Find Mx, My, and (x, y) for the laminas of uniform density p bounded by the graphs of the equations. y = √√√x, y = 0, x = 4 12p 5 Mx = My = (x, y) = 64p 5 1/2, 3/4 5 X X
Find Mx, My, and (x, y) for the laminas of uniform density p bounded by the graphs of the equations. y = √√√x, y = 0, x = 4 12p 5 Mx = My = (x, y) = 64p 5 1/2, 3/4 5 X X
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Calculating Moments and the Center of Mass for Laminas of Uniform Density
To find the moments \(M_x\), \(M_y\), and the center of mass \((\bar{x}, \bar{y})\) for laminas of uniform density \( \rho \) bounded by the given equations, follow the steps below.
Given equations:
\[ y = \sqrt{x}, \ y = 0, \ x = 4 \]
#### Steps:
1. **Moment About the x-axis ( \(M_x\) ) Calculation:**
\[
M_x = \frac{12\rho}{5} \quad \textcolor{red}{\text{Incorrect}}
\]
2. **Moment About the y-axis ( \(M_y\) ) Calculation:**
\[
M_y = \frac{64\rho}{5} \quad \textcolor{red}{\text{Incorrect}}
\]
3. **Center of Mass (\((\bar{x}, \bar{y})\)) Calculation:**
\[
(\bar{x}, \bar{y}) = \left( \frac{12}{5}, \frac{3}{4} \right) \quad \textcolor{green}{\text{Correct}}
\]
#### Explanation:
- **Moments \(M_x\) and \(M_y\)**:
- The expressions for \(M_x\) and \(M_y\) provided in this example are not correct. Correct computation typically involves integrating over the region bounded by the curves to compute the area moments.
- **Center of Mass (\((\bar{x}, \bar{y})\))**:
- The center of mass is accurately computed, giving the coordinates \(( \bar{x}, \bar{y} ) = \left( \frac{12}{5}, \frac{3}{4} \right)\).
- This calculation takes into consideration the correct placement within the context defined by the bounding equations.
### Summary:
While the moments \(M_x\) and \(M_y\) provided seem not correctly computed, the calculation for the center of mass (\((\bar{x}, \bar{y})\)) is accurate. For accurate moments, re-evaluation of the integrals over the defined region is necessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fdef185-de8b-42f5-933e-8018dcf36541%2Fbb29640b-6329-4f4c-95a4-2ed8ba6b8518%2Fhft51pa_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating Moments and the Center of Mass for Laminas of Uniform Density
To find the moments \(M_x\), \(M_y\), and the center of mass \((\bar{x}, \bar{y})\) for laminas of uniform density \( \rho \) bounded by the given equations, follow the steps below.
Given equations:
\[ y = \sqrt{x}, \ y = 0, \ x = 4 \]
#### Steps:
1. **Moment About the x-axis ( \(M_x\) ) Calculation:**
\[
M_x = \frac{12\rho}{5} \quad \textcolor{red}{\text{Incorrect}}
\]
2. **Moment About the y-axis ( \(M_y\) ) Calculation:**
\[
M_y = \frac{64\rho}{5} \quad \textcolor{red}{\text{Incorrect}}
\]
3. **Center of Mass (\((\bar{x}, \bar{y})\)) Calculation:**
\[
(\bar{x}, \bar{y}) = \left( \frac{12}{5}, \frac{3}{4} \right) \quad \textcolor{green}{\text{Correct}}
\]
#### Explanation:
- **Moments \(M_x\) and \(M_y\)**:
- The expressions for \(M_x\) and \(M_y\) provided in this example are not correct. Correct computation typically involves integrating over the region bounded by the curves to compute the area moments.
- **Center of Mass (\((\bar{x}, \bar{y})\))**:
- The center of mass is accurately computed, giving the coordinates \(( \bar{x}, \bar{y} ) = \left( \frac{12}{5}, \frac{3}{4} \right)\).
- This calculation takes into consideration the correct placement within the context defined by the bounding equations.
### Summary:
While the moments \(M_x\) and \(M_y\) provided seem not correctly computed, the calculation for the center of mass (\((\bar{x}, \bar{y})\)) is accurate. For accurate moments, re-evaluation of the integrals over the defined region is necessary.
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