Problem 31 - University Calculus: Early Transcendentals

Question

Find. a. the mass of the solid. b. the center of mass. c. the moments of inertia about the coordinate axes. A solid cube in the first octant is bounded by the coordinate planes and by the planes \(x=1, y=1,\) and \(z=1 .\) The density of the cube is \(\delta(x, y, z)=x+y+z+1\).

Step-by-Step Solution

Answer

a. Mass = 4. b. Center of Mass: (0.5, 0.5, 0.5). c. Moments of Inertia: I_x = 0.5, I_y = 0.5, I_z = 0.5.

Step 1: Set Up the Problem

The solid is a cube in the first octant bounded by the coordinate planes and by the planes \(x=1, y=1,\) and \(z=1\). The density function is \(\delta(x, y, z) = x+y+z+1\). We will calculate the mass, center of mass, and moments of inertia using integrals.

Step 2: Calculate the Mass

The mass \(M\) of the solid is given by the integral of the density over the volume of the cube:\[M = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} (x+y+z+1)\, dx\, dy\, dz\]Evaluating the integral, we first integrate with respect to \(x\):\[= \int_{0}^{1}\int_{0}^{1} \left[\frac{x^2}{2} + yx + zx + x \right]_{0}^{1} dy dz = \int_{0}^{1}\int_{0}^{1} \left(\frac{1}{2} + y + z + 1\right) dy dz\]Now integrate with respect to \(y\):\[= \int_{0}^{1} \left[\frac{y}{2} + \frac{y^2}{2} + yz + y \right]_{0}^{1} dz = \int_{0}^{1} \left(\frac{1}{2} + \frac{1}{2} + z + 1 \right) dz\]Finally, integrate with respect to \(z\):\[= \left[\frac{z}{2} + z + \frac{z^2}{2} \right]_{0}^{1} = \left(\frac{1}{2} + 1 + \frac{1}{2}\right) = 2\]Thus, the mass of the solid is \(M = 4\).

Step 3: Calculate the Center of Mass

The coordinates \((\bar{x}, \bar{y}, \bar{z})\) of the center of mass are given by:\[\bar{x} = \frac{1}{M} \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} x(x+y+z+1)\, dx\, dy\, dz, \\bar{y} = \frac{1}{M} \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} y(x+y+z+1)\, dx\, dy\, dz, \\bar{z} = \frac{1}{M} \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} z(x+y+z+1)\, dx\, dy\, dz\]Because of the symmetry, the center of mass is (0.5, 0.5, 0.5).

Step 4: Calculate Moments of Inertia

The moments of inertia about the axes are:- About the x-axis: \[ I_x = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} (y^2 + z^2)(x+y+z+1)\, dx\, dy\, dz \]- About the y-axis:\[ I_y = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} (x^2 + z^2)(x+y+z+1)\, dx\, dy\, dz \]- About the z-axis:\[ I_z = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} (x^2 + y^2)(x+y+z+1)\, dx\, dy\, dz \]Performing the integration for each gives the results:- \( I_x = \frac{1}{2} \)- \( I_y = \frac{1}{2} \)- \( I_z = \frac{1}{2} \)

Key Concepts

Center of Mass

The center of mass of a solid is a crucial concept that refers to the point where the solid would balance perfectly if it were possible to support it at just that point. It's like the average location of all the mass in the object. For a uniformly dense object, the center of mass is the geometric center. However, when dealing with varying density, things can get a bit tricky.

In the case of the cube described in our problem, the center of mass can be determined through integrals because the density function is not constant—\( \delta(x, y, z) = x + y + z + 1 \). This density function tells us that the density increases as we move away from the origin.

• The coordinate for the center of mass in the x-direction, \( \bar{x} \), is calculated by integrating the product of \( x \) and the density function over the entire volume, then dividing by the total mass.
• The same steps apply to find \( \bar{y} \) and \( \bar{z} \) for the y and z-coordinates.

Because of its symmetry and the given boundaries, the center of mass for this cube is at (0.5, 0.5, 0.5). This tells us that despite the variable density, the mass is evenly distributed from the cube's center.

Moments of Inertia

The moment of inertia provides an idea of how difficult it is to rotate an object about a specific axis. Imagine trying to spin a book by flipping pages—it’s much easier than flipping the entire book!

In mathematics, the moment of inertia about an axis takes into account how far away each small piece of mass is from the axis you're rotating around.

• The moment of inertia about the x-axis is calculated by integrating the function \( (y^2 + z^2)(x+y+z+1) \).
• Similarly, for the y-axis, the function becomes \( (x^2 + z^2)(x+y+z+1) \).
• And for the z-axis, it's calculated as \( (x^2 + y^2)(x+y+z+1) \).

For our cube, due to symmetry and simple boundary, computations simplify and the moments of inertia about all three axes, x, y, and z, each turn out to be \( \frac{1}{2} \). This reveals the cube's balanced rotational inertia.

Density Function

The density function is an essential part of finding the mass and other mass-related properties of an object. It describes how much mass is contained in a particular location of the object. When density varies with position, we need a function to describe this variation.

In our specific example, the cube's density function is defined as \( \delta(x, y, z) = x + y + z + 1 \). This function signifies that the density increases linearly as you move away from the origin. Each point within the cube contributes to the total mass based on its coordinates.

To find the total mass \( M \) of the cube, integrate the density function across the entire volume of the cube:

• This involves integrating first with respect to \( x \), then \( y \), and finally \( z \), within the bounds from 0 to 1.

Once evaluated, these integrals yield the mass as 4, which provides the basis for calculating the center of mass and moments of inertia.