Question

Find the coordinates of the center of mass of the following solids with variable density. The interior of the prism formed by \(z=x, x=1, y=4,\) and the coordinate planes with \(\rho(x, y, z)=2+y\)

Short answer

Answer: The coordinates of the center of mass of the solid are (3.2, 0.1833, 0.4).

Step-by-step solution

Find the limits of integration

We need to analyze the limits of integration for x, y, and z. The limits for x are given by the planes \(x=0\) and \(x=1\). The limits for y are given by the plane \(y=4\) and the y-axis. The limits for z are determined by the equation \(z=x\).

Calculate the total mass(M)

To calculate the total mass(M), we integrate the density function over the entire solid. $$ M = \iiint_S \rho(x,y,z) \,dx \,dy \,dz $$ Using the limits of integration obtained in Step 1, the integral becomes: $$ M = \int_{0}^{1} \int_{0}^{4} \int_{0}^{x} (2+y) \,dz \,dy \,dx $$

Calculate the moments of mass (M_x, M_y, M_z)

To calculate the moments of mass with respect to the coordinate planes, we integrate the product of the density function and the distance from each coordinate plane over the entire solid. $$ M_x = \iiint_S y\rho(x,y,z) \,dx \,dy \,dz $$ $$ M_y = \iiint_S z\rho(x,y,z) \,dx \,dy \,dz $$ $$ M_z = \iiint_S x\rho(x,y,z) \,dx \,dy \,dz $$ Using the limits of integration and the density function from Steps 1 and 2, the integrals become: $$ M_x = \int_{0}^{1} \int_{0}^{4} \int_{0}^{x} y(2+y) \,dz \,dy \,dx $$ $$ M_y = \int_{0}^{1} \int_{0}^{4} \int_{0}^{x} z(2+y) \,dz \,dy \,dx $$ $$ M_z = \int_{0}^{1} \int_{0}^{4} \int_{0}^{x} x(2+y) \,dz \,dy \,dx $$

Compute and simplify the integrals M, M_x, M_y, and M_z

After computing and simplifying the triple integrals from Step 3, we get the following results: $$ M = 10 \\ M_x = 32 \\ M_y = \frac{11}{6} \\ M_z = 4 $$

Calculate the coordinates of the center of mass (x_CM, y_CM, z_CM)

To find the coordinates of the center of mass, we must divide the moments of mass by the total mass: $$ x_{CM}=\frac{M_x}{M}, y_{CM}=\frac{M_y}{M}, z_{CM}=\frac{M_z}{M} $$ Plugging in the values obtained in Step 4, we get $$ x_{CM} = \frac{32}{10}, \quad y_{CM} = \frac{\frac{11}{6}}{10}, \quad z_{CM} = \frac{4}{10} $$ Simplifying the above expressions gives: $$ x_{CM} = 3.2, \quad y_{CM} = 0.1833, \quad z_{CM} = 0.4 $$ Therefore, the coordinates of the center of mass of the solid are \((3.2, 0.1833, 0.4)\).

Key concepts

Triple Integration

Triple integration is a powerful mathematical tool to compute the volume or mass of a three-dimensional region. In the context of finding the center of mass for solids, triple integration helps us accumulate the small masses throughout the space into a total mass and moments needed for finding coordinates of the center of mass.

Here's how it works:

• We define the region of integration using the boundaries provided by the problem. In this case, these are the planes described by equations like \(z = x\) and limits like \(x=0\) and \(x=1\).
• The triple integral lets us evaluate the density function over these bounds. This accounts for every infinitesimally small piece of the solid's volume.
• The innermost integral traverses one direction, computing the integral slice by slice, while the outer integrals progress through the remaining directions.

Breaking down complex volumes through triple integration simplifies calculating properties like mass—ensuring every part is considered, regardless of irregular shape or density.

Variable Density

A solid with variable density has different density values throughout its volume. This contrasts with constant density solids where the distribution of mass is uniform. For the given exercise, the density function is given by \(\rho(x, y, z) = 2 + y\).

• Variable density can depend on any or all spatial coordinates \(x\), \(y\), and \(z\).In this task, the density increases linearly with the \(y\)-coordinate, meaning the mass is greater where \(y\) is higher.
• This variation requires modifying how we compute integrals, as seen in the steps provided. Each part of the solid is weighed according to its respective density value.

When considering variable density in calculations, it gives a more realistic understanding of how materials behave, especially when they are not uniform in weight distribution across the space.

Moments of Mass

Moments of mass are crucial for determining the center of mass of an object. They give us information about how the mass is distributed relative to the coordinate planes.

• **\(M_x\)**: This is the moment of mass with respect to the \(yz\)-plane. It shows how much the mass is spread out along the \(x\)-axis.
• **\(M_y\)**: The moment of mass relative to the \(xz\)-plane, affecting how mass distribution relates to the \(y\)-axis.
• **\(M_z\)**: This represents the moment of mass relative to the \(xy\)-plane, depicting mass distribution along the \(z\)-axis.

To calculate these moments, we integrate the product of the density function and the distance from the respective plane over the whole solid. Mathematically, these are represented by: \[ M_x = \iiint_S y\rho(x,y,z) \,dx \,dy \,dz \ M_y = \iiint_S z\rho(x,y,z) \,dx \,dy \,dz \ M_z = \iiint_S x\rho(x,y,z) \,dx \,dy \,dz \]The coordinate of the center of mass in each direction is found by dividing each moment by the total mass \(M\). This method accounts for different densities and geometric distribution, culminating in calculating a specific point—the center of mass of the solid.