Chapter Introduction
Many Physics problems involve more than one object: interacting bodies, recoil, explosions, collisions, a man on a plank, or bodies connected by a spring. The System of Particles and Centre of Mass approach gives a simpler way to describe what the complete system is doing.
Key question: “What is my system?” Once the system is chosen, its boundary tells us which interactions are internal and which are external. Momentum, impulse and Centre of Mass then become connected descriptions of the same physical system.
Chapter in One View
Central Idea: Internal interactions can change the arrangement and relative motion of the parts. External influence changes the collective motion of the complete system.
- Choose the system and define its boundary.
- Classify interactions as internal or external.
- Use total momentum to describe collective motion.
- Use external impulse to decide whether momentum changes.
- Look at mass distribution to locate the Centre of Mass.
- Use external force to understand COM motion.
- Use the COM frame when common translational motion hides relative motion.
Must-Know Concepts
System of Particles — A chosen group of particles or bodies studied together.
Internal Force — Interaction between members of the chosen system.
External Force — Interaction between the system and something outside it.
Total Momentum — Represents the combined momentum of all particles in the system.
External Impulse — External impulse changes total momentum.
Momentum Conservation — If net external impulse is zero/negligible over the chosen interval → total momentum remains constant.
Centre of Mass (CM) — Mass-weighted position of the complete system.
COM methods: Discrete masses, symmetry, continuous distribution, composite bodies and bodies with holes.
COM motion: External force controls COM acceleration for a complete fixed-mass system in an inertial frame.
COM frame: A frame moving with the Centre of Mass; it removes common translational motion and highlights relative motion.
Formula + Meaning
Momentum Relations
Total Momentum
Meaning: Total momentum of the system. Use: When collective momentum or COM velocity is asked. Condition: M is the total mass of the chosen system. Exam Trap: Do not confuse one particle’s velocity with VCM.
Force and COM Acceleration
External Force Relation
Meaning: Net external force determines COM acceleration. Use: For a complete fixed-mass system. Condition: Inertial reference frame. Exam Trap: Choose the correct system before applying it.
Impulse and Momentum Change
Impulse Relation
Meaning: External impulse produces change in total momentum. Use: For finite interval and external impulse/force. Condition: Use net external impulse on the chosen system. Exam Trap: Momentum is conserved only when net external impulse ≈ 0.
Centre of Mass Position
Discrete System
Meaning: Mass-weighted position of a discrete system. Use: When masses and positions are known. Condition: M is the total mass. Exam Trap: COM is not a literal point where all mass is concentrated.
Continuous Distribution
Meaning: Mass-weighted position for a continuous distribution. Use: When continuous mass description is required. Condition: dm must represent the actual distribution. Exam Trap: First understand density/mass distribution before constructing the integral.
Formula Selection Guide
| Situation | Think This Model | Use This |
|---|---|---|
| Several interacting bodies | Choose the system | Define system boundary |
| Total momentum asked | Collective motion | \(\mathbf{P}=\sum_i m_i\mathbf{v}_i\) |
| Total mass + COM velocity | System momentum | \(\mathbf{P}=M\mathbf{V}_{CM}\) |
| External force on complete system | COM dynamics | \(\mathbf{F}_{\mathrm{ext}}=M\mathbf{A}_{CM}\) |
| Finite interval + external force | Momentum change | \(\mathbf{J}_{\mathrm{ext}}=\Delta\mathbf{P}\) |
| Zero/negligible external impulse | Momentum conservation | \(\mathbf{P}_f=\mathbf{P}_i\) |
| Known masses and positions | Discrete COM | \(\mathbf{r}_{CM}=\frac{\sum_i m_i\mathbf{r}_i}{M}\) |
| Symmetric mass distribution | COM by symmetry | Locate directly |
| Continuous distribution | Distributed mass | \(\mathbf{r}_{CM}=\frac{1}{M}\int \mathbf{r}\,dm\) |
| Composite body | Component COMs | Mass-weighted combination |
| Body with a hole | Missing mass | Negative-mass method |
| Parts move but external force is zero | Collective vs internal motion | Check COM motion |
| Two-body common translation | COM frame | Remove \(\mathbf{V}_{CM}\) |
When to Use / When Not to Use
Momentum Conservation
Use when: Net external impulse is zero or negligible over the selected interval.
Do not use blindly when: A significant external impulse acts.
COM Motion Relation
Use when: A complete fixed-mass system is considered in an inertial frame.
Do not use blindly when: The system boundary or frame conditions are not satisfied.
Discrete COM Formula
Use when: Individual masses and positions are identifiable.
Do not use blindly when: A continuous-density treatment is required.
Symmetry
Use when: The mass distribution has the required symmetry.
Do not use blindly when: Geometry or density breaks that symmetry.
Common Misconceptions
Wrong idea: “If the parts move, the COM must move.” → Correct Physics: Parts can move relative to each other while the COM remains fixed or moves uniformly. → Why it matters: Check external force on the complete system.
Wrong idea: “Momentum is always conserved.” → Correct Physics: It is conserved over an interval only when net external impulse is zero or negligible. → Why it matters: Check external impulse first.
Wrong idea: “Internal forces change total momentum.” → Correct Physics: For the complete system, internal action-reaction contributions cancel. → Why it matters: Define the system boundary first.
Wrong idea: “The COM is where all mass is physically concentrated.” → Correct Physics: It is a mass-weighted position used to represent the system. → Why it matters: Do not confuse representation with physical concentration.
Wrong idea: “COM problems always require integration.” → Correct Physics: Symmetry, discrete and composite methods may be simpler. → Why it matters: Look at mass distribution before calculating.
Exam Traps
- Choosing the formula before choosing the system.
- Using momentum conservation without checking external impulse.
- Confusing individual particle velocity with COM velocity.
- Ignoring symmetry and doing unnecessary calculation.
- Treating a hole as ordinary positive mass.
- Starting a continuous-body problem with integration before identifying dm.
- Forgetting the conditions of the COM motion relation.
- Following every object separately when a complete-system description is simpler.
See This → Think This
| See This | Think This | Recognition Cue |
|---|---|---|
| Several interacting bodies | Choose the system first. | Define the boundary before writing equations. |
| Total momentum asked | Think collective motion. | Use system momentum / COM velocity. |
| No external impulse | Check momentum conservation. | Consider Pf = Pi. |
| Known masses and positions | Think mass-weighted COM. | Use the discrete COM relation. |
| Equal masses | Ordinary average of positions. | Mass weighting becomes a simple average. |
| Symmetric body | Check COM from symmetry. | Locate directly before calculating. |
| Complicated body made of simple pieces | Composite-body method. | Combine component COMs by mass weighting. |
| Body with a cut-out | Missing mass / negative mass. | Subtract the missing mass contribution. |
| Parts move but no net external force | Separate internal motion from COM motion. | Internal motion need not move the COM. |
| Two bodies with common translational motion | Consider the COM frame. | Remove common translation. |
| Explosion, recoil or collision | Identify the system and check external impulse first. | Momentum is a vector. |
Key Diagrams / Physical Situations
Quick Checks
Q: What should you choose before classifying a force? → The system and its boundary.
Q: What changes total momentum? → Net external impulse.
Q: When is momentum conserved? → When net external impulse is zero or negligible.
Q: Equal masses at two positions: which shortcut? → Average of the positions.
Q: Symmetric uniform body: what first? → Check COM from symmetry.
Q: Body with a hole: which method? → Negative-mass method.
Q: Can parts move while COM stays fixed? → Yes, under appropriate zero-external-force conditions.
Q: What does the COM frame remove? → Common translational motion.
Mini Exam Reasoning
Clue → Physics Situation → Model → Relation → Check
Clue: A system is isolated horizontally.
Physics situation: No external horizontal impulse/force. Model: Complete system.
Relation: Momentum conserved / COM has no horizontal acceleration.
Check: Confirm the system boundary.
Clue: A complicated body contains a hole.
Physics situation: Non-uniform shape with a known missing region.
Model: Composite / negative-mass method.
Relation: Mass-weighted COM.
Check: Give the missing mass a negative sign.
Clue: Two bodies separate after an explosion.
Physics situation: Internal redistribution.
Model: Complete system.
Relation: Pf = Pi if external impulse is negligible.
Check: Treat momentum as a vector.
Clue: A two-body problem has large common translational velocity.
Physics situation: Collective + relative motion.
Model: COM frame.
Relation: v′i = vi − VCM.
Check: Relative velocity is unchanged.
One-Minute Recall
- System → Define the boundary.
- Internal force → Interaction within the chosen system.
- External force → Interaction across the boundary.
- Momentum relation → P = Σmivi = MVCM.
- Impulse relation → Jext = ΔP.
- Momentum conservation → Zero/negligible external impulse → Pf = Pi.
- Centre of Mass (COM) → Mass-weighted position.
- Symmetry → Possible direct COM shortcut.
- Composite body → Combine component COMs.
- Body with a hole → Subtract missing mass (negative-mass method).
- Force relation → Fext = MACM.
- Condition → Complete fixed-mass system, inertial frame.
- COM frame → Removes common translational motion: v′i = vi − VCM.
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