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 the wheel, the point A covers `AC = pi` horizontal distance, and BC = 2R vertical distance.

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