Year 10 Physics: Uncertainties

UNCERTAINTIES IN PHYSICS

Definition and Importance of Uncertainties

  • A measurement without uncertainty is considered meaningless.
  • Understanding uncertainties is crucial for accuracy and validity in scientific experiments.

Types of Uncertainties

  • Mistakes:
    • Errors due to equipment misuse, mathematical errors, or misunderstandings.
  • Systematic Errors:
    • Consistent errors resulting from faulty equipment or improper technique. Can be identified and corrected.
  • Random Errors:
    • Naturally occurring variations that cannot be eliminated, present regardless of equipment or methods.

Examples of Errors

  • Worked Example 1: Classifying Errors
    a) Not aligning the wire at the zero mark on a ruler: Mistake
    b) Using a balance accurate to 1g for 2g samples: Systematic Error
    c) Forgetting to zero the balance between measurements: Mistake

Determining Uncertainty in Measurements

  • Limit of Reading: Precision defined by the smallest unit that can be read on the instrument.
    • Example: For a millimeter ruler, the limit of reading is 1mm.
  • Calculating Absolute Uncertainty:
    • If two ends are lined up (e.g., protractor, ruler): Absolute uncertainty = Limit reading.
    • If one scale is used (e.g., thermometer): Absolute uncertainty = Half of the limit reading.
    • For electronic devices (e.g., scale): Absolute uncertainty = Limit reading.

Instrument Examples and Absolute Uncertainty

  • Worked Example 2:
    a) cm ruler: 1 mm (limit reading)
    b) Protractor (degree): 1° (limit reading)
    c) Electronic balance: 0.001g (limit reading)
    d) Analog voltmeter graduated in 10 volts: 5 volts (1/2 scale range).

Average Measurement Calculation

  • Repeating a measurement reduces absolute uncertainty.
  • Notation:
    • Average measurement: x
    • Absolute uncertainty: ±Δ
    • Δ = (max - min) / 2

Worked Example 3: Absolute Uncertainty Calculation

  • Thickness of copper wire measured: 5.84, 5.83, 5.82, 5.85, 5.79, 5.81 mm
  • Calculate mean:
    • Mean = (5.84 + 5.83 + 5.82 + 5.85 + 5.79 + 5.81) / 6
    • Absolute Uncertainty: (Max - Min)/2 = (5.85 - 5.79) / 2

Propagating Uncertainties

  • Absolute Uncertainty: Δ = ± max + min / 2
  • Fractional Uncertainty: Δ / x
  • Percentage Uncertainty: (Δ / x) × 100%

Adding and Subtracting Uncertainties

  • When performing addition or subtraction, absolute uncertainties are added.
  • Worked Example 4:
    • (2.6 ±0.5) + (2.8 ±0.5) = (2.6 + 2.8) ± (0.5 + 0.5)

Multiplying and Dividing Uncertainties

  • For multiplication or division, percentage uncertainties are added.
  • Constants have absolute and percentage uncertainties of zero.
  • Worked Example 5:
    • (6.2 ±0.5) × (2.81 ±0.01)

Powers and Uncertainties

  • When raising measurements to an index, multiply percentage uncertainty by the index's absolute value, then add percentage uncertainties.
  • Worked Example 6:
    • Calculate c = a^n, with uncertainty propagation from provided values.

Overall Uncertainty Calculation

  • Example Problem: Calculate for given measurements involving addition, square root:
    • c = 3.6 ±0.6
    • d = 9.91 ±0.01
    • Apply addition and square root rules carefully to find total uncertainty.

Practice Questions

  • Complete exercises from W1L3 section titled '4. Questions'.