The boundary of a lamina consists of the semicircles y-V1- x and y V9 - x together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is inversely proportional to its distance from the origin. Hint: use polar coordinates K,) - (0,4
The boundary of a lamina consists of the semicircles y-V1- x and y V9 - x together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is inversely proportional to its distance from the origin. Hint: use polar coordinates K,) - (0,4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Mathematics: Calculating the Center of Mass**
The boundary of a lamina consists of the semicircles \( y = \sqrt{1 - x^2} \) and \( y = \sqrt{9 - x^2} \) together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is inversely proportional to its distance from the origin.
**Hint:** Use polar coordinates.
**Solution:**
\[ (\bar{x}, \bar{y}) = \left( 0, 4 \right) \]
*In this problem, we will explore how to use polar coordinates to determine the center of mass for a shape with variable density.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f9c0ebc-093a-4eb3-8ee9-586c61993ff9%2F8f6a1b08-a8e4-4cd5-9b31-542fe8735be8%2Fhdv50ve_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics: Calculating the Center of Mass**
The boundary of a lamina consists of the semicircles \( y = \sqrt{1 - x^2} \) and \( y = \sqrt{9 - x^2} \) together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is inversely proportional to its distance from the origin.
**Hint:** Use polar coordinates.
**Solution:**
\[ (\bar{x}, \bar{y}) = \left( 0, 4 \right) \]
*In this problem, we will explore how to use polar coordinates to determine the center of mass for a shape with variable density.*
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