Problem 5 - Basic Technical Mathematics with Calculus

Question

Find the center of mass (in cm) of the particles with the given masses located at the given points on the \(x\) -axis. $$\text { 31 } \mathrm{g} \text { at }(-3.5,0), 24 \mathrm{g} \text { at }(0,0), 15 \mathrm{g} \text { at }(2.6,0), 84 \mathrm{g} \text { at }(3.7,0)$$

Step-by-Step Solution

Answer

The center of mass is approximately at 1.57 cm on the x-axis.

Step 1: Understand the Problem

The problem provides the masses of particles and their respective positions on the x-axis. We need to find the center of mass, which will be a point also on the x-axis.

Step 2: Write the Formula for Center of Mass on the x-axis

The formula to find the center of mass when particles are aligned on the x-axis is \( \bar{x} = \frac{\sum m_{i} x_{i}}{\sum m_{i}} \), where \( m_{i} \) are the masses and \( x_{i} \) are the positions of the particles along the x-axis.

Step 3: List the Given Data

We have four particles with the following mass-position pairs: \( m_1 = 31 \, \ x_1 = -3.5 \), \( m_2 = 24 \, \ x_2 = 0 \), \( m_3 = 15 \, \ x_3 = 2.6 \), and \( m_4 = 84 \, \ x_4 = 3.7 \).

Step 4: Calculate the Numerator of the Center of Mass Formula

Compute the sum of the products of each mass with its corresponding position, \( \sum m_{i} x_{i} \): \[ \sum m_{i} x_{i} = (31 \times -3.5) + (24 \times 0) + (15 \times 2.6) + (84 \times 3.7) \]Calculate these individually and then sum them up.

Step 5: Calculate the Denominator of the Center of Mass Formula

Compute the sum of the masses, \( \sum m_{i} \): \[ \sum m_{i} = 31 + 24 + 15 + 84 \]Add these masses together.

Step 6: Solve for the Center of Mass \(\bar{x}\)

Divide the result from Step 4 by the result from Step 5 to find the center of mass: \( \bar{x} = \frac{\sum m_{i} x_{i}}{\sum m_{i}} \).

Step 7: Final Calculation

Let's compute these values: - Numerator: \( (31 \times -3.5) + (24 \times 0) + (15 \times 2.6) + (84 \times 3.7) = -108.5 + 0 + 39 + 310.8 = 241.3 \) - Denominator: \( 31 + 24 + 15 + 84 = 154 \) - Center of Mass: \( \bar{x} = \frac{241.3}{154} \approx 1.57 ext{ cm} \).

Key Concepts

Mass Distribution

Imagine scattering small weights at different points along a line, which, in this case, is the x-axis. This layout describes the concept of **mass distribution**. It’s all about how the weight (or mass) is spread out across different locations.
The idea serves as a key factor when determining the center of mass. The mass distribution provides the specifics of where each piece of mass is, and how much it weighs. This becomes crucial because each mass's effect on the center of mass depends on both its size and location.
Understanding this fundamental concept helps in analyzing how various masses influence the overall balance of a system. Essentially, the masses that are farther from the reference point have a larger impact due to their position, which we'll discuss next.

X-axis

In our context, the **x-axis** acts as the horizontal space where these masses are placed. It functions much like a tightrope, with each particle represented as a small weight on it.
On this axis, every point has a specific numerical value that indicates its position from a chosen origin (often the point 0). It’s important, since each mass will affect the balance — or center of mass — differently depending on where it's situated along this line.
By setting the masses along the x-axis explicitly, it simplifies our task to find the center of mass because all other dimensions, such as height or depth, are not involved, making our calculations more straightforward.

Calculation Formula

The **calculation formula** is a mathematical expression used to find the center of mass along the x-axis. It aggregates the effects of all the individual masses at their respective positions. The formula is given by:\[\bar{x} = \frac{\sum m_{i} x_{i}}{\sum m_{i}} \]Here's how it works:

• **Numerator**: Multiply each mass by its position along the x-axis (\(m_{i} x_{i}\)) and sum them all up.
• **Denominator**: Add up all the masses (\(\sum m_{i}\)).

The resulting number \(\bar{x}\) tells us the point where the entire system balances — that is, its center of mass. Using this formula ensures precision in finding that perfect point of equilibrium on the x-axis.

Particle Position

Every mass has a specific **particle position** on the x-axis, contributing uniquely to the overall setup. Each particle's influence on the center of mass varies depending on where it sits.
In a list like our exercise:

• A particle with a strong influence might be further along the axis, as its mass and position combine to produce a larger moment.
• Conversely, particles situated closer to the origin or with less mass have a smaller impact.

These positions, being part of the numerical calculation, make finding the center of mass dependent on not just the size of each mass, but importantly, *where* those masses are situated in the lineup. Visualizing this particle arrangement along the x-axis helps immensely in grasping the system balance.