PHY083 · Measurement & Instrumentation · Classes 11–12 + College
Error propagation
Objective: To see how errors in measured quantities combine into the error of a result, using g = 4π²L/T².
🧰 Apparatus
- Pendulum, stopwatch and metre rule
- Calculator
- Results table with uncertainties
📐 Formulas and SI units
g = 4π²L / T²
Δg/g = ΔL/L + 2ΔT/T
💡 Theory
Every measurement has an uncertainty. For a product or quotient, the relative (fractional) errors add, and a quantity raised to a power multiplies its relative error by that power. For g = 4π²L/T², Δg/g = ΔL/L + 2ΔT/T. The biggest term tells you which measurement to improve.
🪜 Procedure
- Measure L with its uncertainty.
- Time 20 oscillations and find T with its uncertainty (half the reaction time ÷ 20).
- Calculate g.
- Calculate each relative error and add them.
- Find which term dominates and how to reduce it.
📋 Observations Simulated values
✅ Result and conclusion
L = 1.000 ± 0.001 m and T = 2.00 ± 0.01 s give g = 9.87 ± 0.11 m/s²; the timing causes 91% of the error.
🔬 In a real lab
Reaction time (about 0.2 s) is the main error in hand timing, so time many oscillations to share it out.
⚠️ Safety
Keep the swing small and the bob secure on its thread.
❓ Quick check
1. For z = x², the relative error is…
2. Timing 20 swings instead of 1 makes ΔT…
3. Relative errors add for…
🏋️ Practice
EasyΔL/L = 0.5%, ΔT/T = 1%. Δg/g?
Answer: 2.5%
MediumWhich should you improve first?
Answer: T (it is doubled)
Challengeg = 9.8 ± 2.5%. Δg?
Answer: ≈ 0.25 m/s²