System of Particles & COM— Notes, Formulas & Revision

Center of mass (COM) is the mass-weighted average position of a system: R_COM = Σm_i·r_i / Σm_i.

For continuous bodies: R_COM = (1/M)·∫r·dm.

For symmetric uniform bodies: COM lies at the geometric center (rod, sphere, cube, ring).

COM does NOT have to lie INSIDE the body — e.g., COM of a ring or a doughnut is at the geometric center (in the hole).

Behaves as if all external force F_ext acts on a particle of total mass M at the COM: F_ext = M·a_COM.

Internal forces (between parts of the system) DO NOT affect the COM motion. Action-reaction pairs cancel.

If no external force: COM moves with constant velocity (or stays at rest).

Useful for analyzing collisions, explosions, rocket motion — all internal dynamics, no change in COM trajectory.

Motion of the center of mass is determined ONLY by EXTERNAL forces: M·a_COM = F_ext_net.

Internal forces (collisions, springs between parts, explosions) cannot change COM motion.

If F_ext = 0: V_COM = constant. System moves as a single particle.

Projectile that explodes mid-flight: COM continues parabolic trajectory until any fragment lands.

Two-body collision in absence of external force: COM moves with V_COM = (m₁v₁ + m₂v₂)/(m₁+m₂), unchanged by the collision.

Spring oscillation in a closed system: COM stays at rest; the two masses oscillate about it.

Earth-Moon system: Earth + Moon both orbit their COM (barycenter), which lies INSIDE the Earth (~4671 km from Earth's center).

Useful frame: COM frame — moves with V_COM. In this frame total momentum is zero; simpler for many problems.

Linear momentum p of a particle: p = m·v. Vector quantity. Units: kg·m/s.

Newton's 2nd law (general form): F = dp/dt. For constant mass: F = m·a.

Total momentum of a system: P = Σ m_i·v_i.

Momentum of a SYSTEM = M·V_COM. So COM moves as if all momentum were a single point particle.

Conservation: if net external force = 0, total P is conserved.

Internal forces in a system change individual momenta but DO NOT change total momentum.

Photons have momentum p = h/λ (no rest mass).

Relativistic momentum: p = γmv = mv/√(1−v²/c²). Reduces to mv for v ≪ c.

Impulse J is the integral of force over time: J = ∫F·dt. Equal to the change in momentum: J = Δp.

For constant force: J = F·Δt. For variable force: J = average F × Δt = area under F-t curve.

Units: N·s = kg·m/s (same as momentum).

Impulse-momentum theorem: Δp = J. Useful when force varies (collisions, hammer blows).

Long contact time, small force ⇒ same impulse as short contact time, large force.

Engineering use: airbags, crumple zones, gloves, padding extend Δt to reduce peak F for the same Δp.

Impulse is a VECTOR — both magnitude and direction matter.

Impulsive force: very large F acting for very short Δt (impacts, gunshots). Gravity is negligible during such events.

Law of conservation of momentum: in an isolated system (no external force), total linear momentum is constant.

Derived from Newton's 3rd law: internal forces come in equal-and-opposite pairs, so they cancel in total.

In collisions: p_total before = p_total after, ALWAYS (whether elastic or not).

Components: P_x, P_y, P_z all conserved independently — pick axes wisely.

Useful for collisions, explosions, recoil problems (gun-bullet), rocket motion.

Conservation holds in any inertial reference frame; numerical values of P depend on frame, but conservation is universal.

Distinction from energy: momentum is ALWAYS conserved in collisions; kinetic energy is conserved only in ELASTIC collisions.

Photons carry momentum; conservation applies to atomic/nuclear reactions too.

Angular momentum L of a particle about a point O: L = r × p, where r is the position vector from O and p = mv.

Magnitude: L = m·v·r·sin θ = m·v·r_perp, where r_perp = perpendicular distance from O to line of motion.

Units: kg·m²/s = J·s.

Direction: perpendicular to plane containing r and p (right-hand rule).

Torque is the rate of change of angular momentum: τ = dL/dt. If τ_ext = 0, L is conserved.

For a particle in uniform circular motion: L = mvr (constant magnitude, direction along axis).

Conservation of L is a fundamental law — distinct from conservation of linear momentum.

Examples: planets around the Sun (L conserved → Kepler's 2nd law), ice skater pulling in arms (L = Iω constant ⇒ ω rises as I drops).

Rocket propulsion is a clean application of conservation of momentum: rocket ejects exhaust mass backward, gaining forward momentum.

Thrust F = u_e × (dm/dt), where u_e = exhaust velocity relative to rocket, dm/dt = mass-burning rate.

Tsiolkovsky rocket equation: Δv = u_e · ln(m_0/m_f), where m_0 = initial mass, m_f = final mass.

Δv determines mission feasibility. Reaching low Earth orbit: Δv ≈ 9-10 km/s.

Specific impulse I_sp = u_e/g₀ — measure of fuel efficiency. Chemical rockets: I_sp ≈ 300-450 s. Ion thrusters: ~3000+ s.

Mass ratio m_0/m_f determines how much velocity you can build up. Beyond ~10-20, structural and engineering limits dominate.

Multistaging (jettisoning empty fuel tanks) allows higher final Δv with reasonable mass ratios.

Newton's 3rd law in disguise: exhaust gas pushed backward ⇒ rocket pushed forward.