General Physics 1: Uncertainties and Deviations in Measurement

Pre-Assessment

• Complete the pre-assessment on the assessment sheets provided in the learning packet.

Objectives

1. Differentiate accuracy from precision
2. Use short notation in reporting measurement uncertainties
3. Estimate the uncertainty of a derived quantity from estimated values and uncertainties of directly measured quantities
4. Calculate the sum and difference of uncertainty of directly measured quantities

Uncertainty

• Definition: Also known as error, it indicates the maximum difference between the measured value and the true value.
• Example: If the accepted value is 47 meters but the measurement is 44 meters, the uncertainty/error is the difference between these values.

Precision vs. Accuracy

• Precision: Degree to which successive measurements agree with each other.
  • Example: Walking 15 steps consistently from school to a shop.
• Accuracy: Nearness of a measurement to an accepted value.
  • Example: Normal body temperature is 37.0°C; a thermometer reading of 36.9°C is near the accepted value and thus accurate.

Reporting Measurement Uncertainty

• Notation: Write the number followed by ± symbol and a second number indicating uncertainty.
  • Example: Length = 2.51 ± 0.02 meters.
    • Range of true value: 2.51 + 0.02 -> 2.53 meters; 2.51 - 0.02 -> 2.49 meters.
• Shorthand notation: The last value of the uncertainty inside parentheses after the measured value.
  • Example: 2.51(2) meters.

Example Calculation

• Diameter of a steel rod: 56 ± 0.02 mm
  • Higher value: 56 + 0.02 = 56.02 mm
  • Lower value: 56 - 0.02 = 55.98 mm
  • Report: 56(2) mm

Percentage Uncertainty

• Example: 52 seconds ± 10%
  • Conversion: 10% of 52 = 5.2
  • Higher value: 52 + 5.2 = 57.2 seconds
  • Lower value: 52 - 5.2 = 46.8 seconds
  • Report: 5.2 seconds / 52 seconds

Error Propagation Rules

• When derived quantity is a sum, difference, quotient, or product of others.

Calculation of Sum of Uncertainties

1. Given: Mass of object 1 (x) = 80 ± 1 gram; Mass of object 2 (y) = 65 ± 2 grams.
2. Step 1: Total mass: 80 + 65 = 145 grams.
3. Step 2: Higher mass:
   • x: 80 + 1 = 81 grams
   • y: 65 + 2 = 67 grams
   • Total higher mass: 81 + 67 = 148 grams
4. Step 3: Lower mass:
   • x: 80 - 1 = 79 grams
   • y: 65 - 2 = 63 grams
   • Total lower mass: 79 + 63 = 142 grams
5. Step 4: Uncertainty calculation:
   • Total higher mass - total mass: 148 - 145 = 3 grams
   • Total lower mass - total mass: 142 - 145 = -3 grams
   • Report: 145 ± 3 grams

Calculation of Difference of Uncertainties

1. Step 1: Mass difference: 80 - 65 = 15 grams
2. Step 2: Higher mass:
   • x: 81 grams, y: 67 grams
   • Higher mass difference: 81 - 67 = 14 grams
3. Step 3: Lower mass:
   • x: 79 grams, y: 63 grams
   • Lower mass difference: 79 - 63 = 16 grams
4. Step 4: Uncertainty calculation:
   • Higher mass difference - mass difference: 14 - 15 = -1 gram
   • Lower mass difference - mass difference: 16 - 15 = 1 gram
   • Report: 15 ± 1 grams

Further Learning

• Refer to the Propagation of Uncertainties PDF file or scan the provided QR code for more information.

Reflection on Uncertainties

• Quote: "Life is uncertain, and that's part of it. Moving to the unknown can be frightening. No matter how fast the world is changing, uncertainties never go away."
• COVID-19: Reflect on how to deal with uncertainties brought by the pandemic and write your thoughts in the learning packet.

Activities

• Answer Unit 1, activities 2 and 3 on the assessment sheets (page 4).

Final Note

"Be anxious for nothing, but in everything by prayer and supplication with thanksgiving let your requests be made known to God. And the peace of God, which passes all understanding, will guard your hearts and minds through Jesus Christ." - Philippians 4:6–7

Thank you and God bless.