Question

a man of mass M stands at one end of a plank of leangth L which lies at rest on frictionless surface. The man walks to other end of plank.If mass of plank is M/3 then find the distance that man moves relative to ground?

Answer 1

   Let the man stand at the point A where one end of the plank AOB is.  Let O be the center of the plank.   So man is at  L/2 distance from the center O.

   center of mass will be at point P where OP = L/2 * M / (M + M/3)
             OP  =  3 L / 8
   P will be  between  A and O.  Distance AP = L/2 - 3 L/8 = L/8.

   The system consisting of man and plank experience no external force in the horizontal direction.  Hence, CM will remain at the same place.
            
  When the person walks to the other end B, CM remains still at the same point P.  End B is at B' and the end A is at A'.  As the person moves forward towards B, the plank moves backwards to a new position B'PO'A'.

   The distance between the CM and the man remains same..A' P = L/8.  P' will be on the side that the man is.  The other side as compared to the initial position.
 
The actual distance moved by the person relative to the ground,
     or relative to center of mass P =       2 * L/8 = L/4

 distance traveled by the man relative to the plank:  L