What is the displacement of the point of a wheel initially in contact with the ground when the wheel rolls forward half a revolution? Take the radius of the wheel as `R` and the `x`-axis as the forward direction?

Explanation

Refer to Fig. 2 ( HT). 4, when wheel complaetes half revolution, then point (A) on wheel reaches at (C ) . The horizontal distance coverd `= AB= pi R, while vetical distance covered is ` =BC = 2 R`.
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Desplacement of point (A) on the wheel is ` = vec (AC)`
Here, ` Ac = sqrt ( AB^2 = BC^2)= sqrt( pi R)^2 + (2 R)^2)`
` = R sqrt(pi^2 + 4)`
If ` beta` is the angle which ` vec (AC)` makes with ` vec(AB)`, then
` tan beta = (BC)/(AB) = [ ( 2 R)/ (pi R)] = [(2/ (ip)]` :. beta = tan ^(-1)(2/(pi))`.