Calculation error discovered

I was a bit surprised to see that the wheel had to be moved back almost 5 cm when I changed from 45 teeth to 38, and I found myself with a slacking chain. My calculations had indicated 2.6 cm. When calculations and reality doesn't match, rely on reality! I had to cut out real size dummy sprockets from cardboard to see where it went wrong, and I found it.

 

I had came to the conclusion that the wheel would be moved back 2.6 cm (*), but it was the outer line of the sprocket that would move that much. The center would move more like 4.5 cm because of the smaller diameter of the new sprocket. I had missed that part before. I used this insight to make an updated calculation and I got 4.6 cm which is pretty close to the measure I took on the cardboards.

 

(*) Each tooth correspond to 1.5 cm of chain. The perimeter would increase with (45-38) * 1.5 cm = 10.5 cm. The chain only covers about 50% of the sprocket so that means 10.5 cm * 50% = 5.2 cm of chain that is gonna be left over. These 5.2 cm is divided by 2 and you have 2.6cm.

Done