Pendulum Measurement Error Analysis
Example 2.7
Measurements of a simple pendulum's period: 2.63 s, 2.56 s, 2.42 s, 2.71 s, and 2.80 s. Calculate absolute errors, relative error, and percentage error.
| Measurement | Value (s) | Absolute Error (s) |
|---|---|---|
| 1 | 2.63 | +0.01 |
| 2 | 2.56 | -0.06 |
| 3 | 2.42 | -0.20 |
| 4 | 2.71 | +0.09 |
| 5 | 2.80 | +0.18 |
Calculations
Mean period:
\[ T_{mean} = \frac{2.63 + 2.56 + 2.42 + 2.71 + 2.80}{5} = \frac{13.12}{5} = 2.624 \, \text{s} \approx 2.62 \, \text{s} \]
Mean absolute error:
\[ \Delta T_{mean} = \frac{0.01 + 0.06 + 0.20 + 0.09 + 0.18}{5} = \frac{0.54}{5} = 0.108 \, \text{s} \approx 0.11 \, \text{s} \]
Final reported value with proper significant figures:
\[ T = 2.6 \pm 0.1 \, \text{s} \]
(Range: 2.5 s to 2.7 s)
Percentage error:
\[ \text{Percentage error} = \left( \frac{0.1}{2.6} \times 100 \right)\% = 3.85\% \approx 4\% \]
Key Concepts
- Absolute Error: Difference between measured value and true/mean value
- Mean Absolute Error: Average magnitude of absolute errors
- Relative Error: Ratio of mean absolute error to the mean value
- Percentage Error: Relative error expressed as percentage
- Significant Figures: The reliable digits in a measurement
