Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Particles of mass \(m_{1}=5, m_{2}=7\), and \(m_{3}=9\) are located at \(x_{1}=2, x_{2}=-2\), and \(x_{3}=1\) along a line. Where is the center of mass?

Step-by-Step Solution

Answer

The center of mass is at \(x = \frac{5}{21}\).

Step 1: Understand the Center of Mass Formula

The center of mass (CM) of a system of particles is given by the formula: \( x_{CM} = \frac{m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3}}{m_{1} + m_{2} + m_{3}} \). This formula finds the weighted average position of the particles.

Step 2: Substitute the Values into the Formula

Use the given values: \(m_{1} = 5, x_{1} = 2\), \(m_{2} = 7, x_{2} = -2\), \(m_{3} = 9, x_{3} = 1\) and substitute them into the center of mass formula: \( x_{CM} = \frac{5 \times 2 + 7 \times (-2) + 9 \times 1}{5 + 7 + 9}\).

Step 3: Calculate the Numerator

Calculate the weighted positions of each particle and sum them up: \(5 \times 2 = 10\), \(7 \times (-2) = -14\), \(9 \times 1 = 9\). Add these values: \(10 - 14 + 9 = 5\).

Step 4: Calculate the Denominator

Calculate the total mass: \(5 + 7 + 9 = 21\).

Step 5: Solve for Center of Mass

Divide the sum from the numerator by the total mass from the denominator: \(x_{CM} = \frac{5}{21}\). Simplify the fraction, if possible.

Key Concepts

Mass Distribution

Understanding mass distribution is essential when dealing with the center of mass. In our specific example, we have three distinct masses placed along a line. The masses, in this case, are distributed such that each particle contributes proportionally to the total mass distribution. This idea of distribution allows us to calculate where the center of mass lies by determining how each mass affects the overall system.

• Mass distribution signifies how different masses are spread along a certain space or axis.
• It directly influences the location of the center of mass, which acts like a balanced point of the system.

Knowing how mass is distributed between multiple particles is crucial for pinpointing the center of mass. The mass distribution, in this instance, is 5, 7, and 9 along the line at specific positions. Mass distribution reveals the weight each particle brings into the equation, effectively shifting the center of mass depending on their positions and values.

Weighted Average

The concept of a weighted average is key to understanding how the center of mass is calculated. In mathematics, a weighted average allows for calculation which factors in the weight, or importance, of each element.

• Here, the weights are the masses of the particles: 5, 7, and 9.
• These weights are applied to their respective positions: 2, -2, and 1.

A weighted average helps determine where the average position, or balance point, of these masses is located. Mathematically, it ensures that positions tied to larger masses influence the center of mass more heavily than those with lighter masses. This is why we use the formula: \[ x_{CM} = \frac{m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3}}{m_{1} + m_{2} + m_{3}} \] This formula calculates the weighted average of the particle positions, thereby locating the center of mass.

Particles

Particles in physics often refer to discrete amounts of mass situated at particular positions. In the study of mechanics, we simplify objects to point particles to understand complex systems more easily. Our exercise involves three particles, each with unique masses and coordinates:

• Particle 1 with mass 5 at position 2
• Particle 2 with mass 7 at position -2
• Particle 3 with mass 9 at position 1

These particles are crucial as they define the positions we use in calculating the system's center of mass. Every particle contributes to the calculation based on its mass and relative position. Thus, understanding individual particles allows us to appreciate how each shifts the system's overall balance or center of mass.

Coordinate System

The coordinate system is the framework within which the particles' positions are described. In this particular scenario, we consider a one-dimensional coordinate system where particles are positioned along a line at points 2, -2, and 1. This simplified system allows for easy computation of their collective center of mass.

• This setup enables clear identification of each particle's location.
• Positions are represented numerically, aiding in direct substitution into formulas.

Such a system is crucial for accurately calculating positions relative to each other, especially when determining the center of mass. By defining a coordinate system, we can consistently measure and express the interplay between mass and position, leading us to accurate and insightful results about the balance point of the system.