Problem 27 - Essential Calculus

Question

In Exercises 27 and \(28,\) find the center of mass of the point masses lying on the \(x\) -axis. $$ \begin{array}{l} m_{1}=6, m_{2}=3, m_{3}=5 \\ x_{1}=-5, x_{2}=1, x_{3}=3 \end{array} $$

Step-by-Step Solution

Answer

The center of mass of the point masses lies at \(X = -0.857\) on the x-axis.

Step 1: Calculate the total mass.

The total mass \(M\) of the system is simply the sum of the individual masses. Calculate \(M\) using the formula: \(M = m_{1} + m_{2} + m_{3}\). Substituting the provided values obtains \(M = 6 + 3 + 5 = 14\).

Step 2: Calculate the weighted sums.

The center of mass is dependent not only on the masses, but also their positions. Multiply each mass by its corresponding position and calculate the total sum. This sum is given by the formula: \(m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3}\). Substituting the given values obtains \( (6*-5) + (3*1) + (5*3) = -30 + 3 + 15 = -12 \).

Step 3: Calculate the x-coordinate of the center of mass.

The x-coordinate of the center of mass, \(X\), is calculated by dividing the total weighted sum obtained in Step 2 by the total mass. This means \(X = \frac{m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3}}{M}\). Using the previously computed summed weights, and the total mass obtains \(X = \frac{-12}{14} = -0.857\).

Key Concepts

Calculus

Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. It's a tool that allows us to study how things change and accumulate, making it essential for analyzing mass distributions. In our exercise, calculus isn’t used in the form of differentiation or integration, but the concept of a weighted average is a fundamental idea that can also be found in calculus when working with centers of mass or centroids in continuous systems. The center of mass computation is essentially a discrete application where we sum across individual points rather than integrating over a continuous distribution.

If we were dealing with a continuous mass distribution along a line or within a volume, we would use integrals to sum up infinitesimally small pieces of mass over the desired interval or volume to find the center of mass or centroid. This process, while similar in concept, requires a deeper understanding of calculus.

Weighted Sums

A weighted sum is a mathematical concept used to find an average value when different elements have varying levels of importance, or 'weight'. In the context of our center of mass problem, each point mass is assigned a weight that corresponds to its mass, and its position on the coordinate axis is multiplied by its weight to determine its influence on the overall center of mass. This process gives us what's known as the moment about a point.

For example, the contribution of each mass to the center of mass is calculated by multiplying the mass by its distance from a reference point, essentially giving us weighted averages along each axis. Our final center of mass, in this case, the x-coordinate, is found by summing up these individual moments (weighted distances) and dividing by the total mass.

Mass Distribution

Mass distribution refers to how mass is spread out in space. In our case, we're looking at a very simple system where mass is only distributed along the x-axis, at distinct points. More complex systems can have mass distributed along planes or throughout volumes in three dimensions. The concept of the center of mass helps to describe the distribution by providing a single point that characterizes the system's overall balance.

In physics and engineering, understanding mass distribution is crucial since it affects balance, stability, and motion of objects. For the exercise, treating discrete masses with specific coordinates allows us to handle what is otherwise a complex concept involving continuous mass distributions, providing a clear-cut calculation method.

Coordinate System

In mathematics, a coordinate system is a way to uniquely determine the position of a point or a geometric element in a multidimensional space. In our textbook problem, we're using a one-dimensional coordinate system along the x-axis. The system allows us to specify the locations of the point masses easily.

The positions of the masses (\( x_{1}, x_{2}, x_{3} \)) play a vital role in determining the center of mass because they reflect the specific points where mass is concentrated. By using these coordinates in conjunction with the masses, we can apply the concept of weighted sums to find the balance point of the system, which is the center of mass. Understanding how to navigate and use a coordinate system is fundamental for solving problems involving spatial mass distributions.