Section 5.5: Center of Mass and Momentum

The center of mass (CM) of a system of particles is the weighted average of their positions. For particles of masses \(m_i\) at positions \(\vec{r}_i\):

  • \(\vec{R}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i}\)

For continuous bodies, integrals are used:

  • \(\vec{R}_{CM} = \frac{1}{M}\int \vec{r} \, dm\)

Linear momentum of a system:

  • \(\vec{P}_{total} = \sum m_i \vec{v}_i = M \vec{V}_{CM}\)

External forces change the momentum of the CM: \(\vec{F}_{ext} = \frac{d \vec{P}_{total}}{dt}\)

Example 1

Two particles: 2 kg at x=0 m and 3 kg at x=4 m. Find the center of mass.

\(X_{CM} = \frac{2*0 + 3*4}{2+3} = \frac{12}{5} = 2.4 \text{ m}\)

Example 2

A 1000 kg car moves at 20 m/s. Another 1500 kg truck moves at 10 m/s in same direction. Find velocity of CM.

\(V_{CM} = \frac{1000*20 + 1500*10}{1000+1500} = \frac{20000+15000}{2500} = 14 \text{ m/s}\)

Practice Problems

  1. Three masses: 1 kg at 0 m, 2 kg at 3 m, 3 kg at 6 m. Find CM.
  2. Two carts, 5 kg each, moving at 3 m/s and -2 m/s. Find CM velocity.
  3. A 0.5 kg ball at 4 m/s collides with 1 kg ball at rest. Find CM velocity.
  4. Particles: 2 kg at (0,0), 3 kg at (4,0). Find CM coordinates.
  5. A 1000 kg car at 15 m/s collides elastically with 1200 kg truck at 0 m/s. Find CM velocity.
  6. Four particles at corners of a square: 1 kg each. Find CM.
  7. Two ice skaters, 50 kg and 70 kg, push apart. Find velocity of CM after push.
  8. Two 2 kg balls moving in x-y plane. Find CM position.
  9. A system of three particles: masses 1,2,3 kg at x=0,2,5 m. Find CM and velocity if moving.
  10. Four carts on frictionless track. Compute CM velocity and position.
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