Is the Centre of Mass always located within the physical boundaries of the object?

No, not necessarily. While for many common objects like a solid sphere or a cube, the Centre of Mass (CoM) lies within the material of the object, this is not a universal rule. For objects with holes or specific geometries, the CoM can be located in empty space. Classic examples include a ring, a hollow sphere, or a boomerang, where the CoM is situated outside the physical material of the body. It's a mathematical point representing the average mass distribution, not a point that must be occupied by mass.

What is the difference between Centre of Mass and Centre of Gravity?

The Centre of Mass (CoM) is a point representing the average position of all the mass in a system, independent of any gravitational field. The Centre of Gravity (CoG) is the point where the entire weight of the body appears to act. In a uniform gravitational field (which is typically assumed for most NEET problems), the CoM and CoG coincide. However, in a non-uniform gravitational field (e.g., a very tall structure where gravity varies with height), they would be slightly different, with the CoG being slightly lower than the CoM.

How does the Centre of Mass simplify the analysis of complex systems?

The Centre of Mass simplifies analysis by allowing us to treat a complex system of particles or an extended body as a single point mass for translational motion. Newton's second law, $\vec{F}_{ext, net} = M \vec{A}_{CM}$, applies directly to the CoM. This means that the translational motion of the entire system is determined solely by the net external forces, irrespective of internal forces or the system's rotational motion. This separation makes problems involving collisions, explosions, and projectile motion of extended bodies much more manageable.

Can the Centre of Mass of a system change its position?

Yes, the Centre of Mass (CoM) of a system can change its position relative to a fixed external coordinate system if external forces act on the system, causing its acceleration. However, if no net external force acts on the system, its CoM will either remain at rest or move with a constant velocity. The CoM can also change its position *relative to the body itself* if the distribution of mass within the body changes (e.g., a person walking on a boat, or a multi-limbed robot moving its parts).

What happens to the Centre of Mass during an explosion?

During an explosion, the internal forces between the fragments are very large, but they are internal to the system. According to Newton's third law, these internal forces cancel out in pairs. Therefore, if no external forces (like air resistance) are acting on the system, the Centre of Mass of the fragments continues to follow the same trajectory it would have had before the explosion. For example, if a projectile explodes in mid-air, its CoM continues along the original parabolic path, even as the fragments scatter.

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Version 1Updated 22 Mar 2026

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The Centre of Mass (CoM) of a system of particles or a rigid body is a unique point where the entire mass of the system is considered to be concentrated. This hypothetical point represents the average position of all the mass in the system. When analyzing the translational motion of a complex system under external forces, it is often sufficient to consider the motion of its Centre of Mass, as if a…

Quick Summary

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.

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Key Concepts

Calculating CoM for Discrete Particles

When dealing with a system of distinct particles, the Centre of Mass (CoM) is found by taking a weighted…

CoM of Composite Bodies (Subtraction Method)

For bodies with complex shapes or holes, it's often easier to use the 'subtraction method' or 'negative mass'…

Motion of Centre of Mass and Momentum Conservation

The motion of the Centre of Mass (CoM) is governed by the net external force acting on the system, as per…

Discrete Particles: —

\vec{R}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i}

\n- Continuous Bodies:

\vec{R}_{CM} = \frac{\int \vec{r} \,dm}{\int dm}

\n- Velocity of CoM:

\vec{V}_{CM} = \frac{\sum m_i \vec{v}_i}{\sum m_i} = \frac{\vec{P}_{sys}}{M}

\n- Acceleration of CoM:

\vec{A}_{CM} = \frac{\vec{F}_{ext, net}}{M}

\n- CoM of Uniform Rod (length L): At L/2 from an end. \n- CoM of Uniform Semicircular Arc (radius R):

2R/\pi

from center along axis of symmetry. \n- CoM of Uniform Semicircular Plate (radius R):

4R/(3\pi)

from center along axis of symmetry. \n- CoM of Uniform Solid Hemisphere (radius R):

3R/8

from center along axis of symmetry. \n- CoM of Uniform Hollow Hemisphere (radius R):

R/2

from center along axis of symmetry. \n- Conservation: If

\vec{F}_{ext, net} = 0

, then

\vec{V}_{CM}

is constant (momentum conserved). If system is initially at rest,

\vec{V}_{CM} = \vec{0}

always.

For CoM of common shapes, remember: \nRod: L/2 (from end) \nSemi-circular Arc: 2R/pi (from center) \nSemi-circular Plate: 4R/3pi (from center) \nHollow Hemisphere: R/2 (from base) \nSolid Hemisphere: 3R/8 (from base) \nCone: H/4 (from base) \n\nMnemonic: 'Rods Are Simple, Plates Are Heavy, Hemispheres Have Half, Solids Are Three-Eighths, Cones Are Quarters.' (Relates to the complexity/density and fraction of R or H)

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