Problem 256 - Calculus Volume 1

Question

For the following exercises, calculate the center of mass for the collection of masses given. \(m=3\) at \(x=0,1,2,6\).

Step-by-Step Solution

Answer

The center of mass is 2.25 units on the x-axis.

Step 1: Understanding the Problem

We are given a collection of masses where each has a mass of 3 units, located at different positions (0, 1, 2, 6) on the x-axis. We need to calculate the center of mass for these masses.

Step 2: Calculate Total Mass

The total mass of the system is the sum of all individual masses. Since each mass is 3 and there are four masses, the total mass is: \[M = m_1 + m_2 + m_3 + m_4 = 3 + 3 + 3 + 3 = 12.\]

Step 3: Apply Center of Mass Formula

The formula for the center of mass \((\bar{x})\) for a system of particles along a line is given by:\[\bar{x} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{m_1 + m_2 + m_3 + m_4}\]Substitute the given masses and positions into the formula.

Step 4: Calculate Numerator of the Formula

The numerator involves the sum of each mass times its position:\[3 \times 0 + 3 \times 1 + 3 \times 2 + 3 \times 6 = 0 + 3 + 6 + 18 = 27.\]

Step 5: Complete the Calculation

Now compute the center of mass by dividing the total from the previous step by the total mass:\[\bar{x} = \frac{27}{12} = 2.25.\]

Step 6: Conclusion of the Calculation

The center of mass of the system of masses given is 2.25 units along the x-axis.

Key Concepts

Mass Distribution

Mass distribution is about how mass is spread out over certain positions or locations within a system. In this exercise, we have four individual masses, each equal to 3 units, placed at different positions on the x-axis at 0, 1, 2, and 6. Understanding mass distribution is essential because it determines how the system behaves physically. For instance, if all masses were clustered around a specific point, the center of mass would be nearer to that cluster. Here, the positions vary widely, affecting the position of the center of mass.
To visualize it, imagine how the balance of a seesaw changes when you spread people out to different points on it versus when they sit together.

Physics Calculations

Physics calculations often involve straightforward arithmetic when dealing with problems like finding the center of mass. It requires systematic application of mathematical principles. In this problem, the total mass calculation is a simple addition of identical mass units: 3 units each multiplied by the number of masses present.
Then, applying the formula for the center of mass uses multiplication and division. Notice how these fundamental math operations are used repetitively, demonstrating that tough physics problems often break down into simpler math steps. Such calculations don’t just give us numbers but provide insight into physical behaviors and characteristics of systems.

Coordinate System

A coordinate system, particularly the x-axis in this exercise, serves as a reference for the location of masses. It allows us to assign numerical values to the positions where each mass is located. In practical physics and engineering, understanding the reference frame or coordinate system is crucial because it forms the basis for calculating various properties and behaviors of a system
Here, the x-axis helps us determine positions such as 0, 1, 2, and 6, which are essential for calculating the center of mass. It acts as a map that guides us in orientation and measurement, ensuring accurate computations and meanings are derived from these positions.

Weighted Average

A weighted average helps us find a central value from a set of points that have a varying degree of importance or weight. In the context of finding the center of mass, it is a formula where each mass's position contributes differently to the final result based on its value. The positions are multiplied by their respective mass and then averaged.
The formula \[\bar{x} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{m_1 + m_2 + m_3 + m_4}\]captures the essence of a weighted average. This implies that more massive objects or those further away from the origin have a bigger impact on the center of mass. By integrating all these influences, we obtain a balanced point, showing how the masses collectively influence the system's overall center.