Technical question - please help

I struggled to find the words to put this into a Google search term that would throw up what I’m looking for...

On a fixed gear bike, how much further along the dropout will the rear axle move if I up the number of teeth on the sprocket by one tooth (chain length, tyre and everything else remaining the same)? I’m wondering how much closer to the seat tube my tyre will get if I get a bigger sprocket but keep the same number of links in my chain.

Thank you in advance to any nerd geniuses in the house : )

It will be the difference in the radius of the sprocket of the old and new. You could calculate this. The difference I guess will be miniscule. How many teeth on the current sprocket?

Edit: shouldn’t you still be able to put the axle in the same position regardless of the sprocket?!! Unless you’ve got no slack.

Ask tester.. I remember him answering it already somewhere

Good thinking! I feel very unimaginative now. It’s a 17t - does that make any difference? Thank you

Tooth pitch is 12.7mm (0.5"). You can work out a cog's main dimensions from there.

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On that basis I’d guess 2.02 mm. But I’m totally not qualified to answer this.

Thanks all. That makes things a little clearer...from all this, can a concrete distance be deduced that the axle will move along the dropout with every increase in tooth on the sprocket?

No, the difference would be less with every additional tooth

Ok, right, mind-bending stuff

http://www.machinehead-software.co.uk/bike/chain_length/sprocket_radius_table.html

Cog Teeth Radius (Inches) Difference (Inches)
10 0.8090
11 0.8874 0.0784
12 0.9659 0.0786
13 1.0447 0.0787
14 1.1235 0.0788
15 1.2025 0.0789
16 1.2815 0.0790
17 1.3606 0.0791
18 1.4397 0.0791
19 1.5189 0.0792
20 1.5981 0.0792
21 1.6774 0.0793
22 1.7567 0.0793
23 1.8360 0.0793
24 1.9154 0.0793
25 1.9947 0.0794
26 2.0741 0.0794
27 2.1535 0.0794
28 2.2329 0.0794
29 2.3123 0.0794
30 2.3918 0.0794
31 2.4712 0.0794
32 2.5506 0.0794
33 2.6301 0.0795
34 2.7096 0.0795
35 2.7890 0.0795
36 2.8685 0.0795
37 2.9480 0.0795
38 3.0275 0.0795
39 3.1070 0.0795
40 3.1865 0.0795
41 3.2660 0.0795
42 3.3455 0.0795
43 3.4250 0.0795
44 3.5045 0.0795
45 3.5840 0.0795
46 3.6635 0.0795
47 3.7430 0.0795
48 3.8225 0.0795
49 3.9021 0.0795
50 3.9816 0.0795
51 4.0611 0.0795
52 4.1407 0.0795
53 4.2202 0.0795
54 4.2997 0.0795
55 4.3793 0.0795

Lol. So I was right-ish! 0.0791 inches = 2.01 mm.

Where I was wrong though, is that the difference gets larger with every additional tooth. I don't understand that.

I think that is because as the number of teeth increase the polygon we're describing more closely approximates to a circle. A circle has a constant ratio of circumference to diameter - pi. Whereas, for the regular decagon of a 10 tooth sprocket, the ratio of the diameter and perimeter is 3.09, for a hexagon it is 3, and the ratio of the diagonal of a square to the perimeter is 2 time square root of 2 or 2.828. The difference between each difference will get smaller and smaller.

Had to read that about ten times, but I see what you're saying.

I just thought, the size of each additional tooth is constant - whereas the circle is constantly getting bigger. In my head, that means the bigger the 'circle' gets, the less one additional tooth would stretch it. But obviously not.

I wanted to post a follow-up. I think we got off in the wrong track here with the sprocket radius idea. While what we have said about radius is correct it has missed the point that there is also an additional half link of chain used up with each tooth increase.

Anyway, the axle moves about 1/8 inch with each tooth change, or around 3.2mm rather than 2.