1) A lamina occupies the part of the disk x 2 + y 2 9 in the first quadrant. Fin

Question

1) A lamina occupies the part of the disk x2 + y2 9 in the first quadrant. Find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.

2) A lamina occupies the part of the disk x2 + y2 16 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

......When I saw the solutions to the two questions above, while the 'square of its distance' problem used k(x^2+y^2) as p(x, y), the second question used ky as p(x, y).

Could you explain the reasoning behind these two differences? I don't need the solution to the questions above. Thank you!

Explanation / Answer

1)

general point on lamina is (x,y)

distance of point (x,y) from origin =[(x-0)2+(y-0)2]

distance of point (x,y) from origin =[x2+y2]

square of distance of point (x,y) from origin =[x2+y2]

density at any point is proportional to the square of its distance from the origin

=>p(x,y) [x2+y2]

=>p(x,y) = k[x2+y2] , where k is a proportionality constant

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2)

general point on lamina is (x,y)

distance of point (x,y) from x-axis =y ,distance of point (x,y) from y-axis =x

density at any point is proportional to its distance from the x-axis.

=>p(x,y) y

=>p(x,y) = ky  , where k is a proportionality constant

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