Final answer:
To find the center of mass of the lamina, we need to first determine the area that the lamina occupies. The lamina is inside the circle x² + y² = 2y but outside the circle x² +y² = 1. We can integrate the density over the area to find the total mass and the position of the center of mass.
Step-by-step explanation:
To find the center of mass of the lamina, we need to first determine the area that the lamina occupies.
From the given information, we know that the lamina is inside the circle x² + y² = 2y and outside the circle x² + y² = 1.
We can rewrite the equations of the circles as (x - 0)² + (y - 1)² = 1 and (x - 0)² + (y - 0)² = 1 respectively, which represent the equations of circles with centers at (0, 1) and (0, 0) and radii of 1.
Therefore, the lamina is the region between these two circles.
To find the area of this region, we can subtract the area of the smaller circle from the area of the larger circle. Using the formula for the area of a circle, we have:
Area = π(2y)² - π(1)²
Simplifying, we get:
Area = 4πy² - π
Next, we need to determine the mass distribution within this area.
The density at any point is inversely proportional to its distance from the origin, which means the density decreases as the r value increases in polar coordinates.
We can express the density as ρ = k/r, where ρ is the density, k is the constant of proportionality, and r is the distance from the origin.
Now, we can integrate the density over the area to find the total mass and the position of the center of mass. The coordinates of the center of mass are given by the formulas:
x = (1/M) ∫x dM
y = (1/M) ∫y dM
Where M is the total mass, x and y are the coordinates, and dM represents the differential mass within the region.
Since the density ρ is inversely proportional to the distance r, we can express the differential mass as dM = ρ dA, where dA is the differential area.
Substituting the expression for ρ and integrating, we get:
x = (1/M) ∫x (k/r)dA
y = (1/M) ∫y (k/r)dA
To evaluate the integrals, we can convert to polar coordinates and express the differentials as dA = r dr dθ.
Substituting these values and integrating over the appropriate ranges, we can find the coordinates of the center of mass.