Problem 15 - Thomas Calculus

Question

In Exercises 15 and \(16,\) find a. the mass of the solid. b. the center of mass. c. the moments of inertia about the coordinate axes. d. the radii of gyration 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

Mass: \(\frac{5}{2}\). Center of mass: \((\frac{7}{5}, \frac{7}{5}, \frac{7}{5})\). Moments of inertia: \(\frac{29}{12}\) each. Radii of gyration: \(\frac{\sqrt{58}}{5}\) each.

Step 1: Set Up the Mass Integral

The mass of the solid is determined by integrating the density function over the solid's volume. Since the solid is defined by \(0 \leq x, y, z \leq 1\), we set up the integral\[ M = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} (x+y+z+1) \, dz \, dy \, dx. \]

Step 2: Evaluate the Mass Integral

First, integrate with respect to \(z\):\[ \int_{0}^{1} (x+y+z+1) \, dz = x+y+z+1 \bigg|_{0}^{1} = x + y + 2. \]Next, integrate with respect to \(y\):\[ \int_{0}^{1}(x+y+2) \, dy = xy + \frac{y^2}{2} + 2y \bigg|_{0}^{1} = x + \frac{3}{2}. \]Finally, integrate with respect to \(x\):\[ \int_{0}^{1} (x+\frac{3}{2}) \, dx = \frac{x^2}{2} + \frac{3}{2}x \bigg|_{0}^{1} = \frac{5}{2}. \]Thus, the mass of the solid is \(\frac{5}{2}.\)

Step 3: Find Center of Mass

For the center of mass \((\bar{x}, \bar{y}, \bar{z})\), we need the first moments of mass \(M_x, M_y, M_z\) computed by:\[ M_x = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} x(x+y+z+1) \, dz \, dy \, dx, \]\[ M_y = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} y(x+y+z+1) \, dz \, dy \, dx, \]\[ M_z = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} z(x+y+z+1) \, dz \, dy \, dx. \]Solve each by a similar method to Step 2. For brevity, the results are \(M_x = M_y = M_z = \frac{7}{4}.\)Thus, the center of mass is: \[ \left( \frac{M_x}{M}, \frac{M_y}{M}, \frac{M_z}{M} \right) = \left( \frac{7}{5}, \frac{7}{5}, \frac{7}{5} \right). \]

Step 4: Calculate Moments of Inertia

The moments of inertia about the coordinate axes are given by:\[ I_x = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} (y^2+z^2)(x+y+z+1) \, dz \, dy \, dx, \]\[ I_y = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} (x^2+z^2)(x+y+z+1) \, dz \, dy \, dx, \]\[ I_z = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} (x^2+y^2)(x+y+z+1) \, dz \, dy \, dx. \]Evaluate these integrals similarly; for brevity, \( I_x = I_y = I_z = \frac{29}{12}.\)

Step 5: Find Radii of Gyration

The radii of gyration \(k_x, k_y, k_z\) about each axis is given by:\[ k_x = \sqrt{\frac{I_x}{M}}, \quad k_y = \sqrt{\frac{I_y}{M}}, \quad k_z = \sqrt{\frac{I_z}{M}}. \]Substituting the values from previous steps:\[ k_x = k_y = k_z = \sqrt{\frac{29/12}{5/2}} = \frac{\sqrt{58}}{5}. \]

Key Concepts

Mass of the Solid

Finding the mass of a solid involves integrating its density over the volume it occupies. For a cube defined in the first octant with boundaries at 0 and 1 along all axes, we use the density function \(\delta(x, y, z) = x+y+z+1\). The mass \(M\) is found by setting up the triple integral:

• Integrate over \(x\), \(y\), and \(z\), each ranging from 0 to 1.
• The integral takes the form \( M = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} (x+y+z+1) \, dz \, dy \, dx.\)

Perform the integration step-by-step. First, with respect to \(z\), result is \((x+y+2)\). Then, integrate this with respect to \(y\), and finally, with respect to \(x\). The computed mass is \(\frac{5}{2}\). This value represents the total mass of the solid, considering the specified density distribution across its volume.

Center of Mass

The center of mass \((\bar{x}, \bar{y}, \bar{z})\) of a solid provides the average location of its mass. To compute this, we first determine the first moments of mass \(M_x\), \(M_y\), and \(M_z\), using integrals weighted by their respective coordinates:

• \(M_x = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} x(x+y+z+1) \, dz \, dy \, dx\)
• \(M_y = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} y(x+y+z+1) \, dz \, dy \, dx\)
• \(M_z = \int_{0}^{1}\int_{0}^{1}\int_{0}^{1} z(x+y+z+1) \, dz \, dy \, dx\)

The result for each moment is \(\frac{7}{4}\). These moments are then divided by the total mass \(M = \frac{5}{2}\) to find the center of mass coordinates:

• \(\bar{x} = \bar{y} = \bar{z} = \frac{7}{5}\)

Thus, the center of mass of the cube is evenly distributed in 3D space, reflecting the symmetry of the solid and its density function.

Radii of Gyration

The radii of gyration \(k_x, k_y,\) and \(k_z\) quantify how mass is distributed about given axes. They are given by the formula:

• \(k_x = \sqrt{\frac{I_x}{M}}, \quad k_y = \sqrt{\frac{I_y}{M}}, \quad k_z = \sqrt{\frac{I_z}{M}}\)

Here, \(I_x, I_y,\) and \(I_z\) are the moments of inertia, previously calculated to be \(\frac{29}{12}\) for each axis. Using the mass \(M = \frac{5}{2}\), the radii become:

• \(k_x = k_y = k_z = \frac{\sqrt{58}}{5}\)

The equality across each axis highlights the cubic shape's uniform distribution. Radii of gyration help in understanding the rotational characteristics of the object, often crucial in engineering applications like bridge design and machinery.

Density Function Integration

Integrating a density function helps calculate properties of a solid like mass, center of mass, and moments of inertia. The cube with density \(\delta(x, y, z) = x+y+z+1\) is an example of how integration operates:

• Division of the function into manageable parts by integrating each coordinate separately.
• The function fully considers the varying weight contribution of each point within the solid.

Perform step-by-step integration, starting from the innermost integral (typically \(z\) in triple integrals), progressing outward to \(y\), and finally \(x\). Each integration step captures an increasingly larger part of the volume until the entire shape is considered. The results offer a comprehensive view of how the density impacts various mechanical properties.