Question

Particle A has a mass of \(5.0 \mathrm{g}\) and particle \(\mathrm{B}\) has a mass of \(1.0 \mathrm{g} .\) Particle \(\mathrm{A}\) is located at the origin and particle \(\mathrm{B}\) is at the point \((x, y)=(25 \mathrm{cm}, 0) .\) What is the location of the CM?

Short answer

The location of the CM is approximately (4.17 cm, 0).

Step-by-step solution

Understand Center of Mass Formula

The center of mass (CM) of a system of particles is where each particle's position is weighted by its mass. The formula for the CM in the x-direction for two particles is: \[ x_{\text{CM}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \] where \( m_1 \) and \( m_2 \) are the masses, and \( x_1 \) and \( x_2 \) are the x-coordinates of particle A and particle B respectively.

Apply Given Values

Substitute the given values into the formula: \( m_1 = 5.0\,\text{g} \), \( m_2 = 1.0\,\text{g} \), \( x_1 = 0\), and \( x_2 = 25\,\text{cm} \). Thus, the equation becomes: \[ x_{\text{CM}} = \frac{(5.0\,\text{g})(0) + (1.0\,\text{g})(25\,\text{cm})}{5.0\,\text{g} + 1.0\,\text{g}} \]

Calculate the X Coordinate of the CM

Calculate the right-hand side of the equation: \[ x_{\text{CM}} = \frac{0 + 25}{6} = \frac{25}{6} \approx 4.17\,\text{cm} \]. So, the x-coordinate of the CM is approximately 4.17 cm.

Determine the Y Coordinate of the CM

For particles that lie on the x-axis, the y-coordinate of the CM is the weighted average of their y-coordinates. Here, both particles have a y-coordinate of 0, so \( y_{\text{CM}} = 0 \).

Key concepts

Mass Distribution

The concept of mass distribution is crucial when understanding the center of mass in a multi-particle system. It refers to how mass is spread out across different particles or objects. In our exercise, we have two particles. Particle A is heavier with a mass of 5.0 grams, while Particle B is lighter, weighing 1.0 gram. The position and mass of each particle affect the overall center of mass.

• The heavier the particle, the more influence it has on the center of mass.
• A particle located farther from others will shift the center of mass towards itself.

In the exercise, the heavier particle A is located at the origin whereas the lighter particle B is located at (25 cm, 0). This setup helps us see their mass distribution and how it plays a pivotal role in determining the center of mass location.

Coordinate Systems

Understanding coordinate systems is integral in physics whenever you deal with spatial arrangements. For locating the center of mass, we often use a Cartesian coordinate system, which has an x-axis and a y-axis.

• Particle A, with a mass of 5.0 grams, is positioned at the origin, which is the point (0, 0).
• Particle B, slightly lighter at 1.0 gram, is directly placed on the x-axis at the point (25 cm, 0).

This configuration is simpler because both particles sit on the x-axis, meaning the y-coordinates are zero for both. Coordinating this layout is essential for applying the center of mass formula accurately. The use of coordinates allows us to systematically calculate distances and averages, leading to an easier determination of where the combined mass properties will balance out.

Two-Particle System

The two-particle system is one of the simplest multi-object systems to analyze for the center of mass. It involves only two particles whose respective known properties mass and positional coordinates are used to determine a shared center of mass.For a system with two particles:

• Each particle's influence on the center of mass is directly proportional to its mass.
• The formula for calculating the x-component of the center of mass for this system is:\[ x_{\text{CM}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
• Both y-coordinates being zero simplifies the calculation, keeping the y-component of the CM at zero.

By substituting the masses and their respective x-coordinates, as given in the problem, we can solve for the center of mass in the x-direction. This shows how even in small systems, center of mass calculations reveal the collective behavior of multiple particles.