Units and Measurements

Units and Measurement: CBSE Class 11 Physics Comprehensive Notes

Introduction

**Measurement** is the process of comparing a physical quantity with a standard reference called a **unit**. Every physical quantity has two components:

**Numerical value** (how many times)

**Unit** (what is being measured)

For example: A length of 5 metres means the length is 5 times the standard unit metre.

**Physical quantities** are classified into:

**Fundamental (Base) quantities**: Independent quantities that cannot be expressed in terms of other quantities. Examples: length, mass, time, electric current, temperature, amount of substance, luminous intensity.

**Derived quantities**: Quantities expressed as combinations of base quantities. Examples: velocity (length/time), force (mass × length/time²), density (mass/volume).

**Units** are categorized as:

**Fundamental units**: Units of base quantities

**Derived units**: Units of derived quantities, obtained by combining base units

A complete set of base and derived units is called a **system of units**.

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The International System of Units (SI)

**History of Unit Systems:**

Before SI, three major systems were in use:

**CGS system** (Centimetre-Gram-Second): Used in many scientific laboratories

**FPS system** (Foot-Pound-Second): Used in British and American engineering

**MKS system** (Metre-Kilogram-Second): Precursor to SI

**The SI System:**

The **Système International d'Unités** (International System of Units) was adopted internationally in 1960 and redefined in 2018 by the General Conference on Weights and Measures. SI uses a decimal (base-10) system, making conversions simple and convenient.

Seven SI Base Quantities and Units

| Base Quantity | SI Unit | Symbol | 2018 Definition |

|---|---|---|---|

| **Length** | Metre | m | Fixed by speed of light in vacuum (c = 299,792,458 m/s) |

| **Mass** | Kilogram | kg | Fixed by Planck constant (h = 6.62607015 × 10⁻³⁴ J·s) |

| **Time** | Second | s | Fixed by caesium-133 atomic frequency (9,192,631,770 Hz) |

| **Electric Current** | Ampere | A | Fixed by elementary charge (e = 1.602176634 × 10⁻¹⁹ C) |

| **Thermodynamic Temperature** | Kelvin | K | Fixed by Boltzmann constant (k = 1.380649 × 10⁻²³ J/K) |

| **Amount of Substance** | Mole | mol | Contains exactly 6.02214076 × 10²³ elementary entities (Avogadro number) |

| **Luminous Intensity** | Candela | cd | Fixed luminous efficacy of 540 × 10¹² Hz radiation (683 lm/W) |

**Important Note:** Modern SI definitions are based on fundamental constants rather than physical artifacts, ensuring stability and universality.

Supplementary Units

Two additional units are defined for geometric quantities but are **dimensionless**:

**Plane angle (θ)**: Defined as arc length divided by radius

Unit: **radian** (rad)

Formula: θ = s/r (where s = arc length, r = radius)

**Solid angle (Ω)**: Defined as intercepted area divided by square of radius

Unit: **steradian** (sr)

Formula: Ω = A/r² (where A = intercepted area, r = radius)

Derived Units

Derived units are formed from base units. Common examples:

**Velocity** = metre/second (m/s)

**Force** = kilogram × metre/second² = **Newton** (N)

**Pressure** = Newton/metre² = **Pascal** (Pa)

**Energy** = Newton × metre = **Joule** (J)

**Power** = Joule/second = **Watt** (W)

**SI Prefixes** for multiples and sub-multiples (powers of 10):

Deca (da) = 10¹, Hecto (h) = 10², Kilo (k) = 10³, Mega (M) = 10⁶, Giga (G) = 10⁹

Deci (d) = 10⁻¹, Centi (c) = 10⁻², Milli (m) = 10⁻³, Micro (μ) = 10⁻⁶, Nano (n) = 10⁻⁹, Pico (p) = 10⁻¹²

**Real-life example:** A doctor prescribes 500 milligrams (mg) of medicine = 500 × 10⁻³ g = 0.5 g. Here, the SI prefix "milli" (m) = 10⁻³ makes the quantity easier to express and understand.

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Significant Figures

**Definition:** Significant figures (or significant digits) are the digits in a measured value that are known with certainty plus the first uncertain digit. They indicate the **precision** of a measurement.

**Precision** depends on the **least count** of the measuring instrument. For example:

A metre scale with 1 mm divisions (least count = 1 mm) is more precise than one with 1 cm divisions (least count = 1 cm).

If a simple pendulum period is measured as 1.62 s: digits 1, 6 are certain; 2 is uncertain → **3 significant figures**

If length is 287.5 cm: digits 2, 8, 7 are certain; 5 is uncertain → **4 significant figures**

Rules for Counting Significant Figures

1. **All non-zero digits are significant**

Example: 4.237 g has **4** significant figures

2. **All zeros between two non-zero digits are significant** (regardless of decimal position)

Example: 2.308 cm has **4** significant figures (2, 3, 0, 8)

Example: 206 m has **3** significant figures

3. **Zeros to the left of the first non-zero digit are NOT significant** (for numbers < 1)

Example: 0.002308 has **4** significant figures (2, 3, 0, 8)

The leading zeros indicate only the position of the decimal

4. **Trailing zeros in a number WITHOUT a decimal point are NOT significant**

Example: 123 m = 12,300 cm = 123,000 mm all have **3** significant figures

Trailing zeros without decimal convey position, not precision

5. **Trailing zeros in a number WITH a decimal point ARE significant**

Example: 3.500 has **4** significant figures

Example: 0.06900 has **4** significant figures

These zeros explicitly show measurement precision

6. **Leading zeros before the decimal are never significant** (by convention)

Example: 0.1250 has **4** significant figures (1, 2, 5, 0 are significant)

7. **Exact numbers have infinite significant figures**

Mathematical constants (π, e) and defined quantities (2 in r = d/2)

These can be written to any number of decimal places as needed

**Exam Tip:** Change of units does NOT change the number of significant figures. This is why scientific notation is preferred.

Scientific Notation and Significant Figures

**Scientific notation** expresses numbers as: **a × 10ᵇ** where 1 ≤ a < 10 and b is any integer.

**Advantages:**

Eliminates ambiguity about trailing zeros

All digits in the base number (a) are significant

Clearly shows the order of magnitude

**Examples:**

4.700 m = 4.700 × 10⁰ m = 4.700 × 10² cm = 4.700 × 10³ mm (all show **4** significant figures clearly)

Earth's diameter: 1.28 × 10⁷ m (order of magnitude: **7**)

Hydrogen atom diameter: 1.06 × 10⁻¹⁰ m (order of magnitude: **–10**)

**Earth is 17 orders of magnitude larger** (10⁷ ÷ 10⁻¹⁰ = 10¹⁷)

Rules for Arithmetic Operations with Significant Figures

**Rule 1: Multiplication and Division**

Final result should have **as many significant figures as the measurement with the LEAST significant figures**

**Example 1:** Calculate density

Mass = 5.74 g (3 significant figures)

Volume = 1.2 cm³ (2 significant figures)

Density = 5.74 ÷ 1.2 = 4.783... g/cm³

**Result:** 4.8 g/cm³ (rounded to **2** significant figures, matching the least precise input)

**Example 2:** Calculate light-year distance

Speed of light c = 3.00 × 10⁸ m/s (3 significant figures)

One year = 3.1557 × 10⁷ s (5 significant figures)

Distance = 3.00 × 10⁸ × 3.1557 × 10⁷ = 9.47 × 10¹⁵ m

**Result:** 9.47 × 10¹⁵ m (3 significant figures, matching c)

**Rule 2: Addition and Subtraction**

Final result should have **as many decimal places as the measurement with the LEAST decimal places**

This rule differs from multiplication/division

**Example 1:** Sum measurements

436.32 g (2 decimal places)

227.2 g (1 decimal place) ← least

0.301 g (3 decimal places)

Arithmetic sum = 663.821 g

**Result:** 663.8 g (rounded to **1** decimal place)

**Example 2:** Subtract measurements

0.307 m – 0.304 m = 0.003 m = 3 × 10⁻³ m

(Both have 3 decimal places, result has 3)

**Critical Note:** Never use multiplication/division rule for addition/subtraction. The precision of addition/subtraction depends on decimal places, not significant figures.

Rounding Off Uncertain Digits

**Standard Rounding Convention:**

If the digit to be dropped is **> 5**: increase the preceding digit by 1

If the digit to be dropped is **< 5**: keep the preceding digit unchanged

If the digit to be dropped is **= 5**:

If preceding digit is **even**: drop the 5 (keep unchanged)

If preceding digit is **odd**: increase by 1

**Examples:**

2.746 → 2.75 (drop 6 > 5, so round up)

1.743 → 1.74 (drop 3 < 5, so keep unchanged)

2.745 → 2.74 (drop 5, preceding digit 4 is even, so drop it)

2.735 → 2.74 (drop 5, preceding digit 3 is odd, so round up)

**Best Practice for Multi-step Calculations:**

Retain **one extra digit** in intermediate steps

Round to appropriate significant figures **only at the final step**

This prevents cumulative rounding errors

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Worked Examples

Example 1.1: Surface Area and Volume of a Cube

**Given:** Each side of a cube = 7.203 m (4 significant figures)

**Find:** Total surface area and volume

**Solution:**

Surface area = 6l² = 6 × (7.203)²

= 6 × 51.882209

= 311.293254 m²

**= 311.3 m²** (rounded to 4 significant figures)

Volume = l³ = (7.203)³

= 373.714754 m³

**= 373.7 m³** (rounded to 4 significant figures)

**Note:** Both results rounded to 4 significant figures, matching the input precision.

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Example 1.2: Density Calculation

**Given:** Mass = 5.74 g (3 significant figures), Volume = 1.2 cm³ (2 significant figures)

**Find:** Density with appropriate significant figures

**Solution:**

Density = Mass ÷ Volume

= 5.74 ÷ 1.2

= 4.7833... g/cm³

**Least significant figures:** 2 (in volume 1.2 cm³)

**Result = 4.8 g/cm³** (rounded to 2 significant figures)

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Example 1.3: Uncertainty in Arithmetic Operations

**Given:** Length l = 16.2 ± 0.1 cm, Breadth b = 10.1 ± 0.1 cm

**Find:** Area with uncertainty

**Solution:**

Area = l × b = 16.2 × 10.1 = 163.62 cm²

Relative error in l = 0.1/16.2 × 100% = 0.62%

Relative error in b = 0.1/10.1 × 100% = 0.99%

Combined relative error = 0.62% + 0.99% = 1.61% ≈ 1.6%

Absolute error = 1.6/100 × 163.62 ≈ 2.6 cm²

**Result = 163.62 + 2.6 = (164 ± 3) cm²** or approximately **164 cm²**

**Note:** When combining uncertainties in multiplication, add relative errors.

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Dimensions of Physical Quantities

**Dimensional analysis** expresses physical quantities in terms of fundamental dimensions.

**Seven fundamental dimensions:**

**[M]** = Mass

**[L]** = Length

**[T]** = Time

**[A]** = Electric current (Ampere)

**[K]** = Temperature (Kelvin)

**[mol]** = Amount of substance

**[cd]** = Luminous intensity

**Dimensional formula:** Expresses a quantity as a product of fundamental dimensions.

**Examples:**

**Velocity**: Distance/Time = [L]/[T] = **[L T⁻¹]**

**Acceleration**: Velocity/Time = [L T⁻¹]/[T] = **[L T⁻²]**

**Force**: Mass × Acceleration = [M] × [L T⁻²] = **[M L T⁻²]**

**Work/Energy**: Force × Distance = [M L T⁻²] × [L] = **[M L² T⁻²]**

**Pressure**: Force/Area = [M L T⁻²]/[L²] = **[M L⁻¹ T⁻²]**

**Density**: Mass/Volume = [M]/[L³] = **[M L⁻³]**

**Dimensionless quantities** (no dimensions):

Angles, strain, refractive index, specific gravity, relative density

Represented as **[M⁰ L⁰ T⁰]** or simply **[1]**

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Dimensional Analysis and Applications

**Principle of Homogeneity of Dimensions:**

In a valid physical equation, all terms must have the same dimensions. An equation is dimensionally correct if both sides have identical dimensional formulas.

Applications of Dimensional Analysis

**1. Checking Dimensional Consistency:**

Verify if an equation is dimensionally correct.

**Example:** Is s = ut + ½gt² correct?

Left side [s] = **[L]**

Right side ut = [L T⁻¹] × [T] = **[L]** ✓

Right side ½gt² = [L T⁻²] × [T²] = **[L]** ✓

All terms have dimension [L] → **Equation is dimensionally correct**

**2. Deriving Relationships:**

Find unknown exponents in physical relationships.

**Example:** Period of simple pendulum T depends on length (l), mass (m), and g.

Assume: T = k × lᵃ × mᵇ × gᶜ (where k is dimensionless constant)

Dimensional equation:

[T] = [L]ᵃ × [M]ᵇ × [L T⁻²]ᶜ

[T] = [M]ᵇ × [L]ᵃ⁺ᶜ × [T]⁻²ᶜ

Comparing exponents:

For [M]: 0 = b → **b = 0**

For [T]: 1 = –2c → **c = –½**

For [L]: 0 = a + c → a = ½ → **a = ½**

Therefore: **T = k√(l/g)** (exact form: T = 2π√(l/g))

**3. Converting Units:**

Dimensional analysis helps convert between unit systems.

**Example:** Convert 60 km/h to m/s

60 km/h = 60 × (1000 m)/(3600 s) = 60 × 1000/3600 = **16.67 m/s**

**4. Limitations of Dimensional Analysis:**

Cannot find dimensionless constants (e.g., π, 2π, ½ in formulas)

Cannot verify equations with transcendental functions (sin, cos, exponential)

Cannot establish relationships between quantities with same dimensions but different physical meanings

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Summary of Key Concepts

**SI system** is the internationally accepted standard with 7 base units

**Significant figures** indicate measurement precision; use rules carefully for arithmetic operations

**Scientific notation** eliminates ambiguity in significant figures

**Dimensional analysis** verifies equations and derives unknown relationships

**Change of units** does not affect number of significant figures

**Intermediate calculations** should retain extra digits; round only at the end

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Exam-Important Points

1. Write all measured values with **appropriate significant figures**

2. Always use **SI units** unless specified otherwise

3. Express final answers in **scientific notation** when appropriate

4. Apply **multiplication/division rule** (least significant figures) for those operations

5. Apply **addition/subtraction rule** (least decimal places) for those operations

6. Use **dimensional analysis** to check equation validity

7. Remember: exact numbers and mathematical constants have infinite significant figures

8. In multi-step problems, round **only the final answer**, not intermediate results