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13. A lamina occupies the part of the disk a? + y <1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the z-axis.

13. A lamina occupies the part of the disk a? + y <1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the z-axis.

Advanced Engineering Mathematics
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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5, 6, Z, 8, 9, 10, 11, and 12 Find the mass and center of mass of the lamina that occupies the region D and has the given density function p. 5. D= {(x, y) | 1<x< 3,1 < y < 4}; p (x, y) = ky? Answer + 6. D= {(x, y)| 0 < x < a, 0 < y < b}; p(x, y) = 1+ a² + y? 7. Dis the triangular region with vertices (0, 0), (2, 1), (0, 3); p (, y) = x + y Answer 8. Dis the triangular region enclosed by the lines y = 0, y = 2a, and a + 2y = 1; p (x, y) = x 9. Dis bounded by y = 1 – a² and y = 0; p (x, y) = ky Answer 10. Dis bounded by y = x + 2 and y = 2² ; p (x, y) = ka² 11. D is bounded by the curves y = e=®, y = 0, x = 0, x = 1; p (x, y) = xy Answer + 12. Dis enclosed by the curves y = 0 and y = cos æ, –1/2 < x < #/2; p (æ, y) = y 13. A lamina occupies the part of the disk a? + y? <1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the æ-axis. Answer
Transcribed Image Text:5, 6, Z, 8, 9, 10, 11, and 12 Find the mass and center of mass of the lamina that occupies the region D and has the given density function p. 5. D= {(x, y) | 1<x< 3,1 < y < 4}; p (x, y) = ky? Answer + 6. D= {(x, y)| 0 < x < a, 0 < y < b}; p(x, y) = 1+ a² + y? 7. Dis the triangular region with vertices (0, 0), (2, 1), (0, 3); p (, y) = x + y Answer 8. Dis the triangular region enclosed by the lines y = 0, y = 2a, and a + 2y = 1; p (x, y) = x 9. Dis bounded by y = 1 – a² and y = 0; p (x, y) = ky Answer 10. Dis bounded by y = x + 2 and y = 2² ; p (x, y) = ka² 11. D is bounded by the curves y = e=®, y = 0, x = 0, x = 1; p (x, y) = xy Answer + 12. Dis enclosed by the curves y = 0 and y = cos æ, –1/2 < x < #/2; p (æ, y) = y 13. A lamina occupies the part of the disk a? + y? <1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the æ-axis. Answer
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