Center of mass: definition and calculation
The center of mass is the point representing the mean position of the mass in a system, and can be calculated for discrete particles or continuous bodies.
Why this shows up in the exam
NEET tests your ability to define, locate, and use the center of mass in solving problems involving systems of particles.
How NEET tests this
Learn the idea
The centre of mass (COM) is the weighted average position of all the mass in a system; it is found by adding each mass multiplied by its position vector and dividing by the total mass. The key insight is that the COM depends only on the distribution of mass, not on whether material actually occupies that point.
🧠 Memory hook: Think of a seesaw: the heavier child sits closer to the pivot (COM) while the lighter child sits farther away – the pivot balances the weighted distances.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- COM of discrete particles: r_cm = (Σ m_i r_i)/(Σ m_i)
- COM of a continuous body: r_cm = (∫ r dm)/(∫ dm)
- For two particles on a line, distances from COM are inversely proportional to their masses (m1 d1 = m2 d2)
- If the body is uniform and symmetric, COM lies at the geometric centre (e.g., midpoint of a uniform rod)
- COM may lie in a region where there is no material (e.g., centre of a hollow sphere or a ring)
How to approach it
- 1Write down the masses and their position vectors (choose a convenient origin)
- 2Multiply each position vector by its mass and add the products
- 3Divide the resultant vector by the total mass to obtain r_cm
- 4If a distance is unknown, use the relation m1 d1 = m2 d2 for collinear particles
Worked example — watch it click
Assertion A: The centre of mass of a body may lie where there is no mass. Reason R: The centre of mass has nothing to do with the mass.
- A)If both assertion and reason are true, and reason is the correct explanation of assertion.
- B)If both assertion and reason are true, but reason is not the correct explanation of assertion.
- ✅If assertion is true, but reason is false.
- D)If both assertion and reason are false.
The concept behind this problem
The assertion‑reason pair checks whether you recognise that the COM’s location is dictated solely by the mass distribution (true) and that it can be in empty space, while rejecting the mistaken idea that mass plays no role.
Step by step
- 1Assertion A is TRUE: The center of mass can lie where there is no mass (e.g., center of a ring, center of a hollow sphere).
- 2Reason R is FALSE: The center of mass is defined by the mass distribution; it depends entirely on the masses and their positions.
- 3The reason contradicts the fundamental definition of center of mass: r_cm = Σ(m_i × r_i)/Σm_i.
- 4Since A is true but R is false, option (c) is correct.
Watch out
Students often answer that the reason is true because they misread ‘has nothing to do with the mass’ as ‘does not depend on where the mass is placed’.
Common slip-ups that cost marks
- •Treating r as a scalar distance instead of a vector when the system is not collinear
- •Assuming the COM must be inside the material of the body
- •Mixing up the numerator and denominator – forgetting to sum all masses in the denominator
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Practise it
These are real questions from past NEET papers that test this exact idea.
Assertion A: The centre of mass of a body may lie where there is no mass. Reason R: The centre of mass has nothing to do with the mass.
Push further
More challenging4 harder questions built from the past papers above — a step up in difficulty, with distractors designed so you can't get there by elimination. Written and checked by our reviewers, not from a real paper.
Which of the following factors does NOT directly influence the instantaneous position of the center of mass of a rigid body?
More from Motion of System of Particles and Rigid Body
Conservation of momentum and angular momentum
The total linear and angular momentum of a system remains constant in the absence of external forces or torques, including during collisions and rotational motion.
Moment of inertia and radius of gyration
Moment of inertia quantifies how mass is distributed with respect to an axis of rotation, and the radius of gyration is a measure related to this distribution.
Torque and rotational equilibrium
Torque is the rotational analogue of force, causing angular acceleration, and equilibrium occurs when the net torque on a body is zero.
Rotational kinematics and dynamics
Rotational kinematics describes the motion of rotating bodies, while dynamics relates torque, angular acceleration, and rotational kinetic energy.
Centre of Mass of Discrete Particles
For discrete particles, the centre-of-mass position is the vector sum of each mass times its position divided by total mass; this point governs translation even when the particles move relative to one another.
Centre-of-Mass Shift and Internal Rearrangement
When external force is zero, total momentum and centre-of-mass velocity remain constant; therefore mass displacements relative to a fixed frame obey the mass-weighted displacement constraint.