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.*

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