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

In accordance with fig 3.24 during the half revolutin of the wheel, the point A covers, `piR(AC)` horizontal distance while `2R(=CB)` vertical distannce ,
So, here `P = piR, Q= 2R" "and theta= 90^(@)`
`So, " " D=sqrt([(piR)^(2)+(2R^(2))])=Rsqrt(pi ^(2)+4)`
`and" "phi= tan^(-1)[(2R)/(piR)] =tan^(-1)[(2)/(pi)]`
i.e.,displacement has magniude `Rsqrt(pi^(2) + 4)` and makes an angle ` tan^(-1) (2//pi)` with x-axis.