Short on time? This page is the recap for the calculus-based course. For diagrams and worked examples, use the full study guide.
Recap
- Every force is one half of an interaction between two objects.
- If A pushes or pulls on B, then B pushes or pulls on A at the same instant, with equal size and opposite direction.
- The pair is always the same type of force and acts on two different objects.
- Show a pair on two separate free-body diagrams, with equal-length arrows and matching labels.
- Internal forces (between parts of your chosen system) cancel in pairs. They cannot change how the system’s centre of mass moves.
- Tension is the combined effect of string segments pulling on their neighbours.
Key relationships
| Idea | Statement | Notes |
|---|---|---|
| Third law | F_B on A = −F_A on B | names swapped, same type, same instant |
| System of particles | ΣF (all parts) = ΣF_ext | internal pairs add to zero |
| Hanging string at rest | dT/dy = λg | y measured up from the bottom; λ in kg/m |
| Tension along a hanging string | T(y) = T(0) + ∫₀ʸ λg dy′ | largest at the top |
| Ideal string | λ → 0, so T is the same everywhere | also does not stretch |
| Ideal pulley | same T on both sides | massless, frictionless axle |
Assumptions behind the numbers
- g = 9.8 m/s² near Earth’s surface.
- “Light”, “massless” or “ideal” string: ignore its mass and stretch.
- dT/dy = λg holds for a string hanging at rest; other situations need the second law (Topic 2.5).
Mistakes to avoid
- Pairing weight with normal force. Same object, different types: not a pair.
- Bigger mass, bigger force. The two forces in an interaction are always equal in size.
- Cancelling partners on one diagram. They belong on different diagrams.
- Forgetting the system choice. A force is internal or external only relative to the system you chose.
- Assuming equal tension in a heavy rope. Only an ideal string has the same tension throughout.
- Putting the hand’s push on the far block. A force appears only on the object it touches (or attracts).
Quick self-check
- A cyclist’s tyre pushes backwards on the road. What is the partner force? (The road pushes forwards on the tyre, same size, friction.)
- A uniform 4.0 m rope of mass 2.0 kg hangs at rest from a beam. What is the tension 1.0 m below the top? (It supports the 3.0 m below, mass 1.5 kg: T = 1.5 × 9.8 ≈ 15 N.)
- Two skaters, at rest together on smooth ice, push apart. Does their centre of mass move? (No. The pushes are internal and nothing else acts horizontally.)
Next: practice questions.