Short on time? This page is the recap. For explanations, diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- A system is what you choose to study; the rest is the environment. Show the choice with a dashed boundary.
- Internal interactions are between parts inside the boundary. External interactions cross it.
- A system’s properties come from its parts and their interactions. Its internal structure affects how you analyse it.
- If the parts do not matter for the question, treat the system as one object at its center of mass, with the total mass.
- Parts can move differently from each other and from the system as a whole. Changing outside conditions can change the structure (a stack that starts to slide apart).
- Energy or mass can cross the boundary (an open system). Moving the boundary can often make the system closed.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| x_cm = Σmᵢxᵢ ÷ Σmᵢ | m in kg, x in m | locate the center of mass along x |
| y_cm = Σmᵢyᵢ ÷ Σmᵢ | m in kg, y in m | locate it along y (separate calculation) |
| symmetry rule | — | place it on every line of symmetry without arithmetic |
| uniform symmetric part → one particle at its centre | — | handle a plank, rod or plate in a calculation |
| distance ratio = inverse mass ratio (two particles) | — | check a two-particle answer |
Assumptions behind the numbers
- Small objects are particles: all their mass at one point.
- “Uniform” means mass is spread evenly, so the part’s center of mass is at its geometric centre.
- Light frames, rods or strings have negligible mass unless a mass is given.
- The course calculates centers of mass for up to five particles in two dimensions, or for highly symmetric systems.
Mistakes to avoid
- No origin stated. The value of x_cm means nothing without one.
- Midpoint used for unequal masses. The center of mass is nearer the heavier mass.
- Dividing by the number of particles instead of the total mass.
- Forgetting a symmetric part’s own mass, such as the plank under the objects on it.
- Assuming the center of mass is inside the material. A ring’s is in its empty centre.
- Treating a system as one object when its parts move relative to each other in a way that matters.
- Calling an interaction internal without drawing the boundary. It depends on the system you chose.
Quick self-check
- With +x to the right and the origin at a 4.0 kg mass, a 1.0 kg mass sits at x = 1.0 m. Where is the center of mass? (x_cm = 0.20 m)
- Where is the center of mass of a uniform square tile? (Where its diagonals cross: on all four lines of symmetry.)
- Three equal masses sit at x = 0, 0.30 m and 0.90 m. Where is their center of mass? (0.40 m: with equal masses, the plain mean of the positions.)
- A student pushes a box that has a smaller box resting on it. For the system “both boxes”, is the friction between the boxes internal or external? (Internal: both boxes are inside the boundary.)
Next: practice questions.