Question

A wheel of radius r rolls forward half a revolution.The displacement of a point which is at height r from the point which is initially in contact to the ground

Answer 1

the point which is r height from contact point is the centre of the wheel

the centre is rolling it is performing only translation motion

lets imagine that the ring is just coloured by black paint and is making half revolution on white paper so it leaves black mark on paper and the length of that black mark is the distance moved by the centre

as pt A comes from top to bottom

the length of black mark will be the length of arc which is half the perimeter of circle

pi×r

so distance moved by centre is

pi×r