Problem 5 - Single Variable Calculus

Question

Find the center of mass of the system comprising masses \(m_{k}\) located at the points \(P_{k}\) in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. $$ \begin{array}{l} m_{1}=4, \quad m_{2}=3, \quad m_{3}=5 ; \quad P_{1}(-3,-2), \quad P_{2}(-1,2), \\\ P_{3}(2,4) \end{array} $$

Step-by-Step Solution

Answer

The center of mass of the system is located at the point \(R_{CM} =\left(-\frac{5}{12}, \frac{3}{2}\right)\), which means it is positioned at an x-coordinate of \(-\frac{5}{12}\) cm and a y-coordinate of \(\frac{3}{2}\) cm in the coordinate plane.

Step 1: Find the position vectors

We first need to find the position vectors for the given mass points. We can define each position vector as: $$ \vec{r}_k = x_k \hat{x} + y_k \hat{y} = (x_k, y_k), $$ Substitute coordinates for \(P_1\), \(P_2\), and \(P_3\) points to find \(\vec{r}_1\), \(\vec{r}_2\), and \(\vec{r}_3\): $$ \vec{r}_1 = (-3, -2), $$ $$ \vec{r}_2 = (-1, 2), $$ $$ \vec{r}_3 = (2, 4). $$

Step 2: Find the weighted position vectors

Calculate the weighted position vectors using the masses given: $$ m_1\vec{r}_1 = 4(-3, -2) = (-12, -8), $$ $$ m_2\vec{r}_2 = 3(-1, 2) = (-3, 6), $$ $$ m_3\vec{r}_3 = 5(2, 4) = (10, 20). $$

Step 3: Calculate the center of mass

To find the position vector of the center of mass, \(\vec{R}_{CM}\), sum the weighted position vectors and divide by the total mass: $$ \vec{R}_{CM} = \frac{\sum_{k=1}^3 m_k \vec{r}_k}{\sum_{k=1}^3 m_k} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + m_3\vec{r}_3}{m_1 + m_2 + m_3} = \frac{(-12, -8) + (-3, 6) + (10, 20)}{4+3+5}, $$ $$ \vec{R}_{CM} = \frac{(-5, 18)}{12} = (-\frac{5}{12}, \frac{3}{2}). $$

Step 4: Interpret the result

The center of mass is located at the point $$ R_{CM} =\left(-\frac{5}{12}, \frac{3}{2}\right). $$ This means that the center of mass of the system is located at a point with an x-coordinate of \(-\frac{5}{12}\) cm and a y-coordinate of \(\frac{3}{2}\) cm in the coordinate plane.

Key Concepts

Coordinate Plane

In mathematics, the coordinate plane is essential for plotting points and understanding geometric concepts in a two-dimensional space. It consists of two perpendicular lines, known as axes. The horizontal line is the x-axis, and the vertical line is the y-axis. These axes intersect at a point called the origin, denoted as (0,0), which is the reference point for determining the position of other points.

The coordinate plane helps visualize where each point is located using a pair of numbers, or coordinates:

• The x-coordinate indicates the horizontal position.
• The y-coordinate shows the vertical position.

This system allows us to easily plot and identify points, which is critical when calculating the center of mass using position vectors.

Position Vectors

Position vectors are a way to represent points in the coordinate plane using a vector notation. They help us describe the exact location of a point in relation to the origin.

• A position vector for a point labeled as \(P(x, y)\) is written as \(\vec{r} = (x, y)\).
• This vector starts at the origin and ends at the point P, effectively mapping the point's position on the coordinate plane.

Position vectors are crucial in determining relationships between points and in forming the basis for calculating weighted position vectors in a system of masses.

Weighted Position Vectors

Weighted position vectors extend the concept of position vectors by factoring in the mass of each point in a system of masses. This is important when calculating the center of mass because it represents the influence each mass has over the system.

• To find the weighted position vector, multiply the mass of a point \(m_k\) by its position vector \(\vec{r}_k\).
• This results in a new vector \(m_k\vec{r}_k\), which highlights the contribution of that mass to the overall "balance" of the system.

By summing all the weighted position vectors and dividing by the total mass, we can locate the center of mass, effectively finding the balance point of the entire system.

Calculus

Calculus is a branch of mathematics that focuses on change and motion, covering concepts such as derivatives and integrals. It's crucial in many areas, including physics and engineering, for finding solutions to problems involving dynamically changing systems.

While the basic problem of finding the center of mass often involves simple algebra, calculus becomes essential when dealing with continuous mass distributions or varying mass densities over a region. In these cases, integration is used to sum up infinitesimally small mass elements to accurately determine the center of mass.

• Derivatives help understand rates of change, which is often necessary in dynamic systems analysis.
• Integrals provide a method to sum continuous values, necessary for analysis over a range or area.

Understanding calculus concepts can greatly enhance your ability to solve complex problems, including those involving centers of mass in varying or continuous systems.

System of Masses

A system of masses refers to a collection of objects, each with its own mass, placed at specific points in a region. Calculating the center of mass of such a system helps in understanding how these masses balance in space.

In any given system:

• Each mass contributes differently based on its magnitude and position.
• The center of mass is the average position of all the masses, weighted by their respective magnitudes.
• This point is the balance, or "average location," where the system is perfectly balanced if suspended.

By combining the principles of position vectors and weighted vectors, the position of the center of mass can be effectively calculated, offering critical insights into the physical behavior and stability of the system.