Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

The masses and coordinates of a system of particles in the coordinate plane are given by the following: \(2,(1,1) ; 3,(7,1) ;\) 4, \((-2,-5) ; 6,(-1,0) ; 2,(4,6)\). Find the moments of this system with respect to the coordinate axes, and find the coordinates of the center of mass.

Step-by-Step Solution

Answer

The center of mass is approximately at (1, -0.18).

Step 1: Determine the Moment with Respect to the x-Axis

The moment with respect to the x-axis is found by summing up the products of each mass and its y-coordinate. Using the formula: \[ M_x = \sum_{i=1}^{n} m_i y_i \] where \(m_i\) are the masses and \(y_i\) are the y-coordinates of the particles:\[M_x = 2 \times 1 + 3 \times 1 + 4 \times (-5) + 6 \times 0 + 2 \times 6 \]\[M_x = 2 + 3 - 20 + 0 + 12 \]\[M_x = -3\].

Step 2: Determine the Moment with Respect to the y-Axis

The moment with respect to the y-axis is computed by summing up the products of each mass and its x-coordinate. Using the formula: \[ M_y = \sum_{i=1}^{n} m_i x_i \]where \(m_i\) are the masses and \(x_i\) are the x-coordinates of the particles:\[M_y = 2 \times 1 + 3 \times 7 + 4 \times (-2) + 6 \times (-1) + 2 \times 4 \]\[M_y = 2 + 21 - 8 - 6 + 8 \]\[M_y = 17\].

Step 3: Calculate the Total Mass of the System

The total mass of the system is the sum of all individual masses:\[M = 2 + 3 + 4 + 6 + 2 = 17\].

Step 4: Find the Coordinates of the Center of Mass

Using the moments and the total mass, find the coordinates \((\bar{x}, \bar{y})\) of the center of mass with the formulas:\[ \bar{x} = \frac{M_y}{M} \]\[ \bar{y} = \frac{M_x}{M} \]Substituting the values calculated earlier:\[ \bar{x} = \frac{17}{17} = 1 \]\[ \bar{y} = \frac{-3}{17} \approx -0.18\].

Key Concepts

Moments with Respect to Coordinate Axes

The concept of moments with respect to coordinate axes involves calculating the effect of a particle system's distribution in space. To find the moment with respect to the x-axis, each particle's mass is multiplied by its y-coordinate. Think of it as determining the rotational effect around the x-axis. For the y-axis, each mass is multiplied by its x-coordinate. This helps in understanding how the system would rotate around the y-axis. These calculations help pinpoint the system's tendency to balance and rotate, essentially summarizing the system's spatial setup in terms of a central pivot point.

Moments offer great insights into the structural strengths and stability of systems by quantifying how weight is distributed across different axes.

Mass and Coordinates of Particles

Every particle in the system carries a certain mass and occupies a specific position in the coordinate plane. The mass of each particle can be thought of as the weightiness or gravitational effect it contributes to the entire system. Each one is characterized by coordinates

• x, the horizontal position
• y, the vertical position

It's crucial to carefully note these positions as they influence where the center of mass will be located. This data essentially serves as the foundation for any further analysis. Once the individual positions are laid out, it becomes easier to predict how they interact spatially in the plane. Understanding each particle's contribution is the key first step in determining more intricate properties such as moments and the center of mass.

Coordinate Plane

The coordinate plane is a two-dimensional space, defined by x and y axes, where the particles reside. Imagine a big graph paper where each dot or point corresponds to the position of a particle in this space. This plane allows for a visual representation of the spatial arrangement, making it easier to apply mathematical formulas to determine system properties.

By placing mass points on this plane, spatial symmetry or distribution trends emerge. The use of this plane also aids in visualizing the concept of moments and the center of mass. It becomes a crucial interaction field where geometry marries algebra to offer insights into the system dynamics.

Moment Calculations

Moment calculations involve a systematic approach to summing up contributions of masses multiplied by their respective positions. For the x-axis, multiply each mass by its y-coordinate. Similarly, the y-axis moment involves multiplying each mass by its x-coordinate. Sum these products across all particles to get the total moments, noted as \(M_x\) and \(M_y\).

These computed values compress the entire complexity of the particle system into straightforward numbers that can be used in further calculations like finding the center of mass. This makes moment calculations indispensable for understanding the balance and orientation of particle systems in applied settings, like engineering and physics.