# How do you find the center of mass if the density at any point is inversely proportional to its distance from the origin of a lamina that occupies the region inside the circle #x^2 + y^2 = 10y# but outside the circle #x^2+y^2=25#?

##### 1 Answer

Jan 10, 2017

The enter of mass is

#### Explanation:

Consider the first equation:

# x^2+y^2=10y #

We can put this into standard from by completing the square:

# x^2 + y^2 - 10y = 0 #

# :. x^2 + (y-5)^2-5^2 = 0 #

# :. x^2 + (y-5)^2 = 5^2 #

Which is a circle of radius

# x^2+y^2=25 #

Which is a circle of radius

We can plot these curves;

For a more complex problem we would need to use integration to find the Centre of Mass, but because of symmetry the density will be evenly distributed about the lines

Hence the enter of mass is