System of Particles and Rotational Motion

Syllabus Unit: Mechanics

Why it matters

Chapter overview

This topic extends mechanics from single point masses to extended, rotating bodies — the centre of mass that lets a whole system move like one particle, torque and angular momentum as the rotational analogues of force and momentum, moment of inertia as the measure of rotational inertia, and rotational kinetic energy in rolling motion, all central to how NEET tests rigid-body dynamics.

Physics Syllabus Mechanics 30%–35% weightage

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Centre of Mass

1Particle m₁ at x₁Contributes m₁x₁ to the sum

2Particle m₂ at x₂Contributes m₂x₂ to the sum

3External Force F_extOnly this can accelerate the CM

4a_cm = F_ext/MCM moves like a single particle of mass M

1Centre of Mass & Its Motion

Every system of particles, however messy internally, moves as if all its mass were concentrated at one point — the centre of mass — obeying Newton's second law under only external forces.

Learning Objectives

Compute the centre of mass position for a discrete system of particles: x_cm = Σmᵢxᵢ / Σmᵢ.
Locate the centre of mass of common symmetric bodies (rod, ring, disc, sphere) by symmetry.
State that internal forces cannot move the centre of mass; only a net external force can accelerate it.
Apply centre of mass motion to explain phenomena like projectile fragmentation and recoil.

Key Terms

Centre of MassThe unique point where the entire mass of a system can be considered concentrated for describing its overall translational motion.

x_cm FormulaFor discrete particles, x_cm = (Σmᵢxᵢ)/(Σmᵢ), extended similarly for y and z coordinates.

Internal ForcesForces particles of a system exert on each other (e.g. explosion fragments pulling/pushing); these always cancel in pairs by Newton's third law and cannot shift the centre of mass.

External ForceA force from outside the system (e.g. gravity, an external push); this is the only kind of force that can change the centre of mass's velocity.

The centre of mass lets you treat a complicated, many-particle system's overall motion as if it were a single point mass, no matter how the particles move relative to each other.

Locating the centre of mass

x_cm = (Σmᵢxᵢ)/(Σmᵢ) ; similarly for y_cm, z_cm

For symmetric homogeneous bodies, the centre of mass lies at the geometric centre by symmetry — centre of a uniform ring, disc, or sphere; midpoint of a uniform rod.

The key property: internal forces don't matter

Because internal forces between particles of a system always occur in equal-and-opposite Newton's-third-law pairs, they cancel exactly when summed over the whole system — so the centre of mass moves exactly as though the total mass M were a single particle acted on only by the total EXTERNAL force.

F_ext = M·a_cm (Newton's second law applied to the centre of mass)

Key PointClassic application — a projectile explodes mid-air into fragments under purely internal forces (the explosion); the centre of mass of all fragments continues along the SAME parabolic trajectory the unexploded projectile would have followed.

Common MistakeA common NEET trap — the centre of mass of a system need not lie within the material of any of its parts, e.g. for a ring or hollow sphere, the centre of mass is at the empty geometric centre, not on the material itself.