A uniform rod of length l lies on a smooth horizontal table....

A uniform rod of length $l$ lies on a smooth horizontal table. A particle moving on the table strikes the rod perpendicularly at an cnd and stops. Find the distanec, travelled by the ecntre of the rod by the time it turns through an angle $\theta$

Key Concepts

Kinematics of Rotation

The kinematics of rotation provide the relationship between angular displacement, angular speed, and time. This concept is crucial for linking the time taken by the rod to rotate through a given angle to its translational motion, as the center of mass moves at a constant velocity while the rod rotates at a constant angular speed following the impulsive collision.

Rigid Body Motion

Rigid body motion involves both translational and rotational components that occur simultaneously. After the collision, the rod exhibits translational motion (movement of the center of mass) as well as rotation about the center of mass. Analyzing these separate components allows one to relate the angular displacement of the rod to the linear distance traveled by its center.

Conservation of Linear Momentum

This principle states that in the absence of external forces, the total linear momentum of a system remains constant. In the context of the problem, the linear momentum transferred to the rod during the impulsive collision determines the constant translational motion of the rod’s center of mass after the impact.

Angular Momentum and Moment of Inertia

Angular momentum quantifies the rotational equivalent of linear momentum and is conserved (or determined by the impulsive effect) in isolated systems with no external torques. The moment of inertia is the measure of a body's resistance to changes in its rotational motion. When the rod is struck at one end, an angular impulse is imparted, and its moment of inertia about the center of mass governs the resulting angular velocity.

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