Introduction
Total momentum of a system is the vector sum of individual momenta. Linear momentum is conserved when external force is zero. Angular momentum includes both orbital (CM motion) and spin (about CM) components. These conservation laws are fundamental for analyzing collisions, explosions, and rotating systems.
Total Linear Momentum
P = Σ p_i = Σ m_i v_i = M V_CM. Total momentum equals total mass times CM velocity. Conservation: if F_ext = 0, then P = constant. Center of mass moves uniformly when net external force is zero. This applies regardless of complex internal forces and motions within the system.
Total Angular Momentum
L = Σ L_i = Σ r_i × p_i. Decomposition: L = R_CM × P + L_CM where first term is orbital angular momentum (CM motion about origin) and L_CM = Σ (r_i' × p_i') is spin angular momentum about CM. Total L depends on reference point; orbital part changes with origin choice, spin part is origin-independent.
\nIntroduction
Total momentum of a system is the vector sum of individual momenta. Linear momentum is conserved when external force is zero. Angular momentum includes both orbital (CM motion) and spin (about CM) components. These conservation laws are fundamental for analyzing collisions, explosions, and rotating systems.
Total Linear Momentum
P = Σ p_i = Σ m_i v_i = M V_CM. Total momentum equals total mass times CM velocity. Conservation: if F_ext = 0, then P = constant. Center of mass moves uniformly when net external force is zero. This applies regardless of complex internal forces and motions within the system.
Total Angular Momentum
L = Σ L_i = Σ r_i × p_i. Decomposition: L = R_CM × P + L_CM where first term is orbital angular momentum (CM motion about origin) and L_CM = Σ (r_i' × p_i') is spin angular momentum about CM. Total L depends on reference point; orbital part changes with origin choice, spin part is origin-independent.
\nAngular Momentum Conservation
dL/dt = τ_ext. If net external torque is zero, total angular momentum is conserved. Applies to isolated systems and systems with central forces. Internal torques cancel in action-reaction pairs. Conservation holds separately for each component if corresponding torque component is zero.
Torque on a System
τ_ext = Σ r_i × F_i,ext. Includes torque about chosen origin from all external forces. For rotation about CM: τ_CM = dL_CM/dt. External torque changes angular momentum regardless of reference point. Internal torques sum to zero (Newton's Third Law).
Combined Conservation
For isolated systems: both P and L are conserved. Six constants of motion for 3D system (3 components of P, 3 of L). Constraints reduce degrees of freedom. Collisions between isolated systems conserve both momenta. These conservation laws greatly simplify complex multi-body problem analysis.
\nSolved Example: Exploding Projectile
A 10 kg projectile is launched at 50 m/s at 60° above horizontal. At peak height, it explodes into two fragments: 4 kg and 6 kg. The 4 kg fragment moves horizontally at 30 m/s just after explosion. Find velocity of 6 kg fragment and subsequent CM motion. Solution: First find peak height velocity. At peak, v_y = 0, v_x = 50cos60° = 25 m/s. Just before explosion: v = 25\(\hat{\imath}\) m/s. Momentum before: p = 10×25 = 250 kg·m/s \(\hat{\imath}\). After explosion: p = 4×30\(\hat{\imath}\) + 6×v2 = 250\(\hat{\imath}\) → 120 + 6v2 = 250 → v2 = 130/6 = 21.67 m/s \(\hat{\imath}\). The 6 kg fragment moves at 21.67 m/s horizontally, same direction as original motion. CM motion: unaffected by internal explosion. CM continues original parabolic trajectory. At peak, CM velocity is 25 m/s horizontal. CM will land at same point as if no explosion occurred. Internal forces only affect how fragments move relative to CM, not CM motion itself.
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