Flashcards — Chapter 9 🃏

Center of Mass and Momentum — the framework for understanding multi-body systems! Master the center of mass, linear momentum, collisions, and impulse with these cards. 🚀

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📋 Card deck: Key terms & formulas

Section §9.1 — Center of mass

1. Center of mass (discrete) — \( \mathbf{r}_{\text{CM}} = \frac{1}{M}\sum m_i \mathbf{r}_i \) where \( M = \sum m_i \); weighted average position
2. Center of mass (continuous) — \( \mathbf{r}_{\text{CM}} = \frac{1}{M}\int \mathbf{r} \, dm \) for extended bodies; integration replaces summation
3. Cartesian components — \( x_{\text{CM}} = \frac{1}{M}\sum m_i x_i \); similar for \( y \) and \( z \) coordinates
4. Symmetric object — for objects with symmetry (uniform sphere, cylinder, disk), CM is at the geometric center
5. Composite system — CM of an object made of two parts: treat each part as point mass at its own CM

Section §9.2 — Uniform and composite bodies

6. Uniform rod — center of mass is at the midpoint (length \( L \)): \( x_{\text{CM}} = L/2 \)
7. Uniform disk (2D) — CM is at geometric center; also true for uniform rectangular plate
8. Sphere — uniform sphere's CM is at its geometric center, regardless of radius
9. Composite body — find CM of each component, then treat each component as point mass; \( \mathbf{r}_{\text{CM}} = \frac{m_1 \mathbf{r}_1 + m_2 \mathbf{r}_2 + \cdots}{m_1 + m_2 + \cdots} \)
10. Cavity (negative mass) — object with a hole: treat the hole as "negative mass" and subtract its CM contribution

Section §9.3 — Motion of the center of mass

11. Velocity of CM — \( \mathbf{v}_{\text{CM}} = \frac{d\mathbf{r}_{\text{CM}}}{dt} = \frac{1}{M}\sum m_i \mathbf{v}_i \); average velocity weighted by mass
12. Acceleration of CM — \( \mathbf{a}_{\text{CM}} = \frac{d\mathbf{v}_{\text{CM}}}{dt} = \frac{1}{M}\sum m_i \mathbf{a}_i \); average acceleration
13. Newton's second law for CM — \( \mathbf{F}_{\text{ext}} = M\mathbf{a}_{\text{CM}} \); total external force accelerates the CM
14. Internal forces cancel — forces between particles (Newton's third law pairs) do not affect CM motion
15. External force only — only external forces determine CM motion; internal forces are irrelevant
16. Momentum and CM — \( \mathbf{p}_{\text{total}} = M\mathbf{v}_{\text{CM}} \); total momentum equals mass times CM velocity

Section §9.4 — Linear momentum and impulse

17. Linear momentum — \( \mathbf{p} = m\mathbf{v} \) (kg·m/s); measure of how hard it is to stop an object
18. Newton's second law (momentum form) — \( \mathbf{F} = \frac{d\mathbf{p}}{dt} \); force is the rate of change of momentum
19. Impulse — \( \mathbf{J} = \int \mathbf{F} \, dt \) (N·s); change in momentum from a force applied over time
20. Impulse-momentum theorem — \( \mathbf{J} = \Delta \mathbf{p} = \mathbf{p}_f - \mathbf{p}_i \); impulse equals change in momentum
21. Average force and time — \( \mathbf{F}_{\text{avg}} \cdot \Delta t = \Delta \mathbf{p} \); same impulse from different force–time combinations
22. Momentum and energy (difference) — momentum is \( m\mathbf{v} \) (vector, first-order in \( v \)); energy is \( \frac{1}{2}m v^2 \) (scalar, second-order)

Section §9.5 — Conservation of momentum

23. Law of momentum conservation — if \( \mathbf{F}_{\text{ext}} = 0 \), then \( \mathbf{p}_{\text{total}} = \text{constant} \)
24. Isolated system — system with no external forces (or negligible ones); total momentum is conserved
25. Internal forces and momentum — collision forces between two objects are equal and opposite (Newton's 3rd law); momentum is exchanged but total is conserved
26. Momentum before and after — \( \mathbf{p}_{\text{before}} = \mathbf{p}_{\text{after}} \) in any collision or explosion (if isolated)
27. Rocket propulsion — as fuel is expelled (internal force), the rocket accelerates; momentum of expelled gas plus rocket is conserved
28. Explosion — internal chemical forces push fragments apart; momentum is conserved but kinetic energy increases

Section §9.6 — Collisions in one dimension

29. Elastic collision — kinetic energy is conserved: \( KE_{\text{before}} = KE_{\text{after}} \)
30. Inelastic collision — kinetic energy is lost (to deformation, heat, sound): \( KE_{\text{after}} < KE_{\text{before}} \)
31. Perfectly inelastic collision — objects stick together; minimum kinetic energy is lost (consistent with momentum conservation)
32. Elastic collision (two objects) — if \( m_1 \) moving at \( v_1 \) hits stationary \( m_2 \): final velocities depend on mass ratio
33. Equal mass elastic collision — if \( m_1 = m_2 \) and \( m_2 \) is initially at rest, they exchange velocities
34. Coefficient of restitution — \( e = -\frac{v_2 - v_1}{u_2 - u_1} \) (ratio of relative velocities before and after)
35. Elastic collision (coefficient) — \( e = 1 \) for elastic; \( e = 0 \) for perfectly inelastic
36. Momentum conservation in collision — \( m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \) (always holds if isolated)

Section §9.7 — Multi-dimensional collisions and rotations

37. Two-dimensional collision — analyze \( x \) and \( y \) components separately; momentum conserved in each direction
38. Angular momentum — \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \); measure of rotational motion; conserved if torque is zero
39. Torque and angular momentum — \( \boldsymbol{\tau} = \frac{d\mathbf{L}}{dt} \); torque is rate of change of angular momentum
40. Spinning top (gyroscope) — angular momentum vector points along rotation axis; precesses under gravity
41. Conservation of angular momentum — \( \mathbf{L}_{\text{total}} = \text{constant} \) if \( \boldsymbol{\tau}_{\text{ext}} = 0 \); analogous to momentum conservation

Additional section — Important equations

42. Momentum-velocity — \( \mathbf{p} = m\mathbf{v} \) (linear momentum)
43. Impulse-momentum — \( \mathbf{J} = \mathbf{F}_{\text{avg}} \Delta t = \Delta \mathbf{p} \)
44. Total momentum — \( \mathbf{p}_{\text{total}} = \sum m_i \mathbf{v}_i = M\mathbf{v}_{\text{CM}} \)
45. Center of mass (compact) — position of CM is \( \mathbf{r}_{\text{CM}} = \frac{\sum m_i \mathbf{r}_i}{M} \); it moves as if all external force acts there

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