Question

The positions of three particles, written as \((x, y)\) coordinates, are: particle 1 (mass \(4.0 \mathrm{kg}\) ) at \((4.0 \mathrm{m}, 0 \mathrm{m}) ;\) particle \(2(\text { mass } 6.0 \mathrm{kg} \text { ) at }(2.0 \mathrm{m}, 4.0 \mathrm{m}) ;\) particle \(3(\mathrm{mass} 3.0 \mathrm{kg})\) at \((-1.0 \mathrm{m},-2.0 \mathrm{m}) .\) What is the location of the \(\mathrm{cm} ?\)

Short answer

The center of mass is at approximately (1.923 m, 1.385 m).

Step-by-step solution

Understand the Center of Mass Formula

The center of mass (CM) of a system of particles can be found using the weighted average of their positions, considering the mass of each particle. The coordinates of the center of mass (\(x_{cm}, y_{cm}\)) are calculated using the formulas: \[ x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \] and \[ y_{cm} = \frac{\sum m_i y_i}{\sum m_i} \] where \(x_i, y_i\) are the coordinates of the particles, and \(m_i\) are their masses.

Calculate Total Mass

Add up the masses of the individual particles to find the total mass. \[ m_{total} = 4.0 \, \mathrm{kg} + 6.0 \, \mathrm{kg} + 3.0 \, \mathrm{kg} = 13.0 \, \mathrm{kg} \]

Compute Weighted x-coordinate Sum

Calculate the sum of the products of each particle's mass and its x-coordinate: \[ \sum m_i x_i = (4.0 \, \mathrm{kg} \times 4.0 \, \mathrm{m}) + (6.0 \, \mathrm{kg} \times 2.0 \, \mathrm{m}) + (3.0 \, \mathrm{kg} \times -1.0 \, \mathrm{m}) \] Simplify the calculations: \[ = 16.0 \, \mathrm{kg} \, \mathrm{m} + 12.0 \, \mathrm{kg} \, \mathrm{m} - 3.0 \, \mathrm{kg} \, \mathrm{m} = 25.0 \, \mathrm{kg} \, \mathrm{m} \]

Compute Weighted y-coordinate Sum

Calculate the sum of the products of each particle's mass and its y-coordinate: \[ \sum m_i y_i = (4.0 \, \mathrm{kg} \times 0.0 \, \mathrm{m}) + (6.0 \, \mathrm{kg} \times 4.0 \, \mathrm{m}) + (3.0 \, \mathrm{kg} \times -2.0 \, \mathrm{m}) \] Simplify the calculations: \[ = 0.0 \, \mathrm{kg} \, \mathrm{m} + 24.0 \, \mathrm{kg} \, \mathrm{m} - 6.0 \, \mathrm{kg} \, \mathrm{m} = 18.0 \, \mathrm{kg} \, \mathrm{m} \]

Calculate x-coordinate of Center of Mass

Use the formula for the x-coordinate of the center of mass: \[ x_{cm} = \frac{\sum m_i x_i}{\sum m_i} = \frac{25.0 \, \mathrm{kg} \, \mathrm{m}}{13.0 \, \mathrm{kg}} \] Calculate the result: \[ x_{cm} \approx 1.923 \, \mathrm{m} \]

Calculate y-coordinate of Center of Mass

Use the formula for the y-coordinate of the center of mass: \[ y_{cm} = \frac{\sum m_i y_i}{\sum m_i} = \frac{18.0 \, \mathrm{kg} \, \mathrm{m}}{13.0 \, \mathrm{kg}} \] Calculate the result: \[ y_{cm} \approx 1.385 \, \mathrm{m} \]

Combine Coordinates for Final Answer

Combine the results of the x-coordinate and y-coordinate calculations to find the overall center of mass: The location of the center of mass is \((x_{cm}, y_{cm}) = (1.923 \, \mathrm{m}, 1.385 \, \mathrm{m})\).

Key concepts

Particle Systems

In physics, a **particle system** is a model that consists of several individual particles. Each particle represents a distinct object which is often characterized by its mass and position. Particle systems are fundamental because they simplify the complex behaviors of many objects into manageable components. By examining the properties of these components, like mass and position, we can understand and predict the behavior of the entire system. For instance, when studying the motion of celestial bodies or molecules, approaches use particle systems to approximate real-world objects. In our exercise, the particle system consists of three particles with specific masses and positions. These particles allow us to calculate the center of mass, a critical concept helping to identify how the collection behaves as a whole even if the individual particles have vastly different properties.

Coordinate Calculations

Understanding **coordinate calculations** is key to many problems involving particle systems. Coordinates provide the spatial details required to pinpoint particle positions. In this exercise, we have particles located at given coordinates:

• Particle 1 at (4.0 m, 0 m)
• Particle 2 at (2.0 m, 4.0 m)
• Particle 3 at (-1.0 m, -2.0 m)

To find the center of mass, we do not just average these coordinates. We weigh these positions by the mass of each particle. This ensures that the more substantial influence of heavier particles is accounted for. Using expressions like \( x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \) and \( y_{cm} = \frac{\sum m_i y_i}{\sum m_i} \), we calculate the average position of the mass in both x and y directions. This approach balances the system's overall coordinate points in a manner that varies according to each particle's mass.

Physics Problem Solving

**Physics problem solving** involves taking complex situations and breaking them down into simpler, manageable tasks. In our exercise, we go through a series of steps, starting with understanding the problem and what is needed, then gathering necessary data and equations. - Identify what's given: Particle masses and positions are provided. - Develop a strategy: Use given information in the center of mass formula. - Execute the solution: Calculate individual sums for coordinates and total mass. Real-world problems often exhibit similar complexity, requiring step-by-step approaches like this one. By mastering these techniques, one can effectively approach and solve a wide range of problems in physics and related fields. Understanding specific equations, such as the center of mass formula, and applying them in an orderly manner ensures successful outcomes and reinforces the significance of physics principles.