Why does centre of mass of ice-container system shift in absence of any net external force? Consider a cube of ice in a flat based container(the base is very broad).The temperature of the system is at first fixed at a minus Celsius temperature, but then the system is left on a table with the top open to atmosphere.
The ice starts melting, and finally there is only water, spread over the container's base(water doesn't touch walls of the container,so it looks like water spilled on the floor)
Here, the centre of mass of the $H_2O$-container system moves downward because of the lowering of height of water molecules.
But,this system experiences no net external force on it,all the time.Hence, the centre of mass of this system shouldn't accelerate at all!
How is this contradiction sorted out?I feel that since melting is very slow, the centre of mass might move only very slowly, but still that doesn't explain things.For example if the room was very hot, melting wouldn't have been slow, right?
Edit:Many people are probably getting confused regarding what perspective I'm taking.Let the container and water(solid/liquid) be a single system.We can think of them together as a point mass.This point mass is at equilibrium,and at rest,situated at the position of the centre of mass of ice+container system(let's say,at a height 'h' above the table.
There may be internal forces happening inside the point mass,but the net external force(resultant) is zero,all the time.Hence, according to Newton's first law,the point mass must remain at equilibrium and hence,at rest.But,when ice melts,the position of center of mass of ice+container system has moved down!
Hence the point mass has to move down, in the absence of any resultant external force.The gravitational force on ice+container is cancelled by normal reaction of table on container.
 A: For each molecule of water in the ice cube there are two forces to consider: the electromagnetic force and gravity. When the ice cube is frozen, the molecules have low kinetic energy so they stick to their rigid crystalline structure which is sufficient to counter the gravitational force. When they have a bit more kinetic energy (i.e. at higher temperature) they are able to break free of their crystalline structure and so are unable to counter the gravitational force. 
The fact that the water molecules are accelerated is ok - there is additional energy added to the water molecules as they heat up.
A: You can no longer consider the ice block as a system once it starts melting, so it no longer remains a rigid body. Consequently, you cannot apply Newton's law on it.
Peace ;)
A: Simply put, the ice-container system is not closed; there are external gravitational forces.
If you consider the earth and container as one part, and the ice/liquid water as the other, of a new, closed system, as the water melts and drops down, the earth rises (a tiiiiiny bit) and the CofM of this system remains unchanged
