Problem 8 - 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. $$ \begin{array}{l} m_{1}=4, \quad m_{2}=1, \quad m_{3}=2, \quad m_{4}=5 ; \quad P_{1}(-2,3), \\ P_{2}(-1,4), \quad P_{3}(1,4), \quad P_{4}(4,-3) \end{array} $$

Step-by-Step Solution

Answer

The center of mass of the system is located at the point \((1.083, 0.75)\).

Step 1: Identify the masses and coordinates

The given masses are \(m_1 = 4, m_2 = 1, m_3 = 2,\) and \(m_4 = 5\). The coordinates of the points are \(P_1(-2,3), P_2(-1,4), P_3(1,4),\) and \(P_4(4,-3)\).

Step 2: Calculate the total mass of the system

We find the total mass \(M\) by adding the individual masses: \[M = m_1 + m_2 + m_3 + m_4 = 4 + 1 + 2 + 5 = 12\]

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

To find the x-coordinate of the center of mass, \(C_x\), we use the formula: \[C_x = \frac{1}{M}(m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4)\] \[C_x = \frac{1}{12}(4(-2) + 1(-1) + 2(1) + 5(4))\] \[C_x = \frac{1}{12}(-8 - 1 + 2 + 20)\] \[C_x = \frac{1}{12}(13) = 1.083\]

Step 4: Calculate the y-coordinate of the center of mass

Now we calculate the y-coordinate of the center of mass, \(C_y\), using the formula: \[C_y = \frac{1}{M}(m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4)\] \[C_y = \frac{1}{12}(4(3) + 1(4) + 2(4) + 5(-3))\] \[C_y = \frac{1}{12}(12 + 4 + 8 - 15)\] \[C_y = \frac{1}{12}(9) = 0.75\]

Step 5: Combine the x and y-coordinates to find the center of mass

The center of mass \((C_x, C_y)\) is the point: \[(1.083, 0.75)\] So, the center of mass of the system is located at the point \((1.083, 0.75)\).

Key Concepts

Understanding Coordinate Plane

A coordinate plane is a flat two-dimensional surface where points are described using coordinates. Each point on this plane is represented by a pair of numbers, usually written as \(x, y\). The first number tells how far to move from the origin along the x-axis (horizontal direction), while the second number indicates the movement along the y-axis (vertical direction).
The origin, usually at \(0,0\), is the central point where both axes intersect.
This system allows us to pinpoint the exact location of points, like the masses in our problem, P\(1\) at \((-2,3)\), P\(2\) at \((-1,4)\), and so on. Understanding the coordinate plane is crucial for navigating through most geometric and physical problems.

Mass Distribution and Its Importance

Mass distribution tells us how masses are spread out over a system. Each mass has a specific position in the coordinate plane, greatly influencing the overall balance of the system.
In our example, we have different masses located at different points: \(m_1 = 4, m_2 = 1, m_3 = 2,\) and \(m_4 = 5\). Recognizing this means we understand that heavier masses affect the center of mass more significantly than lighter ones.
Understanding mass distribution is crucial when solving for the center of mass because it determines how each mass will pull or shift the center towards itself, depending on its weight and relative position.

The Calculation of Center of Mass

The center of mass is a single point that represents the average position of the entire mass distribution. To compute the center of mass accurately, we use the masses and their coordinates.
First, it's essential to calculate the total mass, denoted as \(M\). In our exercise, \(M = m_1 + m_2 + m_3 + m_4 = 12\).
We then calculate the x-coordinate and y-coordinate of the center of mass. These coordinates tell us where the center of mass lies within the coordinate plane, reflecting the system's equilibrium point. To ensure our results are precise, each step in this calculation, such as summing the products of the mass and its respective coordinates, must be carefully followed.

Finding the X-Coordinate of Center of Mass

To find the x-coordinate of the center of mass, \(C_x\), we employ the formula: \[C_x = \frac{1}{M} (m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4)\]
Here, multiply each mass by its respective x-coordinate, then add those products together. This accounts for the influence each mass has on the x-direction of the center.
Let's look at the example: \[C_x = \frac{1}{12} (4(-2) + 1(-1) + 2(1) + 5(4)) = \frac{1}{12}(-8 - 1 + 2 + 20) = \frac{1}{12}(13) = 1.083\]
This calculation indicates that the center is pulled slightly towards the right due to the higher positive mass values at larger x-coordinates.

Finding the Y-Coordinate of Center of Mass

The y-coordinate of the center of mass, \(C_y\), is determined with the formula: \[C_y = \frac{1}{M} (m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4)\]
This involves multiplying each mass by its respective y-coordinate and summing them up, which aligns the masses' vertical influences.
For instance: \[C_y = \frac{1}{12} (4(3) + 1(4) + 2(4) + 5(-3)) = \frac{1}{12}(12 + 4 + 8 - 15) = \frac{1}{12}(9) = 0.75\]
This tells us that the center of mass's location is slightly above the origin in the y-direction, influenced by the masses positioned at positive y-values.