Dynamics of Systems of Particles

Why it matters

Understanding the dynamics of systems of particles is crucial for designing and analyzing complex mechanical systems such as vehicles, machinery, and structural components. It helps engineers predict how systems will respond to various forces and motions, ensuring safety and efficiency.

Key ideas

• System of Particles: A selected collection of particles. Their relative positions may change; the collection need not form a rigid body.
• Center of Mass: The point representing the average position of the mass in a system of particles.
• Newton's Second Law for a System: The total external force acting on a system of particles is equal to the rate of change of the total momentum of the system.
• Conservation of Momentum: In the absence of external forces, the total momentum of a system remains constant.
• Kinetic Energy of a System: The sum of the kinetic energies of all particles in the system.

Centre-of-mass relations

For a fixed collection of constant-mass particles, M = Σm_i, r_G = Σm_i r_i/M, P = Mv_G, and ΣF_ext = Ma_G. Internal forces cancel pairwise in the linear-momentum balance under Newton’s third law. An open system exchanging mass needs momentum-flux terms.

Kinetic energy separates as K = (1/2)Mv_G² + (1/2)Σm_i|v_i − v_G|². The second term measures motion relative to the centre of mass; K is not generally just (1/2)Mv_G².

For the example below, v_G = 14/5 = 2.8 m/s. Centre-of-mass translation accounts for 19.6 J, and relative motion accounts for 2.4 J, giving 22 J in total.

Formulas

• F_ext = dP/dt
  • F_ext: Total external force (N)
  • dP/dt: Rate of change of total momentum (kg·m/s²)
• P = Σm_i·v_i
  • P: Total momentum (kg·m/s)
  • m_i: Mass of particle i (kg)
  • v_i: Velocity of particle i (m/s)
• K = 1/2 Σm_i·v_i²
  • K: Total kinetic energy (J)
  • m_i: Mass of particle i (kg)
  • v_i: Velocity of particle i (m/s)

Worked example

Given: A system of two particles, A and B, with masses 2 kg and 3 kg, moving along the same +x direction with velocities 4 m/s and 2 m/s respectively, measured in an inertial frame.

1. Calculate the total momentum of the system.

   • Formula: P = Σm_i·v_i
   • Calculation: P = (2 kg × 4 m/s) + (3 kg × 2 m/s) = 8 kg·m/s + 6 kg·m/s = 14 kg·m/s

2. Calculate the total kinetic energy of the system.

   • Formula: K = 1/2 Σm_i·v_i²
   • Calculation: K = 1/2 [(2 kg × (4 m/s)²) + (3 kg × (2 m/s)²)] = 1/2 [32 kg·m²/s² + 12 kg·m²/s²] = 22 J

Final Answer: Total momentum = 14 kg·m/s, Total kinetic energy = 22 J

Common mistakes

• Confusing the center of mass with the geometric center of the system.
• Including internal action–reaction force pairs in the net external force. These cancel for the whole system, although they matter for individual particles and may do net work if separations change.
• Forgetting to square the velocity when calculating kinetic energy.

For GATE ME

Questions often involve calculating the center of mass, total momentum, and kinetic energy of a system of particles. Practice problems that require applying conservation laws and analyzing systems with varying external forces.

Quick check

1. What is the formula for total momentum of a system of particles?
2. How does the total kinetic energy of a system change if all particle velocities double?
3. What happens to the total momentum of a system if no external forces act on it?

Answers: 1. P = Σm_i·v_i 2. It quadruples. 3. It remains constant.