Problem 28 - Essential Calculus

Question

In Exercises 27 and \(28,\) find the center of mass of the point masses lying on the \(x\) -axis. $$ \begin{array}{l} m_{1}=12, m_{2}=1, m_{3}=6, m_{4}=3, m_{5}=11 \\ x_{1}=-6, x_{2}=-4, x_{3}=-2, x_{4}=0, x_{5}=8 \end{array} $$

Step-by-Step Solution

Answer

The center of mass of the given points is located at \(x=0\) on the x-axis.

Step 1: Find the Total Mass (M)

First we start by finding the total mass of the system. This is done by summing up all individual masses: \(M = m_{1} + m_{2} + m_{3} + m_{4} + m_{5} = 12 + 1 + 6 + 3 + 11 = 33 \) kg.

Step 2: Multiply Each Mass With its Corresponding position

Next, multiply each mass with its position: \(x_{1}m_{1} = -6*12, x_{2}m_{2} = -4*1, x_{3}m_{3}= -2*6, x_{4}m_{4}= 0*3, x_{5}m_{5} = 8*11\). This gives us the values -72, -4, -12, 0 and 88 respectively.

Step 3: Sum the Results of Step 2

Now, we add all the values from step 2 together: \(-72 - 4 - 12 + 0 + 88 = 0\). This value indicates the numerator of our center of mass equation.

Step 4: Calculate the Center of Mass

To find the center of mass, we divide the sum from step 3 by the total mass from step 1: \(x_{\text{cm}} = \frac{0}{33} = 0\).

Key Concepts

Point Masses

Point masses are idealized objects where the entire mass is concentrated at a single point in space. This abstraction is useful for simplifying the analysis of mechanical systems.
In practical terms, when calculating the center of mass, each point mass is treated as if its entire weight acts at one coordinate location. This allows us to ignore the dimensions and shapes of these masses and focus solely on their weight and position, greatly simplifying calculations.
Understanding point masses help you in scenarios involving

• particles on a line
• bodies with negligible volume
• systems where mass can be considered concentrated at a point, like stars in astrophysics or atoms in molecules

Recognizing point masses in a problem is the first key step in understanding the system’s behavior and deducing the center of mass.

Mass Distribution

Mass distribution refers to how mass is spread out over space. To determine the center of mass in a linear system like this, it’s crucial to understand where each individual mass is located along the line.
Knowing the positions and values of each point mass along the x-axis allows us to calculate the influence each mass has in determining the center of mass of the system.
Consider

• the influence of larger masses, which can dominate the center of mass location
• how massively weighted points further from the center will skew the center accordingly

Recognizing the balance of mass is crucial: if large masses are distributed unevenly, the center will shift toward the side with more mass. The systematic approach of calculating the influence of each mass involves multiplying the mass by its position, which quantifies its contribution to the overall mass balance on the axis.

x-Axis Calculations

Calculating the center of mass along the x-axis involves breaking down a multifaceted process into straightforward steps.

• First, sum all point masses to determine the total mass of the system. This number serves as the divisor in the final calculation.
• Next, multiply each mass by its position on the x-axis. This step highlights the individual contributions to the "mass moment," showing how each point mass impacts the system's equilibrium.
• Add all these products together to obtain a total mass moment.

The division of the total mass moment by the total mass then provides the x-coordinate of the center of mass.
This calculation is essential in physics and engineering as it helps in understanding how a system will naturally move or balance based on the forces applied or gravitational actions. For instance, in this exercise, the result demonstrates that the masses balance at the origin, meaning their influences counteract perfectly across the x-axis with respect to distribution and weight.