Consider a particle on the rim whose radius from the centre makes an acute angle of with the vertical. Such a point has two velocities: v horizontally due to translation and R at a tangent due to rotation. As the case is of pure rolling, v = R. Hence the net velocity of the particle is the vector sum of v velocity horizontal and a velocity v along the tangent. You can see by drawing the figure that the angle between these two velocities is . Hence the magnitude of velocity of the particle is u = (v2 + v2 + 2 v v cos) = 2v sin(/2) The magnitude of velocity is the speed, whose integral wrt time gives distance covered. s = 0T u dt = 0T 2v sin(/2) dt where T is the time for one complete rotation. Now = t d = dt s = 02 pi 2v sin(/2) / d = 2R02 pisin(/2)d = 8R
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with the vertical.
R at a tangent due to rotation. As the case is of pure rolling, v = 
.
(v2 + v2 + 2 v v cos
T u dt = 0
T 2v sin(
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