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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?

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If some forces pull in an object somewhere else than it's centre of mass (CoM), you might get problems.

Think of a space shuttle floating around in space. Push in its CoM and it translates (accelerates according to Newton's classical linear laws of motion). But push at the edge, and it starts rotating.

But saying that using the CoM can give "incorrect" results is not a good description. It only gives incorrect results if you use wrong models and equations. The CoM is e.g. the point that the space shuttle above will naturally rotate (spin) about. You still use it, you just have to use it correctly, when the situation is different.

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