Problem 3 - Calculus of a Single Variable

Question

Find the center of mass of the point masses lying on the \(x\) -axis. \(m_{1}=1, m_{2}=3, m_{3}=2, m_{4}=9, m_{5}=5\) \(x_{1}=6, x_{2}=10, x_{3}=3, x_{4}=2, x_{5}=4\)

Step-by-Step Solution

Answer

The center of mass of the point masses is at the x-position 4.

Step 1: Identify given masses and their positions

Five point masses have been provided: \(m_{1}=1, m_{2}=3, m_{3}=2, m_{4}=9, m_{5}=5\), along with their respective x-positions: \(x_{1}=6, x_{2}=10, x_{3}=3, x_{4}=2, x_{5}=4\).

Step 2: Calculate the numerator of the formula

Apply each mass and its position into \(m_{i}x_{i}\) and sum up the products to get the numerator of the center of mass formula: \(m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3} + m_{4}x_{4} + m_{5}x_{5} = 1*6 + 3*10 + 2*3 + 9*2 + 5*4 = 80\).

Step 3: Calculate the denominator of the formula

Sum up the provided masses to get the denominator of the center of mass formula: \(m_{1} + m_{2} + m_{3} + m_{4} + m_{5} = 1 + 3 + 2 + 9 + 5 = 20\).

Step 4: Calculate the center of mass

Plug in the values for the numerator (80) and the denominator (20) into the center of mass formula which gives: \(\frac{80}{20} = 4\). Therefore, the center of mass is at the x-position 4.

Key Concepts

Physics Mathematics

To comprehend the intriguing role of mathematics in physics, particularly in locating the center of mass, let's dive into the significance of this concept. The center of mass in a system of particles is essentially the point where the mass distribution of the system is balanced. It's defined by the weighted average position of all the particles, considering their masses.

Mathematically, for a set of particles lying on a straight line, as is the case with our problem, this is given by the formula: \[\text{Center of Mass (COM)} = \frac{\sum m_i x_i}{\sum m_i}\]where \(m_i\) represents the mass of each particle and \(x_i\) denotes their respective positions.

This formula is a prime example of how mathematical operations like summation and division are used to solve physical problems. By applying this formula, one can calculate the point at which one could balance the entire system on the tip of a pencil - if it were possible to make it a single point mass. Understanding this physics-mathematics crossover grants us the privilege to predict behaviors of physical systems and solve complex problems like finding equilibrium points.

Calculus Applications

When we explore calculus applications in physics, we often deal with continuous mass distributions. However, our current exercise involves discrete masses. Nonetheless, the fundamental principles remain connected. In our discrete case, we employed an algebraic calculation to determine the center of mass. In contrast, when dealing with a continuous mass distribution, one would integrate over the continuous mass to find its center of mass.

For instance, we can imagine a rod with varying density. To find this rod's center of mass, calculus steps in as we integrate the product of density and position along the length of the rod. This integral is then divided by the total mass (also found by integrating the density). This approach encapsulates the essence of calculus in physics: providing a technique to deal with infinite points by summing their infinitesimal contributions to a property, like mass, to determine a collective behavior or position, such as the center of mass.

Mechanical Equilibrium

Mechanical equilibrium refers to a state where the sum of forces and the sum of moments (torques) on a system is zero. In essence, it's the condition where a system experiences no net change in motion. For our point masses lying on the x-axis, finding the center of mass is directly related to this concept. If we were to support this system at the center of mass, no net torque would be acting on the system, and it would remain in mechanical equilibrium.

In the broader picture, mechanical equilibrium might involve complex systems with numerous forces acting. However, the underlying principle stands: if a system is in mechanical equilibrium, it's either at rest or moves with constant velocity - both scenarios indicating no acceleration. Understanding the center of mass helps us to pinpoint the location where forces would need to act to maintain equilibrium or where supports should be placed to prevent rotation or translation, maintaining a state of balance in the system.