Centre of Mass — Core Principles

Core Principles

The Centre of Mass (CoM) is a crucial concept in mechanics, representing the average position of all the mass in a system. For discrete particles, its position is calculated as a weighted average of their individual positions, with masses as weights: $\vec{R}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$.

For continuous bodies, this becomes an integral: $\vec{R}_{CM} = \frac{\int \vec{r} \,dm}{\int dm}$. The CoM simplifies the analysis of complex systems by allowing us to treat the entire system's translational motion as if all its mass were concentrated at this single point.

Crucially, the acceleration of the CoM is given by $\vec{A}_{CM} = \frac{\vec{F}_{ext, net}}{M}$, meaning only external forces affect its translational motion. If the net external force is zero, the CoM's velocity remains constant, leading to the conservation of linear momentum for the system.

The CoM does not always lie within the physical boundaries of an object and is distinct from the Centre of Gravity, though they often coincide in uniform gravitational fields.

Important Differences

vs Center of Gravity (CoG)

Aspect	This Topic	Center of Gravity (CoG)
Definition	Centre of Mass (CoM) is the point where the entire mass of the system is considered to be concentrated, representing the average position of all mass.	Centre of Gravity (CoG) is the point where the entire weight of the body appears to act, effectively the point where the resultant gravitational force acts.
Dependence on Gravity	Independent of the gravitational field. It's an intrinsic property of the mass distribution.	Dependent on the gravitational field. Its position can shift if the gravitational field is non-uniform.
Location	A mathematical point, not necessarily within the physical body (e.g., ring, hollow sphere).	A point where the resultant gravitational torque is zero. Can also be outside the body.
Coincidence	Coincides with CoG only in a uniform gravitational field.	Coincides with CoM only in a uniform gravitational field.
Formula (x-coordinate)	$X_{CM} = \frac{\sum m_i x_i}{\sum m_i}$	$X_{CG} = \frac{\sum w_i x_i}{\sum w_i} = \frac{\sum m_i g_i x_i}{\sum m_i g_i}$

While often used interchangeably in introductory physics, the Centre of Mass (CoM) and Centre of Gravity (CoG) are fundamentally distinct concepts. The CoM is a purely kinematic property, representing the average spatial distribution of mass, independent of any external forces. In contrast, the CoG is a dynamic property, representing the point where the net gravitational force acts, and thus depends on the gravitational field. They coincide perfectly only when the gravitational field is uniform across the entire body. For most NEET problems, where objects are small enough for gravity to be considered uniform, this distinction is often overlooked, and they are treated as the same point.