Short on time? This page is the recap for the calculus-based course (separate from Physics 1). For derivations, figures and worked examples, use the full study guide.
Recap
- Period T: time for one full cycle (s). Frequency f: cycles per second (Hz). Angular frequency ω: rad/s.
- Method: apply Newton’s second law (or τ = Iα), reach d²x/dt² = −ω²x, read off ω, then T = 2π/ω.
- Spring: ω² = k/m. Use the effective k for springs in series or parallel. Gravity does not change T.
- Simple pendulum: ω² = g/ℓ for small angles in radians (sin θ ≈ θ). The bob’s mass cancels.
- Neither period depends on the amplitude while the SHM model holds.
Key relationships
| Relationship | Notes |
|---|---|
| f = 1/T; ω = 2πf = 2π/T | units: s, Hz, rad/s |
| d²x/dt² = −ω²x | the constant multiplying x is ω² |
| T = 2π√(m/k) | object on an ideal spring, any orientation |
| T = 2π√(d/g) | vertical spring with static stretch d = mg/k |
| T = 2π√(ℓ/g) | simple pendulum, small angles only |
| T² = (4π²/k)m; T² = (4π²/g)ℓ | straight-line graphs: slope gives k or g |
| k_eff = k₁ + k₂ (parallel); 1/k_eff = 1/k₁ + 1/k₂ (series) | from Topic 2.8 |
Assumptions behind the numbers
- Ideal, massless springs and light, inextensible strings unless stated.
- No friction or air resistance; g = 9.8 m/s² near Earth’s surface.
- A small angle is one where sin θ ≈ θ (θ in radians) is accurate enough.
Mistakes to avoid
- Mixing f and ω. ω = 2πf.
- Forgetting the square root in factor-of-change questions.
- Giving a pendulum a mass dependence or a spring a g dependence.
- Using T = 2π√(ℓ/g) at large angles.
- Dividing one total time by the wrong number of cycles. Count cycles, not swings past the middle.
- Finding k from one reading when the T²–m graph has an intercept.
Quick self-check
- f = 2.5 Hz. Find T and ω. (0.40 s; 16 rad/s)
- m = 0.50 kg on a spring with k = 50 N/m. Find T. (2π√0.010 = 0.63 s)
- A pendulum has ℓ = 2.45 m. Find its small-angle period. (2π√0.25 = 3.1 s)
- A spring oscillator’s mass is multiplied by 9. What happens to T? (× 3)
Next: practice questions.