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Chapter 4: Problem 23

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.

Short Answer

Expert verified
The displacement of the platform is \( s = -\frac{(m_1 - m_2)\cdot x_1}{m_0 + m_1 + m_2} \).

Step by step solution

01

Understand the System

The platform, man, and woman form a closed system where no external horizontal forces act. Therefore, the center of mass of the system does not move horizontally.
02

Write the Center of Mass Equation

The initial position of the center of mass of the system is at rest at position \( s = 0 \). The positions of the man and the woman can be described relative to this initial state. Using the center of mass formula: \[ x_{CM} = \frac{m_0 \cdot s + m_1 \cdot (s + x_1) + m_2 \cdot (s - x_2)}{m_0 + m_1 + m_2} = 0 \]. Here, \(x_1\) and \(-x_2\) are their displacements relative to the platform such that they meet at some point \(a\) (where \(x_1 + a = x_2\)).
03

Solve for Displacement s

Since the center of mass remains stationary, \[ m_0 \cdot s + m_1 \cdot (s + x_1) + m_2 \cdot (s - x_2) = 0 \]. Substitute \(x_2 = x_1\) because they meet, simplifying the equation to: \[ m_0 \cdot s + m_1 \cdot (s + x_1) + m_2 \cdot (s - x_1) = 0 \].
04

Simplify the Equation

Combine like terms to get: \[ (m_0 + m_1 + m_2)\cdot s + (m_1 - m_2)\cdot x_1 = 0 \].
05

Isolate s

Solve for \(s\) to find the platform's displacement: \[ s = -\frac{(m_1 - m_2)\cdot x_1}{m_0 + m_1 + m_2} \]. This is the displacement of the platform when the man and woman meet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Displacement in a System
Imagine you are standing on a skateboard, and a friend is on the other end. When you both move towards each other, your positions change. This change is known as **displacement**. Displacement is the change in position of an object. It's not just how far you've moved; it's also about the direction. In this exercise, displacement is key to understanding how the man and woman are moving on the platform. Since the man and woman start at different ends, their displacements relative to the platform are crucial in calculating the platform's displacement.
  • Displacement is different from distance. While distance is how much ground covered, displacement considers direction.
  • In a system like ours, knowing one person's displacement helps us find the other's movement on the blank canvas of our platform.
Understanding a Closed System
When we refer to a **closed system** in physics, it's like saying the kite in the sky isn't affected by the kids on the ground, only by the wind. For our problem, the platform, man, and woman form a closed system. But what does this mean? It indicates that no external forces, like friction, act horizontally on the system. Thus, the total momentum, or how things move around, stays constant.
  • Closed systems help in predicting movement since no outside force interferes.
  • It ensures that only internal forces influence the behavior of the objects, making calculations straightforward once you understand each part.
Understanding a closed system is crucial because it tells us why the center of mass doesn't budge as long as everything stays within these confines.
A World Without Friction: The Platform
Picture sliding down an ice rink. There's nothing stopping you except the end of the rink. A **frictionless platform** is much like that ice rink. In our problem, the platform's movement is uninhibited by friction, simplifying the calculations. Friction typically provides a resistant force. But here, we can ignore that, which means:
  • The only forces at play are the actions of the man and woman. When they move, it affects the whole system without any friction slowing them down.
  • This type of scenario highlights pure interactions between the masses involved, almost like they're floating in space, offering a perfect scenario to understand fundamental motion concepts.
A frictionless platform is a theoretical concept, but it's handy in understanding basic physics without additional variables interrupting the learning process.

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