Uncertainty in Measurements | 9th Class Physics PTCB | Easy Explanation with Examples & Formula

Uncertainty in Measurements | 9th Class Physics PTCB | Easy Explanation with Examples & Formula

❓ Uncertainty in Measurements: Simple Guide 🌟 Introduction

Imagine trying to measure your exact height 📏. Can you say you're exactly 150 cm? Not 150.1 cm? Not 149.9 cm? 🤔

The truth is: We can never be 100% sure! There's always a small uncertainty (doubt)! ⚠️

🎯 What is Uncertainty?

Uncertainty = How much doubt or error might be in your measurement

┌────────────────────────────────┐ │ Understanding Uncertainty │ ├────────────────────────────────┤ │ │ │ You measure: 10.3 cm 📏 │ │ │ │ But actual might be: │ │ • 10.25 cm │ │ • 10.3 cm │ │ • 10.35 cm │ │ │ │ So we write: 10.3 ± 0.05 cm │ │ ↑ │ │ Uncertainty! │ │ │ └────────────────────────────────┘ 💡 Key Point: ┌────────────────────────────────┐ │ │ │ 🎯 PERFECT measurement │ │ is IMPOSSIBLE! ❌ │ │ │ │ ✅ There's ALWAYS some │ │ uncertainty! │ │ │ └────────────────────────────────┘ 📏 Why Does Uncertainty Happen? Main Reason: Instrument Limitation! 🔧

Every tool has a smallest division (least count). You can't measure smaller than that!

╔════════════════════════════════╗ ║ Ruler Example 📏 ║ ╚════════════════════════════════╝

Ruler with smallest division = 1 mm

Can you measure: ✅ 5.0 cm (yes!) ✅ 5.1 cm (yes!) ✅ 5.2 cm (yes!)

❌ 5.15 cm (NO! Too small!) ❌ 5.23 cm (NO! Too small!)

Your ruler can't "see" anything smaller than 1 mm! 👁️ 📖 Understanding Uncertainty with Example 🎯 Measuring a Line with Ruler ╔════════════════════════════════╗ ║ Line Measurement Example ║ ╚════════════════════════════════╝

Ruler (smallest division = 1 mm):

├─────┼─────┼─────┼─────┼─────┤ 10.0 10.1 10.2 10.3 10.4 10.5 cm

Line end is here: ▼ ├─────┼──░──┼─────┼─────┼─────┤ 10.0 10.1 10.2 10.3 10.4 10.5 cm

Where exactly? 🤔 • Is it 10.2 cm? • Is it 10.25 cm? • Is it 10.3 cm?

Hard to tell exactly! ⚠️ 📐 The Convention (Rule) for Reading 🎯 Two Cases: ╔════════════════════════════════╗ ║ CASE 1: Before Midpoint ║ ╚════════════════════════════════╝

Line end BEFORE middle:

├─────┼──░──┼─────┤ 10.2 ▲ 10.3 10.4 cm │ Before middle!

Read as: 10.2 cm ✅ (Stay at previous division)

──────────────────────────────────

╔════════════════════════════════╗ ║ CASE 2: After Midpoint ║ ╚════════════════════════════════╝

Line end AFTER middle:

├─────┼────░─┼─────┤ 10.2 ▲ 10.3 10.4 cm │ After middle!

Read as: 10.3 cm ✅ (Go to next division) 🎯 Visual Guide - Where to Read? ╔════════════════════════════════╗ ║ Reading Convention Guide ║ ╚════════════════════════════════╝

One small division (1 mm):

├──────────────────────┤ 10.2 Midpoint 10.3 cm ▲ ▲ ▲ │ │ │ Read as Exactly Read as 10.2 cm 10.25 cm 10.3 cm ✅ 🤔 ✅

├─────────┬─────────┤ Zone 1 │ Zone 2 (10.2 cm)│(10.3 cm)

If line in Zone 1 → Read 10.2 ✅ If line in Zone 2 → Read 10.3 ✅ ⚠️ Calculating Uncertainty 🎯 Formula: ┌────────────────────────────────┐ │ Uncertainty = ± (Least Count ÷ 2) │ └────────────────────────────────┘ 📊 Example with Ruler: ╔════════════════════════════════╗ ║ Ruler Uncertainty ║ ╚════════════════════════════════╝

Least Count = 1 mm = 0.1 cm

Uncertainty = ± (0.1 ÷ 2) = ± 0.05 cm

So if you measure 10.3 cm:

Write as: 10.3 ± 0.05 cm ✅

This means actual value is somewhere between: • 10.25 cm (10.3 - 0.05) • 10.35 cm (10.3 + 0.05)

─────────────────────────────────

Visual representation:

10.25 10.3 10.35 ├───────●───────┤ └──uncertainty──┘ 📊 Different Instruments, Different Uncertainties ┌─────────────────────────────────────┐ │ Instrument │ Uncertainty │ ├──────────────────┼──────────────────┤ │ │ │ │ Ruler 📏 │ ± 0.5 mm │ │ (LC = 1 mm) │ (± 0.05 cm) │ │ │ │ │ Vernier 🔧 │ ± 0.05 mm │ │ (LC = 0.1 mm) │ (± 0.005 cm) │ │ │ │ │ Micrometer 🔩 │ ± 0.005 mm │ │ (LC = 0.01 mm) │ (± 0.0005 cm) │ │ │ │ └─────────────────────────────────────┘

Pattern: Smaller least count ↓ Smaller uncertainty ↓ More accurate! ✅ 🔬 Real Example 1: Measuring Wire Diameter ╔════════════════════════════════╗ ║ Measuring Thin Wire 🔌 ║ ╚════════════════════════════════╝

Using Ruler (LC = 1 mm): ───────────────────────── Reading = 2 mm Uncertainty = ± 0.5 mm

Result: 2 ± 0.5 mm Could be: 1.5 mm to 2.5 mm (Very uncertain! ❌)

─────────────────────────────────

Using Micrometer (LC = 0.01 mm): ───────────────────────────────── Reading = 2.15 mm Uncertainty = ± 0.005 mm

Result: 2.15 ± 0.005 mm Could be: 2.145 to 2.155 mm (Much more certain! ✅)

Micrometer is BETTER! 🎯 🕰️ Real Example 2: Pendulum Time ╔════════════════════════════════╗ ║ Timing One Swing ⏱️ ║ ╚════════════════════════════════╝

Method 1: Time 1 swing ❌ ───────────────────────── 🎈 ╱╲ ╱ ╲

Reading = 1.2 seconds Uncertainty = ± 0.1 second

Result: 1.2 ± 0.1 s (Big uncertainty! 8%)

─────────────────────────────────

Method 2: Time 30 swings ✅ ───────────────────────────────── 🎈 🎈 🎈 ... (30 swings)

Total time = 36.0 seconds Uncertainty = ± 0.1 second

Time per swing = 36.0 ÷ 30 = 1.2 seconds

Uncertainty per swing = 0.1 ÷ 30 = ± 0.003 s

Result: 1.2 ± 0.003 s (Much smaller uncertainty! ✅) 💡 Why is Method 2 Better? ┌────────────────────────────────┐ │ Uncertainty gets DIVIDED │ │ when you take average! │ │ │ │ 30 swings = 30 times more │ │ accurate! 🎯 │ │ │ └────────────────────────────────┘ 📱 Digital Instruments Uncertainty 🎯 Special Case! ╔════════════════════════════════╗ ║ Digital Stopwatch 📱 ║ ╚════════════════════════════════╝

Display shows: ┌──────────┐ │ 12.34 s │ └──────────┘

Last digit keeps changing: ┌──────────┐ │ 12.34 s │ → 12.35 s → 12.34 s └──────────┘

The number "flickers"! ✨

Uncertainty = ± 1 in last digit = ± 0.01 s

Write as: 12.34 ± 0.01 s ✅ 📊 Digital vs Analog: ┌─────────────────────────────────┐ │ Analog Stopwatch ⏱️ │ │ (with needles) │ │ Uncertainty = ± 0.05 s │ │ │ │ Digital Stopwatch 📱 │ │ (with numbers) │ │ Uncertainty = ± 0.01 s │ │ │ │ Digital is MORE accurate! ✅ │ │ │ └─────────────────────────────────┘ 📝 How to Write Measurements Properly ✅ Correct Ways: ┌────────────────────────────────┐ │ Proper Notation │ ├────────────────────────────────┤ │ │ │ ✅ 10.3 ± 0.05 cm │ │ ✅ (10.3 ± 0.05) cm │ │ ✅ 10.3 cm (±0.05 cm) │ │ │ │ All show: │ │ • Measurement: 10.3 cm │ │ • Uncertainty: ± 0.05 cm │ │ │ └────────────────────────────────┘ ❌ Wrong Ways: ┌────────────────────────────────┐ │ Incorrect Notation │ ├────────────────────────────────┤ │ │ │ ❌ 10.3 cm │ │ (No uncertainty shown!) │ │ │ │ ❌ 10.3cm ± 0.05 │ │ (Units unclear!) │ │ │ │ ❌ 10.3 ± 0.5 cm │ │ (Wrong uncertainty!) │ │ │ └────────────────────────────────┘ 🎯 Ways to Reduce Uncertainty ┌────────────────────────────────┐ │ How to Reduce Uncertainty │ ├────────────────────────────────┤ │ │ │ 1️⃣ Use better instruments │ │ (smaller least count) 🔧 │ │ │ │ 2️⃣ Take multiple readings │ │ (then average) 📊 │ │ │ │ 3️⃣ Measure larger quantities │ │ (e.g., 30 swings not 1) ⏱️ │ │ │ │ 4️⃣ Use digital instruments │ │ (more precise) 📱 │ │ │ │ 5️⃣ Careful technique │ │ (avoid human error) 👤 │ │ │ └────────────────────────────────┘ 📊 Complete Example: Lab Report ╔════════════════════════════════╗ ║ Student Lab Report 📝 ║ ╚════════════════════════════════╝

Experiment: Measure length of desk Student: Ahmed Date: Today

MEASUREMENTS: ───────────────────────────────── Trial 1: 120.2 cm Trial 2: 120.3 cm Trial 3: 120.1 cm Trial 4: 120.2 cm Trial 5: 120.3 cm

Average = (120.2+120.3+120.1+120.2+120.3) ÷ 5 = 601.1 ÷ 5 = 120.22 cm

Instrument: Metre rule Least Count: 1 mm = 0.1 cm Uncertainty: ± 0.05 cm

FINAL RESULT: ───────────────────────────────── Length = 120.2 ± 0.05 cm ✅

This means actual length is between 120.15 and 120.25 cm

Conclusion: Measurement complete with known uncertainty! 🎯 💡 Important Points to Remember! ┌────────────────────────────────┐ │ Key Takeaways 🎯 │ ├────────────────────────────────┤ │ │ │ ⚠️ NO perfect measurement! │ │ │ │ 📏 Uncertainty = ± (LC ÷ 2) │ │ │ │ 🔧 Better tool = Less uncertain │ │ │ │ 📊 More readings = More accurate│ │ │ │ ✍️ Always write uncertainty! │ │ │ │ 🎲 Last digit may fluctuate │ │ (in digital instruments) │ │ │ └────────────────────────────────┘ 🎮 Quick Practice! ╔════════════════════════════════╗ ║ Question 1: ║ ║ Ruler LC = 1 mm ║ ║ What is uncertainty? ║ ╚════════════════════════════════╝

Answer: ± 0.5 mm (or ± 0.05 cm) ✅

Calculation: Uncertainty = LC ÷ 2 = 1 mm ÷ 2 = 0.5 mm ╔════════════════════════════════╗ ║ Question 2: ║ ║ You measure 15.4 cm with ║ ║ ruler (LC = 1 mm). ║ ║ How to write properly? ║ ╚════════════════════════════════╝

Answer: 15.4 ± 0.05 cm ✅ (Show measurement AND uncertainty!) ╔════════════════════════════════╗ ║ Question 3: ║ ║ Why measure 30 swings ║ ║ instead of 1 swing? ║ ╚════════════════════════════════╝

Answer: Uncertainty gets divided by 30, making result much more accurate! 🎯 ✨ Final Summary ┌─────────────────────────────────┐ │ Uncertainty Summary 📊 │ ├─────────────────────────────────┤ │ │ │ What: Doubt in measurement │ │ │ │ Why: Instrument limitation │ │ │ │ Formula: ± (LC ÷ 2) │ │ │ │ Reduce by: │ │ • Better instruments 🔧 │ │ • Multiple readings 📊 │ │ • Averaging ➗ │ │ │ │ Always write it! ✍️ │ │ Example: 10.3 ± 0.05 cm │ │ │ └─────────────────────────────────┘

You're now an uncertainty expert! 🏆📏

Remember: "Good scientists always show their uncertainty!" 🔬✨