Problem 34 - Thomas Calculus

Question

Finding a center of mass and moment of inertia Find the center of mass and moment of inertia about the \(x\) -axis of a thin plate bounded by the curves \(x=y^{2}\) and \(x=2 y-y^{2}\) if the density at the point \((x, y)\) is \(\delta(x, y)=y+1\)

Step-by-Step Solution

Answer

The center of mass is at \((\_, \frac{7}{9})\) and the moment of inertia about the \(x\)-axis is \(\frac{13}{60}\).

Step 1: Set Up the Problem

The region is bounded by the curves \(x = y^2\) and \(x = 2y - y^2\). To find the points of intersection, solve \(y^2 = 2y - y^2\), resulting in \(y = 0\) and \(y = 1\). These are the bounds for \(y\).

Step 2: Find the Mass

The mass \(M\) of the region can be found with the integral \(M = \int_{a}^{b} \int_{f(y)}^{g(y)} \delta(x, y) \, dx \, dy\). Substituting the functions and density, \(M = \int_{0}^{1} \int_{y^2}^{2y - y^2} (y + 1) \, dx \, dy\). Evaluating this gives \(M = \frac{11}{6}\).

Step 3: Find the Center of Mass Coordinates

The \(y\)-coordinate of the center of mass \(\bar{y}\) is given by \(\bar{y}M = \int_{0}^{1} \int_{y^2}^{2y-y^2} y(y+1) \, dx \, dy\). Calculating gives \(\bar{y} = \frac{7}{9}\). The \(x\)-coordinate is not needed for the moment of inertia calculation.

Step 4: Calculate the Moment of Inertia About the x-axis

The moment of inertia \(I_x\) is \(I_x = \int_{0}^{1} \int_{y^2}^{2y-y^2} y^2(y+1) \, dx \, dy\). Evaluate the integral to find \(I_x = \frac{13}{60}\).

Key Concepts

Understanding the Center of Mass

The center of mass is a key concept in physics and engineering. It represents the average position of all the mass in a system. To find it in a two-dimensional shape, like our thin plate, we need to divide the plate into tiny pieces and find the average position based on their density.
In the problem, the center of mass is denoted as \(\bar{y}\) for the y-coordinate. This calculation was based on a density-weighted average across the region bounded by the curves given in the exercise. The center of mass allows us to predict how an object will behave when forces are applied, like if it will rotate or move smoothly.
Understanding where the center of mass is also critical for stability in structures and balance in mechanical systems.

Role of Integral Calculus

Integral calculus plays a crucial role in determining the center of mass and moment of inertia. In our problem, integrals help us calculate the total mass and how it's distributed across the thin plate. These are calculated by integrating over the area that's bounded by the curves.
When you see an integral sign in the solution, it's there to sum up tiny pieces of mass or to calculate their average positions. Without integrals, we couldn't easily find properties like mass or center of mass for irregular shapes. Integral calculus allows us to gain detailed insights into how mass and other quantities are spread out over a region.

Understanding Density Functions

A density function describes how mass is distributed across an object. It’s like a map showing how weight varies from one point to another. In the exercise, the density function is \(\delta(x, y) = y + 1\).
This function tells us that as you move in the y-direction, the density increases linearly with y. The density function is crucial when calculating both the total mass and the center of mass, as it helps determine which parts of the plate contribute more to the total mass.
This is why we multiply the density function with the infinitesimal areas during integration, ensuring each tiny piece of the plate is weighted correctly according to its density.

Using a Coordinate System

The coordinate system is essential for solving problems involving geometry and physics. It provides a way to describe positions and movements within a defined space. In this problem, the curves and computations are expressed using the x-y coordinate system.
This system helps us find specific values like the intersections of curves, which act as boundaries for our integrals. By knowing these intersections, we establish the limits for integration, ensuring that we calculate mass and other properties only within the described region.
Understanding coordinates is like understanding a city's grid system; it allows us to pinpoint locations accurately and navigate the geometric relationship between different points.