.
The outer part of the wheel goes as you expect. The inner part of the wheel, in addition to rotating, is also “sliding” as it moves. So the distance that the inner part of the wheel travels through the rotation is its circumference (from rotating) plus additional distance because it is essentially pushed (without rotating) at each moment in time.
In physics, we’d say the difference is that the outer circumference is rotating without slipping and the inner circumference is rotating with slipping. Searching these terms should give you a good visual.
Grab a pencil and lay it on a desk, a table, any flat surface with open space. Grab it with 2 fingers by the eraser, and wiggle it side to side- if you’re trying to, a very small motion at one end of the pencil can get the other end of the pencil to move a much larger distance. Because of that, the tip of the pencil is moving much faster than the rest of the object.
This is a lever, fundamentally, and the same principle applies to wheels: the center of the wheel is not spinning very quickly, but the outside of the wheel is moving further than more inward parts of the wheel in the same period of time, and is thus by definition moving faster.
Where the wheel contacts the ground, it isn’t actually moving unless the wheel slips. That is the only restriction a wheel has in how it moves relative to the ground, but it doesn’t continue up into the center of the wheel.
If we have a wheel and rotate it one rotation, it travels the circumference of the wheel. If we take the tire off of that wheel and rotate it once again, it has a smaller radius, and therefore a smaller circumference, and will go less far in that one rotation. This is because we change the point that has that restriction of needing to not move. The other points not having that restriction means that they can move relative to the ground, but there no slipping, since they aren’t touching the ground.
If it helps grasping it, it’s the opposite reason to why if you have a stick on the floor and rotate it around one of its ends that end doesnt move and the other end moves 2x the length of the stick.
Other than all wheel/four wheel drive cars, one set of axles doesn’t turn at all. Yet the axles that don’t turn, like the hubs on those axles, go exactly as far as the rest of the car. You do not rotate at 300 RPM when you’re sitting in the driver’s seat, you need to get past your obsession that wheel parts are different than body parts and passengers.
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