Question

A lamina occupies the region inside the circle x² y ² =12y but outside the circle x ² y² =36. The density at each point is inversely proportional to its distance from the origin. Where is the center of mass?

Answer 1

To find the center of mass of a lamina, we need to calculate the weighted average of the positions of all the points in the lamina.For each quadrant, we can calculate the weighted average of the x and y coordinates to find the center of mass.

Step-by-step explanation:

To find the center of mass of a lamina, we need to calculate the weighted average of the positions of all the points in the lamina. In this case, the density at each point is inversely proportional to its distance from the origin.

We can divide the region inside the first circle but outside the second circle into four quadrants. For each quadrant, we can calculate the weighted average of the x and y coordinates to find the center of mass.

Let's take one quadrant as an example. The equation of the first circle is x² + y² = 12y, which can be rewritten as x² + (y-6)² = 36. This is the equation of a circle with center (0, 6) and radius 6. So the density at each point in this quadrant is inversely proportional to its distance from (0, 6). We can use the formula for the center of mass of a semicircular wire to find the center of mass of this quadrant.