I didn’t respond to this before as for some reason I couldn’t see this post specifically, but now it is showing up.
The larger axle allows a higher torque or horsepower engine to be used on the kart. I wouldn’t run a 30mm axle with a 175cc Shifter engine, I wouldn’t even run a 40mm axle with a KA, although it is possible. It does not mean that changing the axle adds torque or horsepower directly.
This sentence is correct, the edge of the axle is not covering the same distance as the edge of the gear, even though they are moving at the same rotational velocity. Due to the difference in radius, the circumference of the sprocket is larger, and will then have a longer traveling distance than the axle. However, the sprocket is bolted directly to the hub, and therefore any rotation of the sprocket will be equally matched in rotation by the axle, so they will always have the same rotational velocity, or rotations/second.
This part confuses me. If you put a larger axle in than the previous axle, then the dot would be moving a longer distance for an equivalent rotation compared to the smaller axle.
You’re getting lost in the connection from the kart axle to the wheel. In the same way that the sprocket is connected directly to the axle, and the axle will only rotate the same amount as the sprocket does, the wheel is directly bolted to the axle, and will only rotate the same amount as the axle rotates.
Let’s follow this through one rotation from the sprocket to the wheel. The kart is running, and the sprocket moves one full rotation. You agree that the axle will rotate at the same rate as the sprocket, so the axle will also move one full rotation. The wheel is directly connected to the axle, in that the wheel can not rotate without the axle rotating the same amount, just like how the axle cannot rotate without the sprocket rotating. So, when the axle moves one full rotation, the wheel will move one full rotation as well. This means that when the sprocket moves one full rotation, the wheel will always move one full rotation to keep up.
If you change the size of the axle, all that you’re doing is changing the size of the hub that connects the sprocket directly to the axle, and the hubs that connect the wheels directly to the axle. Since changing these hubs still connect the sprocket and wheels directly to the axle, it does not affect the rate at which either of these rotate relative to the axle. It is still a 1-1-1 rotation ratio, going sprocket rotation-axle rotation-wheel rotation.
James is correct, the naming of the parts is likely throwing you off. An axle in a kart is similar to a driveshaft in a car, in that you can change the radius of the part and still have the same number of rotations/second. He is also correct that if you see a difference in, say, top end speed at the end of the straight after changing an axle in a kart, that is going to be due to the kart handling better and exiting the corner faster than it was with the old axle. If you exit a corner faster, you’ll carry that added speed all the way up until either 1) you brake for the next turn or 2) you hit a rev-limiter, depending on your engine.