Question

The masses and coordinates of a system of particles are given by the following: \(5,(-3,2) ; 6,(-2,-2) ; 2,(3,5) ; 7,(4,3)\); \(1,(7,-1) .\) Find the moments of this system with respect to the coordinate axes, and find the coordinates of the center of mass.

Short answer

x-moment = 14, y-moment = 28; Center of mass: \(\left(\frac{2}{3}, \frac{4}{3}\right)\)."

Step-by-step solution

Understanding the Moment of a Particle System

The moment of a particle about an axis is defined as the product of the mass of the particle and its distance from the axis. We will calculate the moment about the x-axis (also known as the y-moment) by multiplying each particle's mass by its y-coordinate. Similarly, the moment about the y-axis (or x-moment) is found by multiplying each particle's mass by its x-coordinate.

Calculate the y-Moment

To find the y-moment, sum the products of each particle's mass and its y-coordinate. Thus,\[ M_y = 5(2) + 6(-2) + 2(5) + 7(3) + 1(-1) \]Calculate each term:- First particle: \(5 \times 2 = 10\)- Second particle: \(6 \times -2 = -12\)- Third particle: \(2 \times 5 = 10\)- Fourth particle: \(7 \times 3 = 21\)- Fifth particle: \(1 \times -1 = -1\)Total y-moment: \[M_y = 10 - 12 + 10 + 21 - 1 = 28\].

Calculate the x-Moment

To find the x-moment, sum the products of each particle's mass and its x-coordinate. Thus,\[ M_x = 5(-3) + 6(-2) + 2(3) + 7(4) + 1(7) \]Calculate each term:- First particle: \(5 \times -3 = -15\)- Second particle: \(6 \times -2 = -12\)- Third particle: \(2 \times 3 = 6\)- Fourth particle: \(7 \times 4 = 28\)- Fifth particle: \(1 \times 7 = 7\)Total x-moment: \[M_x = -15 - 12 + 6 + 28 + 7 = 14\].

Determine the Center of Mass Coordinates

The center of mass (\( \bar{x}, \bar{y} \)) coordinates are given by dividing the moments by the total mass of the system. First, find the total mass:\[ m_{total} = 5 + 6 + 2 + 7 + 1 = 21 \]Calculate the x-coordinate of the center of mass:\[ \bar{x} = \frac{M_x}{m_{total}} = \frac{14}{21} = \frac{2}{3} \]Calculate the y-coordinate of the center of mass:\[ \bar{y} = \frac{M_y}{m_{total}} = \frac{28}{21} = \frac{4}{3} \]Therefore, the center of mass is located at \( \left(\frac{2}{3}, \frac{4}{3}\right)\).

Key concepts

Moments About Coordinate Axes

In physics and engineering, moments are crucial for understanding how forces and objects interact in a space. The moment about a coordinate axis gives you a way to measure how much a system of particles tends to rotate around that axis. For the x-axis, this is often called the y-moment, and for the y-axis, it's called the x-moment.

To find the y-moment, multiply the mass of each particle in the system by its y-coordinate. This product gives you the tendency of each particle to cause rotation about the x-axis. Summing these products for all particles gives the total y-moment. Similarly, to find the x-moment, you multiply each particle's mass by its x-coordinate and sum them up for all particles. This product represents the tendency to cause rotation around the y-axis.

Moments help us investigate the balance and distribution of particles in a space, ensuring systems are stable and well understood.

Moment of a Particle

A moment, in the context of particle mechanics, refers to the measure of a particle's distribution relative to a reference point or axis. Essentially, it's a way of understanding how 'spread out' the mass of a particle system is around an axis or point. With each particle in a system, its moment about an axis is calculated by multiplying its mass by its perpendicular distance from the axis, which is often one of its coordinates.

By calculating the moment of each individual particle, you identify how much influence each particle has over rotational motions in the system. A significant moment indicates that a particle, due to its mass and position, has a high potential to cause rotation about that axis. Conversely, a small moment implies little influence.

In practice, summing these individual moments allows us to find the total moment of the entire system about a specific axis, providing insights into the system's balance and rotational dynamics.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, lets us use coordinates to describe and analyze geometric shapes and their properties. In the context of a system of particles, it provides a way to visualize and calculate the positions of each particle in a coordinate plane and understand how these positions contribute to the overall system's properties.

The coordinates of each particle are used as critical inputs when calculating moments and center of mass, as they determine each particle's relative position. Coordinate geometry uses simple algebraic techniques to find distances, angles, and midpoints, making it an essential tool for calculating moments and the center of mass.

By representing the particles on a coordinate plane, this branch of mathematics makes complex spatial problems more approachable and solvable, employing methods like the Pythagorean theorem and Cartesian coordinates to derive meaningful physical insights.

System of Particles

A system of particles is a collection of masses located at specific points in space. These systems are foundational in physics and engineering because they exist in multiple practical scenarios, from atoms in a molecule to stars in a galaxy. Understanding the system's properties, like its center of mass and moments, is crucial for predicting and explaining physical behavior.

In a particle system, each particle contributes to the overall mass and inertia. The system's total behavior can be understood by summing up the contributions of all individual particles. For instance, finding the center of mass involves summing the products of masses and coordinates and dividing by the total mass. This point acts as an average position of the mass distribution and is where the entire mass of the system could be considered concentrated for analysis.

By analyzing systems of particles, scientists and engineers can solve complex problems involving gravitational forces, rotational motions, and other dynamic behaviors occurring in multi-particle systems. Using systematic methods to quantify these aspects ensures effective design and analysis in various fields.