Problem 27 - Calculus Early Transcendentals

Question

Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The upper half of the ball \(x^{2}+y^{2}+z^{2} \leq 16(\text { for } z \geq 0)\)

Step-by-Step Solution

Answer

Question: Find the centroid of the upper half of a ball with constant density and the equation \(x^2 + y^2 + z^2 \leq 16\). Sketch the region and indicate the centroid location. Answer: The centroid of the upper half of the ball is located at the point \((0, 0, \bar{z})\), where \(\bar{z}\) is the average z-coordinate for all points in the solid. Due to symmetry, \(\bar{x} = \bar{y} = 0\). To calculate \(\bar{z}\), use the integral formula: \(\bar{z} = \frac{1}{V} \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{0}^{4} r \cos (\phi) \, r^2 \sin (\phi) dr d\phi d\theta\) After evaluating this integral and finding \(\bar{z}\), sketch the upper half of the ball with the center at the origin and radius of 4, and mark the centroid location.

Step 1: Set up Spherical Coordinates

In spherical coordinates, we have: \(r^2 = x^2 + y^2 + z^2\) \(\theta\) is the angle in the xy-plane measured from the positive x-axis. \(\phi\) is the angle measured from the positive z-axis to the radial line. With these conventions, we have the following transformation: \(x = r \sin (\phi) \cos (\theta)\) \(y = r \sin (\phi) \sin (\theta)\) \(z = r \cos (\phi)\) For the given solid, since it is an upper half of the ball, we have \(0 \leq r \leq 4\), \(0 \leq \theta \leq 2\pi\), and \(0 \leq \phi \leq \pi/2\).

Step 2: Calculate Mass

Since the density is constant, the mass is simply the volume of the upper half-ball. To find the volume, we integrate over the region: \(V = \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{0}^{4} r^2 \sin (\phi) dr d\phi d\theta\)

Step 3: Calculate Centroid Coordinates

To find the centroid coordinates, we want to find the average x, y, and z coordinates for all points in the solid. We will calculate the individual coordinates as: \(\bar{x} = \frac{1}{V} \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{0}^{4} x \, r^2 \sin (\phi) dr d\phi d\theta\) \(\bar{y} = \frac{1}{V} \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{0}^{4} y \, r^2 \sin (\phi) dr d\phi d\theta\) \(\bar{z} = \frac{1}{V} \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{0}^{4} z \, r^2 \sin (\phi) dr d\phi d\theta\) Due to symmetry, \(\bar{x} = \bar{y} = 0\). So we only need to find \(\bar{z}\).

Step 4: Find \(\bar{z}\)

We can see that: \(\bar{z} = \frac{1}{V} \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{0}^{4} r \cos (\phi) \, r^2 \sin (\phi) dr d\phi d\theta\) Evaluate this integral to find \(\bar{z}\).

Step 5: Sketch the Region and Indicate Centroid Location

Once \(\bar{z}\) is found, we have the centroid location as \((\bar{x}, \bar{y}, \bar{z}) = (0, 0, \bar{z})\). Sketch the upper half of the ball with the center at the origin and the radius of 4, then mark the centroid location.