Problem 24 - Calculus Early Transcendentals

Question

Find the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. The upper half \((y \geq 0)\) of the disk bounded by the circle \(x^{2}+y^{2}=4\) with \(\rho(x, y)=1+y / 2\)

Step-by-Step Solution

Answer

Question: Determine the center of mass (x̄, ȳ) of the upper half of a circle with radius 2, centered at the origin, and with the variable density function \(\rho(x, y) = 1 + \frac{y}{2}\). Solution: 1. Convert the given region and density function into polar coordinates: \(0 \leq r \leq 2\) and \(0 \leq \theta \leq \pi\), \(\rho(r, \theta) = 1 + \frac{r}{2} \sin(\theta)\). 2. Find the total mass M by integrating the density function: $$M = \int_{0}^{\pi}\int_{0}^{2} \rho(r, \theta) \, r \, dr d\theta$$ 3. Calculate the first moments \(M_x\) and \(M_y\) about the x and y axes, respectively: $$M_x = \int_{0}^{\pi}\int_{0}^{2} \rho(r, \theta) \, r \sin(\theta) \, r \, dr d\theta$$ $$M_y = \int_{0}^{\pi}\int_{0}^{2} \rho(r, \theta) \, r \cos(\theta) \, r \, dr d\theta$$ 4. Find the center of mass coordinates (x̄, ȳ) by dividing the first moments by the total mass: $$x̄ = \frac{M_x}{M}$$ $$ȳ = \frac{M_y}{M}$$

Step 1: Identify the region and convert to polar coordinates

Since the region is the upper half of a disk with radius 2, it is reasonable to convert to polar coordinates for the integration. The region can be described as \(0 \leq r \leq 2\) and \(0 \leq \theta \leq \pi\). We also need to convert the density function \(\rho(x, y)\) into polar coordinates using \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\): $$\rho(r, \theta) = 1 + \frac{r}{2} \sin(\theta)$$

Step 2: Find the total mass

To find the total mass M, we need to integrate the density over the region: $$M = \int_{0}^{\pi}\int_{0}^{2} \rho(r, \theta) \, r \, dr d\theta$$ To compute this integral, start by integrating with respect to \(r\) and then integrate with respect to \(\theta\).

Step 3: Calculate the first moments

In order to find the center of mass, we need to calculate the first moments about the x and y axes. This requires integrating the product of the density and the distance from the axis over the region. $$M_x = \int_{0}^{\pi}\int_{0}^{2} \rho(r, \theta) \, r \sin(\theta) \, r \, dr d\theta$$ $$M_y = \int_{0}^{\pi}\int_{0}^{2} \rho(r, \theta) \, r \cos(\theta) \, r \, dr d\theta$$ To compute these integrals, follow the same procedure as in calculating the total mass.

Step 4: Find the center of mass coordinates

Divide the first moments by the total mass to get the center of mass coordinates: $$x̄ = \frac{M_x}{M}$$ $$ȳ = \frac{M_y}{M}$$ Compute the numerical coefficients to find the center of mass (x̄, ȳ).