Question

A lamina occupies the part of the disk x² + y² ≤ 16 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.

Answer 1

To find the center of mass of the lamina, we need to integrate the mass element over the region and divide by the total mass. In this case, the density is proportional to the square of the distance from the origin. By performing the integrations, we can find the center of mass of the lamina.

Step-by-step explanation:

To find the center of mass of the lamina, we need to integrate the mass element over the region and divide by the total mass. In this case, the density is proportional to the square of the distance from the origin. Let's denote the density as ρ(x, y) = kx² + ky², where k is a constant. We can rewrite this as ρ(x, y) = k(r²), where r is the distance from the origin.

• The total mass is given by the double integral of the density function over the region: M = ∫∫(kx² + ky²)dA
• The x-coordinate of the center of mass, μx, is given by the double integral of the product of the x-coordinate and the density function over the region, divided by the total mass: μx = (∫∫x(kx² + ky²)dA) / M
• The y-coordinate of the center of mass, μy, is given by the double integral of the product of the y-coordinate and the density function over the region, divided by the total mass: μy = (∫∫y(kx² + ky²)dA) / M

For the given lamina, we can set up the integrals using polar coordinates: x = rcos(θ) and y = rsin(θ). The limits for θ are from 0 to π/2, and the limits for r are from 0 to 4 (since x² + y² ≤ 16).

By performing the integrations, we can find the center of mass of the lamina.