Problem 80 - Thomas Calculus

Question

a. Find the center of mass of a thin plate of constant density covering the region between the curve \(y=1 / \sqrt{x}\) and the \(x-\) axis from \(x=1\) to \(x=16\) . b. Find the center of mass if, instead of being constant, the density function is \(\delta(x)=4 / \sqrt{x}\) .

Step-by-Step Solution

Answer

a. (7, 0), b. \( \left( \frac{15}{\ln 16}, 0 \right) \).

Step 1: Understand the Problem Requirements

We are asked to find the center of mass for a thin plate covering a region between a curve and the x-axis. This involves integrating over the region to find both mass and moments.

Step 2: Set Up the Integral for Total Mass with Constant Density

For part (a) where density is constant, assume density \( \rho \) = 1 for simplicity. The mass \( M \) is the integral of the density over the area. Thus, \( M = \int_{1}^{16} (1/\sqrt{x}) dx \).

Step 3: Calculate Total Mass for Constant Density

Evaluate the integral \( M = \int_{1}^{16} \frac{1}{\sqrt{x}} \, dx \). By substitution, \( \int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} \), so \( M = [2\sqrt{x}]_{1}^{16} = 8 - 2 = 6 \).

Step 4: Set Up Integrals for Moments with Constant Density

For the x-coordinate of the center of mass, calculate the moment about the y-axis: \( M_{x} = \int_{1}^{16} x \cdot \frac{1}{\sqrt{x}} \, dx \).

Step 5: Calculate x-Moment for Constant Density

Evaluate \( M_{x} = \int_{1}^{16} x \cdot \frac{1}{\sqrt{x}} \, dx = \int_{1}^{16} \sqrt{x} \, dx = \frac{2}{3}x^{3/2} \) to get \( M_{x} = \left[\frac{2}{3}x^{3/2}\right]_{1}^{16} = \frac{2}{3}[64 - 1] = \frac{126}{3} = 42 \).

Step 6: Calculate Center of Mass for Constant Density

The x-coordinate of the center of mass is \( \bar{x} = \frac{M_{x}}{M} = \frac{42}{6} = 7 \). For constant density, the y-coordinate doesn’t affect the integration here. The center of mass is (7, 0).

Step 7: Modify Density Function and Set Up New Mass Integral

For part (b), change the density to \( \delta(x) = \frac{4}{\sqrt{x}} \). Find the new mass \( M = \int_{1}^{16} \frac{4}{\sqrt{x}} \frac{1}{\sqrt{x}} \, dx = \int_{1}^{16} \frac{4}{x} \, dx \).

Step 8: Calculate Total Mass for Variable Density

Evaluate \( M = 4 \int_{1}^{16} \frac{1}{x} \, dx = 4[\ln x]_{1}^{16} = 4(\ln 16 - \ln 1) = 4 \ln 16 \).

Step 9: Set Up Integrals for Moments with Variable Density

Find the x-moment \( M_{x} = \int_{1}^{16} x \cdot \frac{4}{\sqrt{x}} \frac{1}{\sqrt{x}} \, dx = \int_{1}^{16} 4 \, dx = 4[x]_{1}^{16} \).

Step 10: Calculate x-Moment for Variable Density

Evaluate \( M_{x} = 4[16 - 1] = 4 \times 15 = 60 \).

Step 11: Calculate Center of Mass for Variable Density

The x-coordinate of the center of mass is \( \bar{x} = \frac{M_{x}}{M} = \frac{60}{4 \ln 16} = \frac{15}{\ln 16} \). For the y-coordinate, it again does not affect the integration. The center of mass is \( \left( \frac{15}{\ln 16}, 0 \right) \).

Key Concepts

Calculus Integration

In calculus, integration is a fundamental concept often used to find areas under curves or to calculate total values such as mass or moments of mass in physical systems. Specifically, an integral sums up infinitesimally small quantities over a given range to produce a cumulative outcome. Integration involves the calculation of definite integrals over specific intervals. In the context of the center of mass, it's used to aggregate the effects of density and shape across a region. For example, in the given exercise, we utilized the integral to determine the mass of a solid by integrating the density across the defined area of the plate. With integration,

• We established the total mass by integrating the density function.
• We computed moments by integrating weighted positions, which contribute to the understanding of the overall balance of the shape.

The definite integrals in this problem, like \( \int_{1}^{16} \frac{1}{\sqrt{x}} \, dx \), are calculated to determine these quantities accurately. Mastering integration provides a powerful tool for analyzing variable regions and forms in physics and engineering.

Variable Density

When dealing with physics problems involving objects like plates, density is not always constant. It may vary with position, which is what we refer to as variable density. This characteristic means that some parts of the object could be denser than others, affecting calculations of mass and other properties. In the given problem, the density function is altered from constant to a function \( \delta(x) = \frac{4}{\sqrt{x}} \) for part (b). This variable density means:

• The density increases or decreases along the region's span.
• The calculation of mass is not straightforward like in the constant density case and requires careful integration handling.

When density varies, the total mass \( M \) is determined by integrating the density function over the specified range, which involves evaluating expressions like \( \int_{1}^{16} \frac{4}{x} \, dx \). This ensures that each part of the region contributes to the mass based on its density at that point, affecting the final calculation of the center of mass.

Moments of Mass

In the context of finding a center of mass, moments of mass are essential concepts. They represent the rotational analogs of mass that account for the distribution of mass at various points relative to a chosen axis. There are several key points to remember:

• The moment about an axis provides insight into the balance or rotational behavior of an object.
• In this exercise, the moment about the y-axis \( M_x \) was calculated to find the x-coordinate of the center of mass.

To compute moments, you integrate the position-weighted density across the region. For instance, when density was constant, the x-moment was \( M_x = \int_{1}^{16} x \cdot \frac{1}{\sqrt{x}} \, dx \). For the variable density, it shifted to \( M_x = \int_{1}^{16} 4 \, dx \). These integrals provide the lever effect that a mass distribution exerts relative to a point or axis. Understanding and applying the concept of moments allows us to determine how different parts of a system influence its overall behavior and is crucial in statics and dynamics in physics and engineering.