In a physics lesson the other day our teacher gave us the definition of centre of mass as being 'the point at which the mass of an object can be thought to be concentrated'. Upon further investigation, I found the more rigorous definition 'the unique point where the weighted relative position of the distributed mass sums to zero'.
Since we can use the centre of mass as a modelling assumption to solve mechanics questions, but it is clearly not true in reality that the entire mass of an object is centred on one point, I began to wonder when assuming a centre of mass would give us incorrect results? My gut feeling was that the centre of mass wouldn't work in the same way for systems not explained adequately by classical mechanics, but I'm not sure whether all problems within classical mechanics can be solved using a centre of mass.
Can anyone give examples (and if possible explain them) of when this assumption would produce incorrect predictions?