Centre of Mass and Reference Point

Question

When calculating the centre of mass for a small body compared to the earth, the equation $$\vec r_c = \frac {\sum_i m_i \vec r_i} {\sum_i m_i}$$ is used, where $\vec r_c$ is the position vector for the centre of mass from a reference point $O$, $\vec r_i$ is the position vector for each particle in the body, $m_i$ is the mass for each particle and $\sum_i$ is the sum that extends over all particles comprising the body.

My question is, when you calculate the centre of mass of a body, what do you choose as the reference point $O$? $\vec r_c$ should be the point where as if the whole weight of the body acts there and the point that gives the correct torque of the body when the weight of the body is applied there.

Since there is a consideration of torque involved, should the reference point $O$ be a pivot point? But how do you define a pivot point for a body, for example a stationary book on a table or a floating balloon?

Answer 1

The definition of centre of mass, with respect to the parts of the total body, is independent of the choice of reference. Of course, the centre of mass position vector with respect to another body depends on where that other body is.

Answer 2

The principle of moments states that for an object to be at equilibrium the algebraic sum of clockwise and anti-clockwise moments is equal to zero about any point(taking suppose clockwise positive and anti clockwise negative).

Hence it doesn’t really matter which point you take the pivot but in order to solve problems we often take certain points as pivots some moments of some forces become zero so we can solve problems with more variables. In principle if the object is at equilibrium(or even in an accelerated frame where there is relative rest, you’d need pseudo forces) you can essentially take any point to be the pivot and it would satisfy the principle of moments.

Thus, the center of mass just mentions a reference point $O$ as a formality as the center of mass of an object would be the same position for the object(in a uniform gravitational field) regardless of where $O$ really is.