Whats the center of gravity/mass of this figure? I've tried figuring this out, but I can't seem to find a way to find it, since I dont have any other information to calculate something. The only thing I could find out was that its possible to use symmetries to find out the center of mass/gravity.

My intuition tells me it should be C however I am not sure. Is there any way to check it?
 A: It can be shown by exclusion:
A and D are not possible, because they are to far of.
You can draw a ellipsoid around B, which contains the area left of B and is symmetric. Still, you are left with the two tails on the right.
only C is left, therefore its C.
Maybe one can use the fact, that this figure is "stackable" to archive a more rigorous proof of that.
A: Symmetries:
Can't do without more information.
If have an analytical formula for the shape (parabola?), you can use the integral version of the center of mass equation to obtain an exact center of mass.
Experimental method:

*

*Print it and use a pair of scissors to cut it out.

*Suspend it at any point on a string.

*Draw a line with a pencil along the string and down

*Do 2) and 3) again by suspending it from a different angle.
The point where the two drawn lines intersect, is the center of mass.

Analytical method:
Obtain the center of mass $CM_x$ and $CM_y$ for the x- and y-direction respectively using
$$\begin{array}{l}
C{M_x} = \frac{{\sum\limits_i^N {{m_i}{x_i}} }}{{\sum\limits_i^N {{m_i}} }}\\
C{M_y} = \frac{{\sum\limits_i^N {{m_i}{y_i}} }}{{\sum\limits_i^N {{m_i}} }}
\end{array}$$
where i is the index for each pixel and N is the number of pixels.
Can be done quickly with little programming.
Edit: Solution below at coordinates (246,171)
It's between B and C...
I suppose there is some uncertainty associated with the quality of the image.

A: It's a multiple choice, with three obviously wrong answers. You can't show the exact position of the centre of mass without equations for the curves, but you can think about where the intuitive middle is, allowing that matter further from the centre has greater moment.
