Question

Find the coordinates of the center of mass of the following solids with variable density. The region bounded by the paraboloid \(z=4-x^{2}-y^{2}\) and \(z=0\) with \(\rho(x, y, z)=5-z\)

Short answer

Question: Find the coordinates of the center of mass of the given solid with density function ρ(x, y, z) = 5-z and region bounded by the paraboloid z = 4-x²-y² and z = 0. Answer: The coordinates of the center of mass of the given solid are (0, 0, z_cm), where z_cm is given by the expression: \(z_{cm} = \frac{\int_{0}^{4}\int_{0}^{2\pi}\int_{0}^{\sqrt{4-z}} z(5-z)r\:dr\:d\theta\:dz}{\int_{0}^{4}\int_{0}^{2\pi}\int_{0}^{\sqrt{4-z}} (5-z)r\:dr\:d\theta\:dz}\) After evaluating the integral expressions, we will find the z-coordinate of the center of mass.

Step-by-step solution

STEP 1: Convert density to mass and set up integral bounds

In our case, we have the density function \(\rho(x, y, z)=5-z\) and the region bounded by the paraboloid \(z=4-x^{2}-y^{2}\) and \(z=0\). We will first convert the density into mass dm by multiplying the density by the volume element. Using cylindrical coordinates \(x = r\cos{\theta}\), \(y = r\sin{\theta}\) and \(z = z\), the volume element is \(dV = r\:dr\:d\theta\:dz\). Next, we will set up the integral bounds for r, θ, and z as we will be integrating over cylindrical coordinates to find the mass.

STEP 2: Find mass of the solid

To find the mass of the solid, integrate the product of the density function and the volume element over the entire solid. This gives: \(M = \int\int\int_{solid} 5-z\:rd\:rd\theta\:dz\) \(r\) has a range of \(0\) to \(\sqrt{4-z}\), \(\theta\) is from \(0\) to \(2\pi\), and \(z\) ranges from \(0\) to \(4\). So, the mass of the solid is: \(M = \int_{0}^{4}\int_{0}^{2\pi}\int_{0}^{\sqrt{4-z}} (5-z)r\:dr\:d\theta\:dz\)

STEP 3: Find the first Cartesian moments

The first Cartesian moments can be found by integrating the product of the density function, the respective coordinate variable, and the volume element over the solid. For x and y, this requires calculating: \(M_x = \int\int\int_{solid} x(5-z)r\:dr\:d\theta\:dz\) \(M_y = \int\int\int_{solid} y(5-z)r\:dr\:d\theta\:dz\) For z: \(M_z = \int\int\int_{solid} z(5-z)r\:dr\:d\theta\:dz\)

STEP 4: Find the coordinates of the center of mass

To find the center of mass coordinates, we will divide the first moments by the mass M. This gives: \(x_{cm} = \frac{M_x}{M}\) \(y_{cm} = \frac{M_y}{M}\) \(z_{cm} = \frac{M_z}{M}\) By symmetry, we see that \(x_{cm}\) and \(y_{cm}\) are both zero. Therefore, calculate \(z_{cm}\): \(z_{cm} = \frac{M_z}{M} = \frac{\int_{0}^{4}\int_{0}^{2\pi}\int_{0}^{\sqrt{4-z}} z(5-z)r\:dr\:d\theta\:dz}{\int_{0}^{4}\int_{0}^{2\pi}\int_{0}^{\sqrt{4-z}} (5-z)r\:dr\:d\theta\:dz}\) Now, after evaluating the above integral expressions, we will find the coordinates of the center of mass of the given solid.

Key concepts

Variable Density

Understanding variable density is essential when calculating the center of mass for objects with non-uniform distribution of matter. In this exercise, the density function is given by \( \rho(x, y, z) = 5 - z \), indicating that density varies with the height \( z \) in the solid. This means that as you move along the z-axis, the density decreases linearly.
The density at a given point tells us how much mass exists in a small region around it. In three-dimensional space, this density function helps determine how much each small piece contributes to the total mass and the mass distribution of the object. When the density is uniform, calculations are straightforward. But with a variable density, integrating the density function over the volume becomes necessary to find quantities like total mass.

Cylindrical Coordinates

Cylindrical coordinates are especially useful in solving problems with rotational symmetry, such as those involving paraboloids. In this coordinate system, a point in space is represented by \((r, \theta, z)\), where \( r \) is the radial distance from the z-axis, \( \theta \) is the angle around the z-axis from the positive x-axis, and \( z \) is the height from the xy-plane.
Using cylindrical coordinates simplifies the integration process for certain solids because it aligns more naturally with their geometry. Here, the use of \( x = r\cos{\theta} \), \( y = r\sin{\theta} \), and \( z = z \) allows us to express the bounds and the integrals in a more manageable form. This coordinate system is particularly beneficial when dealing with circular symmetry, as seen in a paraboloid.

Integrals

Integrals are the main mathematical tools needed to calculate the mass and center of mass of an object, especially when it involves variable densities. In this exercise, the integrals are set up to compute the mass and the first moments about the Cartesian axes.
To find the mass \( M \), you integrate the product of the density function and the infinitesimal volume element \( r\,dr\,d\theta\,dz \) over the entire solid. The triple integral \( M = \int_0^4 \int_0^{2\pi} \int_0^{\sqrt{4-z}} (5-z)r\,dr\,d\theta\,dz \) comprehensively covers the entire volume and accounts for the variable density.
Similarly, integrals are used to determine the moments \( M_x, M_y, \) and \( M_z \), which are necessary for calculating the center of mass coordinates.

Paraboloid

A paraboloid in three-dimensional geometry is a surface generated by rotating a parabola around its axis. The specific paraboloid in this problem is defined by \( z = 4 - x^2 - y^2 \), which is a downward-opening paraboloid centered at the origin. The intersection of this paraboloid with \( z = 0 \) defines our solid's boundary.
This shape is symmetrical around the z-axis, which simplifies many computations, especially for finding moments. Due to this symmetry, both \( x_{cm} \) and \( y_{cm} \) will be zero, as the mass distribution equally spreads around the axis. The particular equation \( z = 4 - r^2 \) when converted into cylindrical coordinates also helps establish the limits for integration, with \( r \) ranging from 0 to \( \sqrt{4-z} \). Understanding the geometric and mathematical properties of paraboloids is crucial for solving related problems efficiently.