The man of mass m1 and the woman of mass m2 are standing on...

The man of mass $m_{1}$ and the woman of mass $m_{2}$ are standing on opposite ends of the platform of mass $m_{0}$ which moves with negligible friction and is initially at rest with $s=0 .$ The man and woman begin to approach each other. Derive an expression for the displacement $s$ of the platform when the two meet in terms of the displacement $x_{1}$ of the man relative to the platform.

Key Concepts

Center of Mass

The center of mass is the weighted average position of all the masses in a system. In scenarios with internal movements, such as the man and woman moving on the platform, the overall center of mass remains fixed if there are no external forces. This invariant property provides a key relationship between the individual displacements and the platform’s displacement.

Reference Frames

Understanding the problem requires considering different frames of reference. In an inertial frame (the ground frame), the platform and the individuals are seen moving, while in the platform’s frame, the relative motion of the individuals is observed. Analyzing the problem from these perspectives is essential for correctly applying the conservation principles and relating the displacements.

Relative Motion

Relative motion involves analyzing the movement of an object with respect to another moving object or frame. In the problem, the displacement of the man relative to the platform is used to derive the platform’s displacement by applying the condition of a fixed center of mass, linking the internal movements to the overall motion of the system.

Conservation of Momentum

This concept states that in an isolated system with no external forces, the total momentum remains constant. In problems like this, where individuals move on a frictionless platform, only internal forces act, ensuring the system’s overall momentum (and hence the center of mass) does not change over time.

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