TSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP III 1997 question 5 solution
Wlog, the wheel has radius 1. Let the angle turned through be t. If the wheel wasn't rolling, the position of a point on the rim is (-sin t, 1-cos t). As the wheel is rolling, we have to add t to the x coord to get (t-sin t, 1 - cos t).
Distance travelled by point on rim =
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(note that we must take the positive root).
This = ![\displaystyle 2\int_0^{2\pi} \sin \frac{t}{2} dt = -4\left[ \cos \frac{t}{2}\right]_0^{2\pi} = 8.](http://thestudentroom.co.uk/../latexrender/pictures/a2ce3626a92c2349d177fd7f27060c03.png "\displaystyle 2\int_0^{2\pi} \sin \frac{t}{2} dt = -4\left[ \cos \frac{t}{2}\right]_0^{2\pi} = 8.")
Distance traveled by center = 2 pi. So ratio is 
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Solution by DFranklin
