Short on time? This page is the recap for the calculus-based course (separate from Physics 1, which does not combine springs). For explanations, diagrams and worked examples, use the full study guide.
Recap
- An ideal spring has negligible mass and exerts a force proportional to its change in length from its relaxed length. A spring with real mass, or a curved force–extension graph, is nonideal.
- A massless spring pulls with the same size of force at both ends.
- The spring force points back towards the relaxed length: a stretched spring pulls, a compressed spring pushes.
- For a hanging block, equilibrium is at stretch s₀ = mg/k. The net force points towards that equilibrium.
- k is the slope of the force–extension graph.
Key relationships
| Relationship | Meaning |
|---|---|
| |F_s| = k|Δx| | size of an ideal spring force; k in N/m |
| F_s,x = −kx | direction: back towards the relaxed position (x = 0) |
| s₀ = mg/k | stretch of a hanging block at equilibrium |
| 1/k_eq = 1/k₁ + 1/k₂ + … | series (same force); k_eq < smallest k |
| k_eq = k₁ + k₂ + … | parallel (same extension) |
| k_piece = nk | one of n equal pieces of a cut spring |
Assumptions behind the numbers
- Springs are ideal unless stated. g = 9.8 m/s².
- Combinations are all series or all parallel; mixed arrangements are not expected.
- In a parallel set-up, the springs have equal extensions (for example, the bar they hold stays level).
Mistakes to avoid
- Centimetres with N/m. Convert to metres.
- Using the spring’s length instead of its change in length.
- Series and parallel swapped. Ask “same force or same extension?”, not “how does it look?”
- Thinking series springs are stiffer. They are softer than the softest one.
- Spring force versus net force for a hanging block.
- Assuming F = kx for a curved graph. Use the local slope instead.
Quick self-check
- A spring with k = 400 N/m is compressed by 2.5 cm. What force does it exert? (400 × 0.025 = 10 N, pushing outwards)
- Two 200 N/m springs: what is k_eq in series and in parallel? (100 N/m; 400 N/m)
- A 0.50 kg block hangs from a 98 N/m spring. What is its equilibrium stretch? (4.9 ÷ 98 = 0.050 m)
Next: practice questions.