Short on time? This page is the recap. For explanations, the graph and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Period T: time for one full cycle (s). Frequency f: cycles per second (Hz). They are reciprocals.
- An object on an ideal spring oscillates faster with a stiffer spring and slower with more mass. Gravity does not change its period.
- A small-angle pendulum oscillates slower with a longer string and faster with stronger gravity. The bob’s mass does not change its period.
- For SHM, the period does not depend on the amplitude.
- Time many cycles and divide, starting and stopping as the object passes equilibrium.
Key relationships
| Relationship | Depends on | Does not depend on / use |
|---|---|---|
| T = 1/f | frequency f (Hz) | converts between T and f |
| T_s = 2π√(m/k) | mass m, spring constant k | not g, amplitude, or vertical vs horizontal |
| T_p = 2π√(ℓ/g) | length ℓ (pivot to centre of bob), g | not bob mass or amplitude (small angles) |
| T_s² = (4π²/k) m | — | plot T² against m: straight line, slope 4π²/k |
| T_p² = (4π²/g) ℓ | — | plot T² against ℓ: straight line, slope 4π²/g |
Assumptions behind the numbers
- The spring is ideal (massless, obeys Hooke’s law); the pendulum string is light and does not stretch.
- The pendulum swings through a small angle, so the SHM model holds.
- No friction or air resistance, so the amplitude stays constant.
- g = 9.8 m/s² near Earth’s surface.
Mistakes to avoid
- Forgetting the square root. Double the mass gives √2 times the period, not twice.
- Putting g in the spring equation, or m in the pendulum equation.
- Timing one cycle. Reaction time swamps it. Time ten or more.
- Counting “one” as you start the stopwatch. You then time one cycle fewer than you count.
- Plotting T against ℓ and expecting a straight line. Plot T².
- Reading the slope from data points. Use two points on the best-fit line.
- Mixing up T and f. Check: a long period means a low frequency.
Quick self-check
- A vibrating platform has f = 4.0 Hz. What is its period? (0.25 s)
- The mass on a spring is multiplied by 9. What happens to the period? (It triples.)
- What is the period of a pendulum 0.25 m long? (2π√(0.25 ÷ 9.8) ≈ 1.0 s)
- A 0.20 kg mass oscillates on a spring with k = 20 N/m. What is its period? (2π√(0.010) ≈ 0.63 s)
- Which graph is a straight line through the origin for a pendulum: T against ℓ, or T² against ℓ? (T² against ℓ)
Next: practice questions.