Question

A lamina occupies the part of the disk x2+y2≤4 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis. (xˉ,yˉ)=

Answer 1

The center of mass of the given lamina is at \((\bar{x}, \bar{y}) = \left(\frac{8}{3\pi}, \frac{16}{9\pi}\right)\).

Step-by-step explanation:

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Answer 2

The center of mass of the lamina is (xˉ, yˉ) = (0, 4/3).

To find the center of mass of the lamina, we need to consider the distribution of mass within the region. In this case, the lamina occupies the part of the disk
`x^(2) +y^(2)`≤ 4 in the first quadrant.

Given that the density at any point is proportional to its distance from the x-axis, we can determine the center of mass by integrating the position coordinates (x, y) over the lamina and dividing by the total mass.

Using symmetry, we can conclude that the center of mass lies on the y-axis. Additionally, due to the given density distribution, the center of mass will be at a height proportional to the moments of the lamina.

By calculating the moments of the lamina and applying the formula for the center of mass, we find that the x-coordinate of the center of mass (xˉ) is 0, and the y-coordinate (yˉ) is 4/3.

Complete question:-
A lamina occupies the part of the disk x2+y2≤4 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis. (xˉ,yˉ)=
`((27)/(8) , (27k)/(16) X)`.