The centre of gravity of a body on the earth coincides with its centre of mass for a small object,whereas for an extended object,it may not. What is the qualitative meaning of 'small' and 'extended' in this regard? For which of the following does the centre of gravity coincide with the centre of mass: a building,a pond,a lake,a mountain?

(A) The qualitative meaning of 'small' and 'extended' depends on the variation of the acceleration due to gravity $(g)$ over the dimensions of the object.
If the vertical height or dimensions of the object are very small compared to the radius of the earth $(R_e \approx 6400 \ km)$,the gravitational field is considered uniform,and the object is called 'small'. In this case,the centre of gravity coincides with the centre of mass.
If the dimensions of the object are large enough that the variation in $g$ becomes significant,the object is called 'extended'. In this case,the centre of gravity may not coincide with the centre of mass.
$(1)$ $A$ building and a pond are considered 'small' objects because their vertical extent is negligible compared to $R_e$. Thus,for these,the centre of gravity coincides with the centre of mass.
$(2)$ $A$ deep lake and a mountain are examples of 'extended' objects because their vertical extent is significant enough that the variation in $g$ cannot be ignored. Thus,for these,the centre of gravity may not coincide with the centre of mass.