UNCERTAINTY IN MEASUREMENTS

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ERROR ANALYSIS | UNCERATAINTY IN MEASUREMENTS
PHYSICS | Class 11 | CBSE & SEBA Board

ERROR ANALYSIS | Chapter 1

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📏 Section 1.3.3 – Rules for Determining the Uncertainty in Arithmetic Calculations

📘 Chapter: Units and Measurements (CBSE Class 11 Physics)

🔎 Why Error Propagation Matters

Every physical measurement carries uncertainty. When these measurements are used in calculations, their uncertainties must be systematically combined so that the final result honestly reflects experimental limitations.

This section explains how to combine errors in:

• Addition & Subtraction
• Multiplication & Division
• Powers and multi-variable expressions
• Multi‑step calculations

🔢 1. Multiplication and Division of Quantities

🔑 Rule

When quantities are multiplied or divided, their relative (percentage) errors are added.

📐 Mathematical Form

If:
\( Q = A \times B \) or \( Q = \dfrac{A}{B} \)

Then:
\( \dfrac{\Delta Q}{Q} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B} \)

📌 Worked Example (Area of a Rectangle)
Given:
Length: \( l = 16.2 \pm 0.1 \) cm
Breadth: \( b = 10.1 \pm 0.1 \) cm

Step 1: Relative Errors

• \( \dfrac{0.1}{16.2} \times 100 \approx 0.6% \)
• \( \dfrac{0.1}{10.1} \times 100 \approx 1.0% \)

Step 2: Area Calculation

\( A = l \times b = 16.2 \times 10.1 = 163.62 \) \( cm^2 \)

Step 3: Total Relative Error

0.6% + 1.0% = 1.6%

Step 4: Absolute Error in Area

\( \Delta A = \dfrac{1.6}{100} \times 163.62 \approx 2.6 \) \( cm^2 \)

✅ Final Result

\( \boxed{A = (164 \pm 3)} \) \( cm^2 \)

➕ 2. Addition and Subtraction of Quantities

🔑 Rule For addition or subtraction, absolute errors are added, and the result is reported with the least number of decimal places.

📐 Mathematical Form

If:
\( Q = A + B \quad or \quad Q = A - B \)
Then: \( \Delta Q = \Delta A + \Delta B \)
📌 Example
12.9 g - 7.06 g = 5.84 g
12.9 → 1 decimal place
7.06 → 2 decimal places
➡ Result must have 1 decimal place
✅ Final Answer \( \boxed {5.8} \) g

⚖️ 3. Effect of Magnitude on Relative Error

🔍 Key Idea

The same absolute error produces different relative errors depending on the magnitude of the measurement.

| Measurement	| Absolute Error	| Relative Error |
1.02 g	±0.01 g	≈ 1%
9.89 g	±0.01 g	≈ 0.1%

🧠 Conclusion
👉 Smaller measurements are more affected by the same absolute uncertainty.

🧮 4. Multi‑Step Calculations & Rounding Errors

⚠️ Common Mistake
Rounding off at each step causes error accumulation.
🔑 Correct Rule
* Keep one extra significant figure during intermediate steps
* Round off only in the final answer

📌 Example

Reciprocal of 9.58:

\( \Rightarrow \dfrac{1}{9.58} = 0.1044 \quad \) (retain extra digit)
If rounded early: \( \dfrac{1}{0.104} = 9.62 \neq 9.58 \)
➡ Shows why premature rounding is dangerous.

📘 Combination of Errors – Formula Sheet

🔧 Definitions

Absolute error: \( \Delta x \)
Relative error: \( \delta x = \dfrac{\Delta x}{x} \)

Percentage error: \( \delta x \times 100 \)

📌 (A) Addition / Subtraction

\( \Delta Q = \Delta A + \Delta B \)

📌 (B) Multiplication / Division

\( \dfrac{\Delta Q}{Q} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B} \)

📌 (C) Powers

If:
\( Q = A^n \)
Then:
\( \dfrac{\Delta Q}{Q} = |n| \cdot \dfrac{\Delta A}{A} \)

📌 (D) Multiple Variables with Powers

If:
\( Q = \dfrac{A^p B^q}{C^r} \)
Then:
\( \dfrac{\Delta Q}{Q} \) = \( p\dfrac{\Delta A}{A} \) + \( q\dfrac{\Delta B}{B} \) + \( r\dfrac{\Delta C}{C} \)

✅ Best Practices (Exam‑Oriented)

1. Carry extra digits in intermediate steps.
2. Round off only at the end
3. Use least significant figures / decimal places correctly.
4. Exact numbers (2, π, etc.) have zero uncertainty

📝 CBSE Tip

📌 Error propagation is high‑weightage for numericals + theory questions in Class 11 exams.

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