Q.Distinguish between the centre of mass and the centre of gravity of a body. Under what condition on the surrounding gravitational field do the two coincide?

The centre of mass is defined purely from how a body's mass is distributed, $\vec R = \sum_i m_i\vec r_i/M$, with no reference to gravity at all. The centre of gravity is instead defined as the single point at which the body's total WEIGHT may be taken to act, for the purpose of computing the net gravitational torque on the body.

If the acceleration due to gravity $g$ has the same value and direction at every part of the body (an excellent approximation for any body of ordinary, laboratory or everyday size on the Earth's surface), then every mass element's own weight is simply $m_ig$ with the identical $g$ throughout, and the resulting mass-weighted (and hence weight-weighted) average point comes out at exactly the same location as the ordinary centre of mass. The two points genuinely separate only for a body so large, or so positioned, that $g$ measurably varies in strength or direction across it.

Compare the two defining formulas and note that both reduce to the same mass-weighted average when $g$ is the same constant for every mass element.

• Treating the two terms as always meaning exactly the same thing without ever stating the uniform-$g$ condition under which this is true.
• Believing centre of gravity depends on the body's material/mass distribution differently from centre of mass -- both are built from the same mass distribution, only weighted by $g_i$ instead of just $1$.