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 ` pi R (=AC )` horizontal distance while 2R =(BC ) vertical distance while 2R (=BC ) vertical distance
so here `P = pi R , Q = 2R `
` therefore ` displacement `b= sqrt( p^2 +q^2)`
` = sqrt( (pi R^2 ) +(2R)^2 ) = R sqrt( pi^2 +4)`
` and theta = Tan^(-1) (Q/P)= tan ^(-1) ((2R)/(pi R))= tan^(-1)"" ((2)/(pi)) ` with x axis