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 R` horizontal distance, and BC = 2R vertical distance.

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