Q.Why do we need to know the centre of mass of an object? For which objects, its position may differ from that of the centre of gravity?

[!TLDR] The c.m. lets Newton's point-mass laws be applied to any extended, finite-sized real object by treating it as a single equivalent point; c.g. differs from c.m. only for objects so large (comparable to Earth's radius) that gravity varies noticeably across the object. [!ANSWER] To apply Newton's laws to exte …

Newton's laws of motion, as originally formulated, apply strictly to POINT masses. Real objects always have finite size and an extended distribution of mass, so without some way to reduce them to an 'equivalent point', Newton's laws could not be directly applied to them at all. The CENTRE OF MASS solves exactly this problem: it is the single point at which the entire mass of the object can be treated as concentrated, letting Newton's second law be applied to that one point ($\vec{F}_{ext}=M\vec{a}_{cm}$) to correctly describe the object's overall translational motion, regardless of how complicated its internal shape or mass distribution is (4.13, 4.13.3).

The centre of GRAVITY coincides EXACTLY with the centre of mass whenever the gravitational field g can be treated as constant/uniform across the object -- true for essentially every object we ever encounter in daily life on Earth, since g varies negligibly over the small size of ordinary objects. The two would differ only for an object so enormous that its size is comparable to the EARTH's own radius (thousands of kilometres) -- at that scale, the gravitational field is measurably stronger at the object's lower portions (closer to Earth's centre) than at its upper portions, so the weighted-average location of the gravitational PULL (the c.g.) would sit slightly lower than the plain mass-weighted-average location (the c.m.). No such object is realistically ever encountered, so f …

Explain the c.m.'s role in extending point-mass Newtonian mechanics to extended bodies, and state the precise condition (non-uniform g over an astro …

Claiming c.g. and c.m. 'always' differ for large objects, without specifying that the size must be comparable to the EARTH's radius for the effect to be non-negligible -- ordinary larg …