A uniform rod of length 2 a is held with one end resulting...

A uniform rod of length $2 a$ is held with one end resulting on a smooth horizontal table making an angle $\alpha$ with the vertical. When the rod is released: (a) its centre of mass moves vertically downwards on a straight line (b) its centre of mass remains in rest (c) the-rod rotates about a vertical axis (d) both (a) and (c) are correct

A uniform rod of length $2 a$ is held with one end resulting on a smooth horizontal table making an angle $\alpha$ with the vertical. When the rod is released:
(a) its centre of mass moves vertically downwards on a straight line
(b) its centre of mass remains in rest
(c) the-rod rotates about a vertical axis
(d) both (a) and (c) are correct

Key Concepts

Center of Mass Motion

In rigid body dynamics, the center of mass is the point where the body's mass can be considered to be concentrated and where external forces act according to Newton's second law. Regardless of any rotational motion the body undergoes, the center of mass will accelerate in the direction of the net external force. For a system subject only to vertical forces (such as gravity and a vertical normal reaction from a frictionless surface), the center of mass will move vertically along a straight line.

Rigid Body Rotation

Rigid body rotation describes how all points in a solid move in a correlated manner, maintaining fixed distances between them as the body rotates about an axis. The motion of rotation is decoupled from the translation of the center of mass, meaning that even if the body rotates, the translational motion of the center of mass depends solely on the sum of external forces acting on the body.

Constraint and Reaction Forces

Constraint or reaction forces, such as those provided by a smooth (frictionless) surface, restrict the motion of a body without doing work in certain directions. When a body is in contact with a frictionless surface, the reaction forces are typically vertical, ensuring that no net horizontal force acts on the body. This contributes to a scenario where, under gravity, the center of mass follows a vertical path, independent of any rotational effects initiated by the body's geometry.

Transcript

Please give Ace some feedback