Question

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4. A lamina occupies the region inside the circle x2 + y2 = 2y but outside the circle x2 + y2 =1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin

Answer 1

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solution : carte sian The and relation polar between Coordinates is, 2 rcoso y =rsing és 82= n²+yz. Hence the equation of the the circle 2²ty2 =24 be comes,

х2 = arsind. r = => 2 sing ว x² + y2 and be comes, re=1 ㅋ rol ران let is find the Now Point inters section cirles between this two from ů and ll 1 can write 1 = asino => sino b. now here circle oco <QD for these tho Hence sino 솔 2 → & = 50 6 T. and (A-7) - 2 I and sa g we can Hence define the region inside the circle x²+y2 =24 outside the circle x² + y²=1 and as, SA М» (8,0): rcis asind,

Here it is given that, density s at any Point (x, y) is inversely Proportional to its distance from origin. distance from origin of any point is 22 +42 p . S x vazny ſa²+ y 2 K =7 S K is constant of proportionality7 1x²+4² => S - A [since 82 se²+y² Therefore mass is given by m IS where dA=&drdo off sum 2sing => m -1/54 rdrdo. 7 El asino 51 G = k dr do 요 6

역 Us 2 asino K [*] 6. K 2 sino -1) do 6 K [86039 .- 278 D 6 SR 6 et esos e (ako 17 - uş 2m 23 6 С -K [ -3 EN All 6 + - 40 -253 + Ele -* | 2K [ V3 D 7 m 3 Due to the domains tea) and symmetry of the function that Miny O En hre know - and, M Iss § y dA.

[ 2 EN xlov - Eros f ( t ) + (6 4 2 + tsson 6+ ()s-] (E14 [3561+ visos + 65015.] * = 6. Genis-QUISE = = b cuist Immuus che Z (3 sino -sin38 - sino) do. = us 473 op (ou!5 - 8 ms) */N 5 (s/Erre 2 sino (4 since I do. = K 6 Us Verein ist [ ]os K 11 2sino. dl=) BP IP prish ES Y 2 US asino (sle SP e P e Bruse к 1) Quist

7 Mx = 313 2 V5 2 6√3 2/3 2 2 = = [ ☆ [313 - vo] K [-se k [ va = L K 11 . VBK: if center of Mass be a ý then as1 Me 3k 2k [FB- Elm 10 3 Ginie 33 k 2K (313-) 3/3 2(353 π and Ik since My 33 center of Mass is (o 0 9.213.23=T