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.
Why this shows up in the exam
People walking on boats or carts · Mass transfer within an isolated platform · Keeping a two-particle centre of mass fixed
Learn the idea
Internal motion shifts parts relative to the centre of mass but cannot move an isolated system centre of mass. Internal motion shifts parts relative to the centre of mass but cannot move an isolated system centre of mass. 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: Inside motions balance by mass times displacement.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- sum m_i Delta r_i = M Delta r_cm — Mass-weighted displacement identity for a fixed set of particles.
- Delta r_cm = 0 => sum m_i Delta r_i = 0 — Constraint for internal rearrangement of an isolated system.
How to approach it
- 1Define all displacements in the same frame
- 2Check whether external horizontal impulse vanishes
- 3Apply the mass-weighted displacement relation with signs
Common slip-ups that cost marks
- •Assuming the heavier body moves the same distance
- •Applying the fixed-centre condition when external impulse exists
- •Confusing motion relative to the platform with motion relative to ground
🌟 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 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.