What depth should water be filled to in a cylinder flask to make it the most stable As said in title, a cylinder flask with a mass of 100g and radius of 3 has a center of balence 10cm above the base. Assuming negligible wall thickness, to what depth should water be added to this flask to make it the most stable. 
The question doesn't give any other information and I'm at a loss of how to approach this question. Any help or tips in the right direction would be great  
 A: Goutham is quite correct in some ways but overlooks something. Look at the diagram below:

We known the centre of gravity (COG) of the empty cylinder is $z_1=10\:\mathrm{cm}$ and the mass of the empty cylinder is $100\:\mathrm{g}$.
If we fill the cylinder up with water to height $2z_2$ then the COG of the water is $z_2$ and the mass of water is (assuming the radius is $3\:\mathrm{cm}$ and the density of water is $1\:\mathrm{g/cm^3}$):
$$m_2=18\pi z_2.$$
The overall COG $z$ of the cylinder plus water can then be calculated as:
$$z=\frac{1000+18\pi z_2^2}{10+18\pi z_2}.$$
As Goutham points out, stability is highest for lowest $z$ and $z_{min}$ can be found for:
$$\frac{dz}{dz_2}=0.$$
Unfortunately that really only holds when the added mass is a non-flowing material. For a liquid, the moment you tilt the cylinder $z_2$ starts moving in the same direction, as well as downward.
So Goutham's approach is at best approximate. An accurate prediction would have to take into account the effect of tilt on $z_2$ and thus also $z$. For tall, thin cylinders the effect of tilt on $z$ is of course smaller than for stubby, wide ones.
A: The flask become most stable when its centre of gravity is at the smallest height. If you start pouring water, you will notice that the effective centre of gravity gets down to a lower postion. As you keep on filling, it would be at the lowest height for some level of water and rises again, afterwards. You will have to find that point of minimum height. Just frame an equation relating height of water and height of resultant centre of gravity, find the derivative and equate to zero
