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 the x - axis as the forward direction ?

Explanation

From figure, during half revolution of thewheel, the point A covers AC = `piR` horizontal distance, and BC = 2R vertical distance.

`x=piR` and y=2R, Displacement
`s=sqrt(x^(2) + y^(2)) = sqrt((piR)^(2) + (2R)^(2)) =Rsqrt(pi^(2) +4)`
and `theta =tan^(-1)(y/x) =tan^(-1)((2r)/(piR)) =tan^(-1)(2/pi)`
with x-axis.
Displacement has magnitude `Rsqrt(pi^(2)4)` and makes an angle `tan^(-1)(2/ipi)` with x-axis.