A uniform rod is placed vertically on a smooth surface and...

A uniform rod is placed vertically on a smooth surface and then released. Then:
(a) the centre of mass of rod follows straight line path
(b) the centre of mass of rod follows circular path
(c) the instantaneous axis is passing through the contact point
(d) all the above

Key Concepts

Instantaneous Center of Rotation

For bodies in planar motion, the instantaneous center of rotation is a point that, at a given moment, has zero velocity. Its position is determined by both the translation of the center of mass and the rotation about the center of mass. Without constraints forcing a specific point (such as a frictional contact sticking to the surface), the instantaneous center may not coincide with the point of contact with the supporting surface.

Frictionless Constraint Forces

When a rigid body is in contact with a smooth (frictionless) surface, the only reaction force available is the normal force, which acts perpendicular to the surface. This means there is no frictional force to prevent horizontal motion, so points of contact with the surface do not remain stationary and are free to move horizontally. This characteristic alters both the translational and rotational aspects of the body's motion compared to cases with friction.

Center of Mass Motion

In rigid body dynamics, the center of mass (COM) moves as if all the mass were concentrated at that point and all external forces were applied there. The COM’s trajectory is determined solely by the net external force (gravity and any applied forces), irrespective of any rotational motion. In this context, since only vertical forces (gravity and the normal reaction) act, the COM follows a straight (vertical) line.

Decomposition of Rigid Body Motion

Any motion of a rigid body in a plane can be split into a translational motion of the center of mass and a rotation about the center of mass. This separation simplifies analysis as the translational part follows Newton’s laws (with the COM accelerating according to net force), while the rotation is governed by the sum of torques about the COM.

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