A man of mass m is standing at one end of of a plank of mass M. The length of the plank is L and it rests on a frictionless horizontal ground. The man walks to the other end of the plank. Find displacement of the plank and man relative to the ground.

Explanation

Denoting x-coordinates of the man, mass center of plank and mass center of the man-plank system by `x_(m) x_(p)` and `x_(c )`, we can write the following equation.
`(Sigmam_(i))vec(r)_(c ) = Sigma m_(i)vec(r)_(i) rarr" " (m+M)vec(x)_(c ) = m vec(x)_(m)+ Mvec(x)_(p)` Net force on the system relative to the ground is zero. Therefore mass center of the system which is at rest before the man starts walking, remains at rest `(Delta vec(x)_(c ) = 0)` after while the man walks on the plank.
`(Delta bar(x)_(c ) = 0) rarr " " m Delta bar(x)_(m) + M Delta vec(x)_(p) = 0`
The man walks displacement `(Delta bar(x)_(m))/(p = -L hat(i))` relative to the plank. Denoting displacements of the man and the plank relative to the ground by `Delta vec(x)_(m)` and `Delta vec(x)_(p)`, we can write
`Delta vec(x)_(m//p) = Delta vec(x)_(m) - Delta vec(x)_(p) rarr " " Delta bar(x)_(m) - Delta bar(x)_(p) = - L hat(i)`
From the above equations (1) and (2), we have
`Delta vec(x)_(m) = -(MLhat(i))/(m+M)`
The plank moves a distance `(mL)/(m+M)` towards right relative to the ground.
`Delta vec(x)_(m) = - (ML hat(i))/(m + M)`
The plank moves a distance `(mL)/(m + M)` towards right relative to the ground.