Problem 6 - 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. $$\begin{aligned} 550 \mathrm{g} \text { at }(-42,0), 230 \mathrm{g} \text { at }(-27,0), 470 \mathrm{g} \text { at }(16,0), 120 \mathrm{g} \text { at } (22,0) \end{aligned}$$

Step-by-Step Solution

Answer

The center of mass is approximately at \(-17.7\, \text{cm}\) on the x-axis.

Step 1: Calculate Total Mass

First, calculate the total mass of all particles. Add up all the given masses: \(550\, \text{g} + 230\, \text{g} + 470\, \text{g} + 120\, \text{g} = 1370\, \text{g}\).

Step 2: Compute Moments

Next, compute the moment of each particle, which is the product of its mass and its position. For each particle, compute: \(550 \cdot (-42) = -23100\), \(230 \cdot (-27) = -6210\), \(470 \cdot 16 = 7520\), and \(120 \cdot 22 = 2640\).

Step 3: Summing Moments

Sum all the computed moments to find the total moment: \(-23100 + (-6210) + 7520 + 2640 = -24250\).

Step 4: Calculate Center of Mass

To find the center of mass, divide the total moment by the total mass. The formula is \(x_{cm} = \frac{\text{Total Moment}}{\text{Total Mass}}\). So, \(x_{cm} = \frac{-24250}{1370} \approx -17.7\, \text{cm}\).

Key Concepts

Total Mass Calculation

The concept of total mass calculation is foundational to understanding the center of mass, which is heavily used in physics. When we talk about total mass, we are simply referring to the sum of the masses of all particles in a system.
To calculate this, you add up each individual mass. In the given problem, we find the total mass by adding each provided mass:

• Mass 1: 550 grams
• Mass 2: 230 grams
• Mass 3: 470 grams
• Mass 4: 120 grams

Thus, the total mass is simply the sum: 550 + 230 + 470 + 120 = 1370 grams.
This step is crucial because it gives us a baseline for further calculations like moments and eventually the center of mass. It sets the stage for understanding how the different particles collectively contribute to the system's behavior.

Moments of Particles

The moment of a particle is a key concept in both physics and mathematics, representing the tendency of that particle to cause rotation around a point or axis. It is calculated as the product of the particle's mass and its position.
Mathematically, for each particle on the x-axis, the moment is given by:
\( ext{Moment} = ext{Mass} imes ext{Position} \).
For example, in the exercise, we computed the moments as:

• Particle 1: 550 g at -42 cm ⇒ Moment = 550 * (-42) = -23100
• Particle 2: 230 g at -27 cm ⇒ Moment = 230 * (-27) = -6210
• Particle 3: 470 g at 16 cm ⇒ Moment = 470 * 16 = 7520
• Particle 4: 120 g at 22 cm ⇒ Moment = 120 * 22 = 2640

Knowing the moment for each particle informs us of its influence on the overall center of mass. The farther a particle is, the greater its impact due to the multiplier of its position.

Summation of Moments

Once individual moments are calculated, the next step is to find the total moment by summing these values. This total moment describes how the entire system of particles behaves in terms of rotation around the origin.
The calculation is simple:
We add each of the moments like this:
\( -23100 + (-6210) + 7520 + 2640 = -24250 \).
This single value, \(-24250\), reveals the collective influence of all particles on the x-axis.
It's important to remember that positive and negative moments may cancel each other out, depending on their positions on the axis. The summation gives us an overall rotational tendency, leading directly into the calculation of the center of mass.

Physics Applications in Mathematics

Physics applications in mathematics, especially concepts like the center of mass, bridge theory and real-world application.
The exercise of finding the center of mass is a prime example of this, integrating mathematical calculations with physical principles.
In physics:

• The center of mass is vital in predicting how objects behave when subjected to forces.
• For engineers, understanding this concept is essential in designing stable structures or products.

Mathematically:

• The process involves straightforward arithmetic, reliant on core principles like total mass and moments.
• This is widely applicable across different fields, showing how an abstract concept has practical utility.

The real power lies in making predictions about stability and motion, applicable to both simple systems like particles on an axis and complex engineering solutions.