A meter stick is placed vertically at the origin on a frictionless surface. A gentle push in $ + X$ direction is given to the top most point of the rod when it has fallen completely $X$ coordinate of center of rod is at:
(a) Origin
(b) $ - 0.5m$
(c) $ - 1m$
(d) $ + 0.5m$

Hint: A meter stick is placed in a vertical direction at the origin on a frictionless surface. A gentle push in positive $X$-direction is given to the topmost point of the rod when it has fallen completely, $X$- coordinate of the center of mass of rod

Complete step by step solution:

It looks like there is no external force applied on the rod along the x-axis.
After reading the statement “A gentle push in positive x-direction is given to the topmost point of the rod.”
But we know, if the external force acting on the particle is zero, the position of the center of mass remains the same. It doesn't change.
So, new $X$- coordinate of the center of mass of rod = initial x - coordinate of the center of mass of the rod.
We have the initial center of the mass \[ = {\text{ }}\left( {0,{\text{ }}0.5} \right)\]
That is initial $X$- coordinate of com \[ = {\text{ }}0\]
So, new $X$ - coordinate of com \[ = {\text{ }}0\]
But \[y\]-coordinate has changed because of a gravitational force acting along the \[y\]-axis. And it will be changed from \[0.5\] to zero as shown in the figure.
\[{\text{X }} = {\text{ }}0\] and \[{\text{Y }} = {\text{ }}0\] that is origin
Therefore the new center of mass of the rod is at the origin.

Hence, the correct answer is option (a).

Note:To know the frictionless surface we have to take the example of a skater. If there are no forces acting upon the skater and he is on a frictionless surface, then that means he has no net acceleration. If an object moves with the velocity and if there is no acceleration, then the velocity remains constant.