Section 5.7: Systems of Particles

A system of particles is a collection of particles whose total motion can be analyzed using center of mass (CM) concepts and the net external forces.

Center of Mass:
  • \( \vec{R}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i} \)
  • Velocity of CM: \( \vec{V}_{CM} = \frac{\sum m_i \vec{v}_i}{\sum m_i} \)
  • Acceleration of CM: \( \vec{A}_{CM} = \frac{\sum m_i \vec{a}_i}{\sum m_i} \)

Newton's Second Law applies to the CM:

  • \( \vec{F}_{ext} = M_{total} \vec{A}_{CM} \)
  • Internal forces cancel in the sum over the system.

Example 1

Two particles of mass 3 kg and 5 kg are at positions \( \vec{r}_1 = 2\hat{i} \) m and \( \vec{r}_2 = 5\hat{i} \) m. Find the CM.

\( \vec{R}_{CM} = \frac{3*2 + 5*5}{3+5} \hat{i} = \frac{6+25}{8} \hat{i} = 3.875 \hat{i} \) m

Example 2

Three particles (2 kg, 4 kg, 6 kg) have velocities \( \vec{v}_1 = 2\hat{i} \), \( \vec{v}_2 = -1\hat{i} \), \( \vec{v}_3 = 3\hat{i} \) m/s. Find velocity of CM.

\( \vec{V}_{CM} = \frac{2*2 + 4*(-1) + 6*3}{2+4+6} = \frac{4-4+18}{12} = 1.5 \hat{i} \text{ m/s} \)

Practice Problems

  1. Two particles, masses 5 kg & 10 kg, positions \(x_1=1\), \(x_2=4\) m. Find CM.
  2. Three particles, masses 2 kg, 3 kg, 5 kg at \((0,0),(2,1),(4,0)\) m. Find CM coordinates.
  3. Two particles with velocities \(v_1=3\), \(v_2=-2\) m/s, masses 3 kg & 2 kg. Find CM velocity.
  4. A system of 4 particles, total mass 12 kg, CM velocity 2 m/s. Find total momentum.
  5. Two particles collide internally. Show CM motion remains unaffected.
  6. Rocket expels gas. Show CM of rocket+gas obeys \( F_{ext}=M_{total}A_{CM} \).
  7. Three particles on a line: masses 1,2,3 kg, positions 1,3,6 m. Find CM.
  8. Two masses connected by spring. Compute CM velocity if system slides frictionless.
  9. Particle system with velocities along x: 2, -1, 4 m/s; masses 1,2,3 kg. Find CM velocity.
  10. Four particles at corners of square, equal masses. Find CM location.