Two-particle system. A \(2 \mathrm{~kg}\) particle is placed at \(x=2 \mathrm{~m}\) and a \(4 \mathrm{~kg}\) particle is placed at \(x=6 \mathrm{~m}\). (a) Where is the center of mass of this two-particle system?

The center of mass (COM) is a fundamental concept in physics and mechanics. It refers to a point in a system of particles where the whole mass of the system can be considered to be concentrated. For a system composed of different masses at varying positions, the center of mass can be calculated using a weighted average of their positions. This simplifies the study of motion because it allows us to analyze the motion of one single point instead of multiple points. Essentially, the motion of the entire system can be described by the motion of the center of mass. The formula used for finding the x-coordinate of the center of mass in a two-particle system is:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
Here, \( m_1 \) and \( m_2 \) are the masses of the particles, and \( x_1 \) and \( x_2 \) are their respective positions. By substituting the given values, you can find the exact position of the center of mass along the x-axis.

A two-particle system is one of the simplest systems used to understand the principles of mechanics and the concept of the center of mass. In this exercise, we're focusing on a system with two particles of different masses placed at specific positions along a line. Given:

• A 2 kg particle at 2 meters
• A 4 kg particle at 6 meters

The formula for the center of mass is used to calculate the point where these two masses balance each other out. Essentially, this point tells us where you could balance a rod on your finger if the rod had the same mass distribution as described.
By substituting the values in the formula:
\[ x_{cm} = \frac{(2 \text{ kg})(2 \text{ m}) + (4 \text{ kg})(6 \text{ m})}{2 \text{ kg} + 4 \text{ kg}} = 4.67 \text{ m} \]you find that the center of mass of this system is at 4.67 meters from the origin. This calculated point is the effective position where the combined mass behaves as if it were concentrated at that single point.

Mechanics is a branch of physics that deals with the motion of objects and the forces acting on them. One fundamental concept within mechanics is the center of mass. Understanding how to calculate the center of mass allows us to predict and describe the motion of objects more accurately. In the context of this exercise, mechanics principles are applied to a two-particle system.
Key points to grasp include:

• The center of mass depends on both the masses of the particles and their positions.
• By treating multiple particles as a single point at the center of mass, complex motion becomes simpler to analyze.
• The center of mass is crucial in mechanics as it simplifies the equations of motion and is used in various applications from simple physics problems to sophisticated engineering designs.

By understanding and applying these basic principles, you can better understand the behavior of different systems in real-world situations like vehicles, machinery, and even in sports and daily life activities.