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.
Why this shows up in the exam
Locating the balance point of separated masses · Reducing a many-particle translation problem to one point · Finding centre-of-mass velocity from particle velocities
Learn the idea
A system balances translationally as though its total mass were concentrated at the mass-weighted average position. A system balances translationally as though its total mass were concentrated at the mass-weighted average position. Start from a clear axis, origin, body, and reference frame; the geometry and constraints then decide which rotational law is safe to use.
🧠 Memory hook: Heavy masses pull the average position closer.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- r_cm = (sum m_i r_i)/(sum m_i) — Mass-weighted position for particles measured in one inertial frame.
- M v_cm = sum m_i v_i — Total linear momentum relation for a fixed-mass system.
How to approach it
- 1Choose one origin and axes
- 2Form each mass-position product componentwise
- 3Divide by total mass and check that the result lies in the expected region
Common slip-ups that cost marks
- •Averaging coordinates without mass weights
- •Mixing position vectors from different origins
- •Using signed coordinates as unsigned distances
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
Masses 1 kg and 3 kg lie at x = 0 and x = 4 m. Find the x-coordinate of their centre of mass.
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.
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.
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 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.