Centre of Mass
Location for discrete and continuous bodies and motion of the centre of mass. (Physics › System of Particles and Rotational Motion, NEET UG syllabus.)
What is Centre of Mass?
A unique point in a system of particles or a body where the entire mass of the system is considered to be concentrated for the purpose of describing its translational motion.
Key formula / rule: Position of Centre of Mass (Discrete Particles)
Key points
- Define the Centre of Mass for a system of particles and a rigid body.
- Calculate the position of the Centre of Mass for discrete particle systems in 1D, 2D, and 3D.
- Calculate the position of the Centre of Mass for simple continuous bodies (e.g., rod, ring, disc) using integration.
- Determine the velocity and acceleration of the Centre of Mass.
Common exam trap
Confusing Centre of Mass with Centre of Gravity; they are identical only in a uniform gravitational field.
Definitions
- Term
Centre of Mass (COM)
- Meaning
A unique point in a system of particles or a body where the entire mass of the system is considered to be concentrated for the purpose of describing its translational motion.
- Term
System of Particles
- Meaning
A collection of two or more particles or bodies interacting with each other, which can be treated as a single entity for analysis.
Learning objectives
Define the Centre of Mass for a system of particles and a rigid body.
Calculate the position of the Centre of Mass for discrete particle systems in 1D, 2D, and 3D.
Calculate the position of the Centre of Mass for simple continuous bodies (e.g., rod, ring, disc) using integration.
Determine the velocity and acceleration of the Centre of Mass.
Relate the external forces acting on a system to the motion of its Centre of Mass.
Apply the concept of Centre of Mass to solve problems involving conservation of linear momentum.
Formulae
- Name
Position of Centre of Mass (Discrete Particles)
- Note
rᵢ are position vectors of individual particles, mᵢ are their masses. RCM is the position vector of the COM.
- Expression
RCM = (m₁r₁ + m₂r₂ + ... + mₙrₙ) / (m₁ + m₂ + ... + mₙ) = (Σmᵢrᵢ) / (Σmᵢ)
- Name
Position of Centre of Mass (Continuous Body)
- Note
r is the position vector of an infinitesimal mass element dm. Integration is over the entire body. Mtotal is the total mass.
- Expression
RCM = (∫r dm) / (∫dm) = (∫r dm) / Mtotal
- Name
Velocity of Centre of Mass
- Note
vᵢ are velocities of individual particles. VCM is the velocity vector of the COM.
- Expression
VCM = (m₁v₁ + m₂v₂ + ... + mₙvₙ) / (m₁ + m₂ + ... + mₙ) = (Σmᵢvᵢ) / (Σmᵢ)
- Name
Acceleration of Centre of Mass
- Note
aᵢ are accelerations of individual particles. ACM is the acceleration vector of the COM.
- Expression
ACM = (m₁a₁ + m₂a₂ + ... + mₙaₙ) / (m₁ + m₂ + ... + mₙ) = (Σmᵢaᵢ) / (Σmᵢ)
- Name
Newton's Second Law for a System of Particles
- Note
Fext is the net external force acting on the system. Mtotal is the total mass of the system. This shows that the COM moves as if all external forces act on it.
- Expression
Fext = Mtotal * ACM
Prerequisites
Basic vector algebra (addition, scalar multiplication)
Newton's Laws of Motion
Basic differentiation and integration (for continuous bodies)
Understanding of position, velocity, and acceleration vectors
Common mistakes
Confusing Centre of Mass with Centre of Gravity; they are identical only in a uniform gravitational field.
Incorrectly applying the formula for discrete particles to continuous bodies without integration.
Forgetting to consider all masses and their respective position vectors in calculations.
Assuming COM always lies within the physical boundaries of the object (e.g., a ring's COM is at its centre, outside the material).
Ignoring the vector nature of position, velocity, and acceleration in COM calculations.
Keywords
Centre of Mass
COM
System of Particles
Translational Motion
Mass Distribution
Weighted Average
External Force
Internal Force
Momentum Conservation
Practice preview
A man of mass 60 kg is standing on a raft of mass 120 kg. The raft is initially at rest on a frictionless lake. The man starts walking on the raft at a velocity of 1 m/s relative to the raft. What is the velocity of the …
hard
For a uniform rod of length L, where is its Centre of Mass located?…
easy
Two particles of masses 1 kg and 2 kg are located at positions (1, 2) m and (-2, 3) m respectively. What is the position vector of their Centre of Mass?…
medium