What is a homogeneous curve or a circle?

Question

I am facing a problem which asks to find coordinates of mass center of a homogeneous curve L. Obviously, this should be done by the line integral but the part I do not understand at all is the homogeneous curve? What does it mean?

Alternatively, there comes another problem which asks coordinates of a homogeneous circle

Thank you for help.

P.s. I cannot find anything about this online neither in English or in my native language

Answer 1

Homogeneous curve means that we can assume the linear mass constant along the curve, that is

$$dm=\rho ds$$

with $\rho$ (kg/m) constant.

Since the value of $\rho$ doesn't affect the position of the center of mass we can assume $\rho=1$ and thus evaluate the line integral neglecting any information on the mass.

Answer 2

For an homogeneous circle, by symmetry, the mass center must be the geometric center.

For an arbitrary curve, you will obtain the average $x$ and average $y$ by integration of $ds, x\,ds$ and $y\,ds$. The constant linear mass cancels out.