Introduction

The study of physics is based on the measurement of physical quantities. Measurement of any physical quantity involves the comparison of that quantity with a certain basic, arbitrarily chosen, internationally accepted reference standard called a unit.

The result of a measurement is always expressed by a numerical measure (a number) followed by the unit. For example, if the length of a room in an Indian house is measured to be $5$ metres, the number $5$ is the numerical measure and 'metre' is the unit.

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Fundamental and Derived Quantities

In nature, there are a large number of physical quantities, but they are all interrelated. Therefore, we do not need a separate unique unit for every single quantity. These quantities are categorized into two types:

1. Fundamental (Base) Quantities

The physical quantities which are independent of each other and cannot be further simplified are called fundamental or base quantities. The units defined for these quantities are known as fundamental or base units. Examples include length, mass, and time.

2. Derived Quantities

All other physical quantities whose units can be expressed as combinations of the fundamental units are called derived quantities. Their units are known as derived units. For example, Speed is derived from length and time.

A complete set of these units, including both the base units and derived units, is known as a system of units.

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Comparison of Base and Derived Units

The following table illustrates the distinction between the two types of units with examples.

Type of Quantity	Example Quantity	Standard Unit (SI)
Fundamental	Length	Metre ($m$)
Fundamental	Mass	Kilogram ($kg$)
Derived	Area (Length $\times$ Breadth)	Metre$^2$ ($m^2$)
Derived	Speed (Distance / Time)	Metre / Second ($m/s$)
Total Base Quantities	7	Fixed by SI

The International System of Units

Before the global adoption of a single system, scientists across the world relied on different localized systems. The most notable among these were the CGS system, the FPS system (often called the British system), and the MKS system. These systems primarily focused on three base quantities: length, mass, and time.

System Name	CGS	FPS	MKS
Length Unit	centimetre ($cm$)	foot ($ft$)	metre ($m$)
Mass Unit	gram ($g$)	pound ($lb$)	kilogram ($kg$)
Time Unit	second ($s$)	second ($s$)	second ($s$)

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The Advent of SI Units

The Système Internationale d’ Unites (SI) was established to provide a universal language for science and trade. Managed by the International Bureau of Weights and Measures (BIPM), it was significantly revised in November 2018 to define units based on fundamental physical constants rather than physical prototypes. This ensures that the units are stable over time and can be reproduced anywhere in the universe.

Advantages of the SI System

1. Decimal Nature: SI is a metric system. Multiples and sub-multiples are related by powers of $10$, making calculations straightforward.

2. Universal Acceptance: It is used in all scientific, technical, and industrial work globally, including all research conducted by ISRO and DRDO in India.

3. Coherence: All derived units are obtained by simple multiplication or division of base units without introducing numerical factors.

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The Seven SI Base Units

The following table provides the complete list of SI base quantities, their names, symbols, and the rigorous scientific definitions adopted recently.

Base Quantity	Unit Name	Symbol	Definition and Fixed Constant
Length	metre	m	Defined by the fixed numerical value of the speed of light in vacuum $c$ to be $299,792,458$ when expressed in $\text{m s}^{-1}$.
Mass	kilogram	kg	Defined by the fixed numerical value of the Planck constant $h$ to be $6.62607015 \times 10^{-34}$ when expressed in $\text{J s}$ (or $\text{kg m}^2 \text{s}^{-1}$).
Time	second	s	Defined by the fixed numerical value of the caesium frequency $\Delta \nu_{Cs}$ (ground-state hyperfine transition) to be $9,192,631,770$ when expressed in $\text{Hz}$.
Electric current	ampere	A	Defined by the fixed numerical value of the elementary charge $e$ to be $1.602176634 \times 10^{-19}$ when expressed in unit $\text{C}$ (or $\text{A s}$).
Thermodynamic Temperature	kelvin	K	Defined by the fixed numerical value of the Boltzmann constant $k$ to be $1.380649 \times 10^{-23}$ when expressed in unit $\text{J K}^{-1}$.
Amount of substance	mole	mol	One mole contains exactly $6.02214076 \times 10^{23}$ elementary entities (Avogadro constant $N_A$).
Luminous intensity	candela	cd	Defined by the luminous efficacy of monochromatic radiation of frequency $540 \times 10^{12} \text{ Hz}$ to be $683 \text{ lm W}^{-1}$.

Mathematical Logic Behind the Definitions

The modern SI system is unique because it defines units by "fixing" the value of nature's constants. This ensures that the units remain invariant and can be reproduced in any laboratory, whether in New Delhi or on the Moon.

1. The Metre and Speed of Light

The metre is no longer a distance between two marks on a bar. It is derived from the constant speed of light ($c$).

$$1 \text{ metre} = \left( \frac{c}{299,792,458} \right) \text{ seconds}$$

2. The Kilogram and Planck Constant

The kilogram is now linked to the Planck constant ($h$). The relation between energy ($E$), mass ($m$), and frequency ($\nu$) is given by Einstein's and Planck's formulas:

$$E = mc^2 \quad \text{and} \quad E = h\nu$$

$$mc^2 = h\nu \implies m = \frac{h\nu}{c^2}$$

By fixing $h$ and $c$, the unit of mass (kg) is precisely determined without the need for a physical weight.

3. The Mole and Avogadro Number

The Mole is used to measure the amount of substance. It is strictly numerical. If you have $6.02214076 \times 10^{23}$ atoms of Gold, you have exactly $1 \text{ mole}$ of Gold.

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Supplementary Units: Angles

In addition to the seven fundamental base units, the SI system includes two supplementary units used to measure angles. These units are unique because they represent ratios of similar physical quantities (length/length or area/area$^2$), making them dimensionless. However, they are assigned units to distinguish them from pure numbers.

1. Plane Angle ($d\theta$)

A Plane Angle is a two-dimensional angle formed at the centre of a circle by an arc. It is defined as the ratio of the length of the arc ($ds$) to the radius ($r$) of the circle.

Mathematical Formula and Unit

The plane angle $d\theta$ is expressed as:

$$d\theta = \frac{ds}{r}$$

The SI unit for a plane angle is the radian, denoted by the symbol rad.

Derivation of Total Angle and Conversions

For a complete circle, the total arc length is equal to the circumference, which is $2\pi r$. Therefore, the total plane angle subtended by a circle at its centre is:

$$\theta_{total} = \frac{2\pi r}{r} = 2\pi \text{ rad}$$

Since a full circle is also $360^{\circ}$, we can derive the following conversion factors:

1. $2\pi \text{ rad} = 360^{\circ}$

2. $\pi \text{ rad} = 180^{\circ}$

3. $1 \text{ rad} = \frac{180}{\pi} \approx 57.3^{\circ}$

Angle in Degrees ($^{\circ}$)	Angle in Radians (rad)
$30^{\circ}$	$\frac{\pi}{6}$
$45^{\circ}$	$\frac{\pi}{4}$
$60^{\circ}$	$\frac{\pi}{3}$
$90^{\circ}$	$\frac{\pi}{2}$
$180^{\circ}$	$\pi$
$360^{\circ}$	$2\pi$

2. Solid Angle ($d\Omega$)

A Solid Angle is the three-dimensional equivalent of a plane angle. It is the angle subtended at a point (the apex) by a portion of a spherical surface. It is defined as the ratio of the intercepted area ($dA$) of the spherical surface to the square of its radius ($r^2$).

Mathematical Formula and Unit

The solid angle $d\Omega$ is expressed as:

$$d\Omega = \frac{dA}{r^2}$$

The SI unit for a solid angle is the steradian, denoted by the symbol sr.

Derivation of Total Solid Angle for a Sphere

For a complete sphere, the total surface area is $4\pi r^2$. Therefore, the total solid angle subtended by a sphere at its centre is:

$$\Omega_{total} = \frac{4\pi r^2}{r^2} = 4\pi \text{ sr}$$

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The Dimensionless Nature of Angles

Although radians and steradians are units, the quantities they measure have no dimensions. This is because they are calculated as the ratio of quantities with identical dimensions.

For Plane Angle:

$$\text{Dimension of } \theta = \frac{[\text{Length}]}{[\text{Length}]} = [L^0]$$

For Solid Angle:

$$\text{Dimension of } \Omega = \frac{[\text{Area}]}{[\text{Length}]^2} = \frac{[L^2]}{[L^2]} = [L^0]$$

In dimensional analysis, these are represented as $[M^0 L^0 T^0]$.

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Units Retained for General Use

While the SI system is the international standard for scientific and industrial work, several non-SI units remain in frequent use across the globe and within the Indian domestic context. These units are retained because they are deeply embedded in daily life, commerce, and specific fields of study like medicine, agriculture, and astronomy. For convenience, their values are precisely defined in terms of SI base units.

Comprehensive List of Non-SI Units

The following table provides the complete list of units that are outside the SI system but are officially recognized for continued use. This includes units for time, angle, mass, area, and pressure.

Physical Quantity	Unit Name	Symbol	Value in SI Units
Time	minute	min	$60 \text{ s}$
Time	hour	h	$60 \text{ min} = 3600 \text{ s}$
Time	day	d	$24 \text{ h} = 86400 \text{ s}$
Time	year	y	$365.25 \text{ d} \approx 3.156 \times 10^7 \text{ s}$
Angle	degree	$^{\circ}$	$1^{\circ} = (\pi/180) \text{ rad}$
Volume	litre	L	$1 \text{ dm}^3 = 10^{-3} \text{ m}^3$
Mass	tonne	t	$10^3 \text{ kg} = 1000 \text{ kg}$
Mass	quintal	q	$100 \text{ kg}$
Mass	carat	c	$200 \text{ mg} = 2 \times 10^{-4} \text{ kg}$
Pressure	bar	bar	$0.1 \text{ MPa} = 10^5 \text{ Pa}$
Pressure	standard atmosphere	atm	$101325 \text{ Pa} = 1.013 \times 10^5 \text{ Pa}$
Area	hectare	ha	$1 \text{ hm}^2 = 10^4 \text{ m}^2$
Area	are	a	$1 \text{ dam}^2 = 10^2 \text{ m}^2$
Radioactivity	curie	Ci	$3.7 \times 10^{10} \text{ s}^{-1}$

Key Categories

I. Units of Time

The SI unit for time is the second. However, for longer durations, we use minutes, hours, and days. The value of a Year is typically taken as $365.25$ days to account for leap years in long-term calculations.

$$\text{1 Year} \approx 365.25 \times 24 \times 60 \times 60 \text{ s} \approx 3.156 \times 10^7 \text{ s}$$

II. Units of Area (Agriculture)

In Indian agriculture and land records, the Hectare is a common unit. It represents the area of a square with sides of $100$ metres.

$$\text{1 Hectare} = 100 \text{ m} \times 100 \text{ m} = 10^4 \text{ m}^2$$

Similarly, the Are represents a square of $10$ metres side length ($100 \text{ m}^2$).

III. Units of Mass (Trade)

For heavy goods like grains and steel in Indian markets, Quintal ($100 \text{ kg}$) and Metric Tonne ($1000 \text{ kg}$) are used. In the jewellery industry, the Carat is used for measuring gemstones.

$$\text{1 Carat} = 200 \text{ mg} = 0.2 \text{ grams}$$

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Derived Units and Scientific Notation

While the seven base units form the foundation of the SI system, most physical quantities we encounter are derived quantities. These are expressed as combinations of the base units. To handle extremely large or small values, the SI system employs a standard set of prefixes and specific guidelines for symbolic representation.

SI Derived Units

Derived units are obtained by performing mathematical operations (multiplication or division) on the base units. Some derived units are expressed directly in terms of base units, while others have been given special names in honour of famous scientists.

A. Units Expressed in Base Units

The following vertical table lists common physical quantities and how their units are derived from the seven fundamental pillars.

Physical Quantity	SI Unit Name	Symbol
Area	square metre	$m^2$
Volume	cubic metre	$m^3$
Speed, velocity	metre per second	$m/s$ or $m \, s^{-1}$
Acceleration	metre per second square	$m/s^2$ or $m \, s^{-2}$
Density	kilogram per cubic metre	$kg/m^3$ or $kg \, m^{-3}$
Momentum	kilogram metre per second	$kg \, m \, s^{-1}$
Moment of inertia	kilogram square metre	$kg \, m^2$

B. Units with Special Names

Certain quantities are used so frequently that their complex base-unit combinations are replaced by single symbols. For example, the unit of Force ($kg \, m/s^2$) is called the Newton ($N$).

Quantity	Name	Symbol	In Base Units
Frequency	hertz	$Hz$	$s^{-1}$
Force	newton	$N$	$kg \, m \, s^{-2}$
Pressure, Stress	pascal	$Pa$	$kg \, m^{-1} s^{-2}$
Energy, Work	joule	$J$	$kg \, m^2 s^{-2}$
Power	watt	$W$	$kg \, m^2 s^{-3}$
Electric Charge	coulomb	$C$	$A \, s$
Electric Potential	volt	$V$	$kg \, m^2 s^{-3} A^{-1}$
Resistance	ohm	$\Omega$	$kg \, m^2 s^{-3} A^{-2}$

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Mathematical Derivations of Derived Units

To find the unit of any derived quantity, we simply substitute the units of the base quantities into the defining formula.

1. Derivation of the Joule (Energy)

Energy or Work is defined as Force $\times$ Displacement.

$$\text{Unit of Work} = \text{Unit of Force} \times \text{Unit of Displacement}$$

$$\text{Unit of Work} = (kg \, m/s^2) \times m$$

$$\text{Unit of Work} = kg \, m^2 s^{-2}$$

This composite unit is defined as $1$ Joule ($J$).

2. Derivation of the Pascal (Pressure)

Pressure is defined as Force per unit Area.

$$\text{Unit of Pressure} = \frac{\text{Force}}{\text{Area}}$$

$$\text{Unit of Pressure} = \frac{kg \, m \, s^{-2}}{m^2}$$

$$\text{Unit of Pressure} = kg \, m^{-1} s^{-2}$$

This is defined as $1$ Pascal ($Pa$).

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Common SI Prefixes

When measuring quantities like the distance between Indian cities (very large) or the size of a virus (very small), using standard units is inconvenient. Prefixes are used to indicate multiples or sub-multiples of $10$.

Factor	Prefix	Symbol	Factor	Prefix	Symbol
$10^{12}$	Tera	$T$	$10^{-3}$	milli	$m$
$10^{9}$	Giga	$G$	$10^{-6}$	micro	$\mu$
$10^{6}$	Mega	$M$	$10^{-9}$	nano	$n$
$10^{3}$	kilo	$k$	$10^{-12}$	pico	$p$
$10^{2}$	hecto	$h$	$10^{-15}$	femto	$f$

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Guidelines for Symbols

To ensure international consistency, strict rules must be followed when writing symbols for units and physical quantities.

Rules for Physical Quantities

1. Italics: Symbols for physical quantities are printed in italics (e.g., $v$ for velocity, $m$ for mass).

2. Vectors: Vectors are printed in bold upright type or indicated with an arrow (e.g., $\vec{F}$).

3. Chemical Elements: Symbols for elements like $Ca$ or $Fe$ are written in Roman upright type and never followed by a full stop.

Rules for SI Units

1. Capitalization: Unit names (like newton, joule) are never capitalized. Unit symbols are capitalized only if named after a scientist (e.g., $N$ for newton, $W$ for watt), with the exception of $L$ for litre.

2. Plurals: Symbols remain unaltered in the plural. We write $25 \, cm$, not $25 \, cms$.

3. Punctuation: Do not use a full stop at the end of a unit symbol unless it is at the end of a sentence.

4. Prefix Placement: Prefixes are attached directly to the unit without a space (e.g., $km$, $ns$).

5. The Mass Exception: Prefixes are always attached to 'gram', never to 'kilogram'. For example, $10^{-6} \, kg$ is written as $1 \, mg$ (milligram), not $1 \, \mu kg$.

Measurement of Length

The measurement of length involves determining the distance between two points in space. In physics, the range of lengths varies from the extremely minute (sub-atomic levels) to the incredibly vast (astronomical levels). Because of this vast range, a single instrument cannot be used for all measurements. We categorize these into Direct Methods and Indirect Methods.

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1. Direct Methods of Measurement

The measurement of length is a fundamental task in physics. Depending on the magnitude of the length and the accuracy required, we choose different instruments. Direct methods involve the physical use of an instrument to read the value against a standard scale.

The most important concept in direct measurement is the Least Count (LC). It is the smallest value that can be measured accurately with a given instrument. A smaller least count implies higher precision.

1. Metre Scale

The Metre Scale is the most basic instrument used in Indian schools and laboratories. It is typically a wooden or plastic strip of $1 \text{ m}$ length, graduated in centimetres and millimetres.

• Graduation: Each centimetre is divided into $10$ equal parts (millimetres).
• Least Count: Since the smallest division is $1 \text{ mm}$, the $LC = 0.1 \text{ cm} = 10^{-3} \text{ m}$.
• Range: $10^{-3} \text{ m}$ to $1 \text{ m}$ (or more for measuring tapes).

2. Vernier Callipers

When we need to measure the thickness of a small object or the internal/external diameter of a tube with greater precision, we use the Vernier Callipers. It overcomes the limitation of the metre scale where values between two millimetre marks cannot be read accurately.

Construction and Principle

It consists of two scales:

1. Main Scale (MS): Fixed scale graduated in $mm$.
2. Vernier Scale (VS): A sliding scale that moves over the main scale.

Derivation of Least Count (LC)

The Least Count is the difference between the value of one Main Scale Division ($MSD$) and one Vernier Scale Division ($VSD$).

$LC = 1 \text{ MSD} - 1 \text{ VSD}$

Usually, $n$ divisions on the Vernier Scale coincide with $(n-1)$ divisions on the Main Scale.

$n \text{ VSD} = (n-1) \text{ MSD}$

$1 \text{ VSD} = \left( \frac{n-1}{n} \right) \text{ MSD}$

Substituting this back into the $LC$ formula:

$LC = 1 \text{ MSD} - \left( \frac{n-1}{n} \right) \text{ MSD}$

$LC = \left( 1 - \frac{n-1}{n} \right) \text{ MSD} = \frac{1}{n} \text{ MSD}$

Standard Case: In most Indian labs, the smallest $MSD = 1 \text{ mm}$ and the Vernier scale has $10$ divisions.

$LC = \frac{1 \text{ mm}}{10} = 0.1 \text{ mm} = 0.01 \text{ cm} = 10^{-4} \text{ m}$.

3. Screw Gauge

The Screw Gauge works on the principle of a screw rotating in a nut. It is used for extremely precise measurements, such as the diameter of a thin copper wire or the thickness of a glass slide.

Key Terms and Derivation

• Pitch: It is the linear distance moved by the screw on the main scale when the circular head is given one full rotation.
• Circular Scale: The head of the screw is divided into equal parts (usually $50$ or $100$).

The Least Count is derived as the ratio of the pitch to the total number of divisions on the circular scale.

$LC = \frac{\text{Pitch of the Screw}}{\text{Total number of divisions on Circular Scale}}$

Calculation: If the pitch is $1 \text{ mm}$ and there are $100$ divisions on the circular scale:

$LC = \frac{1 \text{ mm}}{100} = 0.01 \text{ mm} = 0.001 \text{ cm} = 10^{-5} \text{ m}$.

4. Spherometer

A Spherometer works on the same principle as the screw gauge. It consists of a metallic frame with three fixed legs and a central moving screw. It is primarily used to measure the radius of curvature of spherical surfaces like lenses or mirrors.

The $LC$ is calculated identically to the screw gauge:

$LC = \frac{\text{Pitch}}{\text{Number of circular divisions}}$

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Summary Table of Direct Instruments

Instrument	Least Count ($m$)	Application
Metre Scale	$10^{-3} \text{ m}$	General cloth/paper measurement
Vernier Callipers	$10^{-4} \text{ m}$	Internal/External diameters
Screw Gauge	$10^{-5} \text{ m}$	Diameter of thin wires
Spherometer	$10^{-5} \text{ m}$	Curvature of lenses

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2. Indirect Methods for Large Distances

For distances that are beyond the reach of physical scales, such as the distance to a mountain peak, a planet, or a star, we use Indirect Methods.

• These methods rely on geometrical properties and trigonometry.
• The most fundamental indirect method used in astronomy is the Parallax Method.
• It is used for measuring distances of objects that are less than $100$ light-years away.

The Concept of Parallax

• Parallax: It is the apparent shift in the position of an object relative to a distant background when viewed from two different positions.
• Observation: If you hold a pen in front of you and look at it first with your left eye (closing the right) and then with your right eye (closing the left), the pen appears to move against the wall.
• Basis ($b$): The distance between the two points of observation (e.g., the distance between your two eyes) is called the Basis.

Mathematical Derivation for Distance ($D$)

To find the distance $D$ of a planet $S$, follow these steps:

1. Observe the planet from two different observatories $A$ and $B$ on the Earth, separated by a distance (basis) $b$.
2. Measure the angle $\angle ASB = \theta$ between the two directions along which the planet is viewed. This $\theta$ is called the parallactic angle.
3. Since the planet is very far away, $D$ is very large compared to $b$. Therefore, the ratio $b/D$ is very small, and $\theta$ is also very small.
4. For such small angles, we consider $AB$ as an arc of a circle with centre $S$ and radius $D$.
5. From the definition of an angle in radians:

   $\text{Angle } (\theta) = \frac{\text{Arc Length } (b)}{\text{Radius } (D)}$

6. Rearranging to find the distance $D$:

   $D = \frac{b}{\theta}$

Important Note: In the above formula, the angle $\theta$ must be expressed in radians.

Determination of Angular Size and Linear Diameter

• Once distance $D$ is known, we can find the linear diameter $d$ of the planet.
• We measure the angle $\alpha$ subtended by the planet’s diameter at the Earth using a telescope.
• This angle $\alpha$ is known as the Angular Diameter.
• The relationship is given by:

  $\alpha = \frac{d}{D} \implies d = \alpha \times D$

Angle Conversions for Calculations

In physics and astronomy, angles are treated as ratios of lengths. To perform calculations using the parallax formula $D = \frac{b}{\theta}$, the angle must be in the SI unit, which is the Radian ($rad$).

The fundamental relationship between degrees and radians is based on a full circle:

• A complete rotation equals $360^\circ$.
• In radians, a complete rotation equals $2\pi \ rad$.
• Therefore, $360^\circ = 2\pi \ rad \implies 180^\circ = \pi \ rad$.

1. Calculating $1$ Degree ($1^\circ$) in Radians

To find the value of $1^\circ$, we use the basic proportionality:

1. We have the relation: $180^\circ = \pi \ rad$.
2. $1^\circ = \frac{\pi}{180} \ rad$.
3. Substituting the value of $\pi \approx 3.14159$:

   $1^\circ = \frac{3.14159}{180} \ rad$

4. Result: $1^\circ \approx 1.745 \times 10^{-2} \ rad$.

2. Calculating $1$ Minute of arc ($1'$) in Radians

A degree is further divided into $60$ smaller parts called minutes.

1. Relation: $1^\circ = 60'$.
2. Therefore, $1' = \frac{1^\circ}{60}$.
3. Using the radian value of $1^\circ$ calculated above:

   $1' = \frac{1.745 \times 10^{-2}}{60} \ rad$

4. Result: $1' \approx 2.908 \times 10^{-4} \ rad$ (commonly rounded to $2.91 \times 10^{-4} \ rad$).

3. Calculating $1$ Second of arc ($1''$) in Radians

A minute is further divided into $60$ smaller parts called seconds.

1. Relation: $1' = 60''$.
2. Therefore, $1'' = \frac{1'}{60}$.
3. Using the radian value of $1'$ calculated above:

   $1'' = \frac{2.908 \times 10^{-4}}{60} \ rad$

4. Result: $1'' \approx 4.847 \times 10^{-6} \ rad$ (commonly rounded to $4.85 \times 10^{-6} \ rad$).

Summary of Angle Conversions

This table is essential for solving problems in Chapter 2 of Class 11 Physics (Units and Measurement):

Angle Unit	Symbol	Value in Radians
Degree	$1^\circ$	$1.745 \times 10^{-2} \ rad$
Minute	$1'$	$2.91 \times 10^{-4} \ rad$
Second	$1''$	$4.85 \times 10^{-6} \ rad$

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3. Estimation of Very Small Distances

To measure extremely small sizes, such as those of a molecule ($10^{-8} \text{ m}$ to $10^{-10} \text{ m}$), we cannot use ordinary tools or even standard optical microscopes. This requires indirect chemical methods or high-resolution electronic instruments.

Limitations of Optical Microscopes

• Nature of Investigation: An optical microscope uses visible light to observe a system.
• Resolution Limit: The ability of a microscope to "resolve" (distinguish between two close points) is fundamentally limited by the wave-like features of light.
• Wavelength Constraint: The resolution of an optical microscope is roughly equal to the wavelength of light used.
• Measurement Gap:
  • Visible light has a wavelength range of approximately $4000 \text{ Å}$ to $7000 \text{ Å}$.
  • Since $1 \text{ Å} = 10^{-10} \text{ m}$, the smallest wavelength ($4000 \text{ Å}$) is still $4000$ times larger than a typical molecule ($1 \text{ Å}$).
• Conclusion: Any particle or structure smaller than the wavelength of visible light cannot be resolved or seen clearly, making optical microscopes unsuitable for measuring atoms and molecules.

Advanced Microscopic Techniques

To overcome the resolution barriers of visible light, scientists use alternative beams with much smaller wavelengths.

1. Electron Microscopes

• Mechanism: Instead of visible light, these instruments use a beam of electrons.
• Wave Nature: Electrons also behave as waves (as per De Broglie's hypothesis), and their wavelength can be much smaller than that of visible light.
• Resolution:
  • The wavelength of an electron beam can be as small as a fraction of an $Å$.
  • Modern electron microscopes have achieved a resolution of $0.6 \text{ Å}$.
• Utility: These microscopes can almost resolve individual atoms and molecules within a material.

2. Tunnelling Microscopy

• Development: This is a more recent advancement in microscopy.
• Performance: It provides a limit of resolution that is even better than an $Å$.
• Outcome: It allows for highly accurate estimations of the sizes and arrangements of molecules.

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The Oleic Acid Method

To estimate the size of a molecule, which is typically in the range of $10^{-8} \text{ m}$ to $10^{-10} \text{ m}$, we cannot use direct measurement. Instead, we use a chemical method involving Oleic Acid to form a mono-molecular layer.

Experimental Procedure

The estimation involves the following steps to create a extremely thin film of known concentration:

1. First Solution Preparation:
   • Take $1 \text{ cm}^3$ of pure oleic acid.
   • Dissolve it in alcohol to prepare a solution of $20 \text{ cm}^3$.
2. Second Dilution:
   • Take $1 \text{ cm}^3$ of the above solution.
   • Dilute it again with alcohol to make a final volume of $20 \text{ cm}^3$.
   • Concentration: The final concentration of oleic acid is now $\frac{1}{20 \times 20} = \frac{1}{400} \text{ cm}^3$ of oleic acid per $\text{ cm}^3$ of solution.
3. Film Formation:
   • Take a large trough filled with water and lightly sprinkle lycopodium powder on its surface.
   • Add $n$ drops of the prepared solution to the water surface.
   • The oleic acid spreads quickly, and the alcohol evaporates, leaving a thin circular film of pure oleic acid.
4. Measurement:
   • Quickly measure the diameter of the circular film.
   • Calculate the Area ($A$) of the film.

Mathematical Derivation

We assume that the film formed on the water surface is exactly one molecule thick. This thickness ($t$) is treated as the diameter of the oleic acid molecule.

Step 1: Calculate Total Volume of Solution

If we drop $n$ drops of the solution into the trough, and the volume of each drop is $V$:

$\text{Total Volume of solution} = nV \text{ cm}^3$

Step 2: Calculate Volume of Pure Oleic Acid

Since the concentration is $\frac{1}{400}$:

$\text{Volume of Oleic Acid in the film} = \frac{nV}{400} \text{ cm}^3$

Step 3: Calculate Thickness (Molecular Diameter)

The film spreads over an area $A$. The relationship between Volume, Area, and Thickness is:

$\text{Volume} = \text{Area} \times \text{Thickness}$

$t = \frac{\text{Volume of Oleic Acid}}{\text{Area}}$

Substituting the volume from Step 2:

$t = \frac{nV}{400A} \text{ cm}$

By using this formula, the thickness $t$ is found to be approximately $10^{-9} \text{ m}$.

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4. Range of Lengths

The universe exhibits a staggering range of lengths, spanning from the incredibly small sub-atomic particles to the vast extent of the observable universe. In Physics, we deal with lengths as small as $10^{-14} \text{ m}$ (the size of a tiny nucleus) to as large as $10^{26} \text{ m}$ (the extent of the observable universe).

Because these numbers are either extremely small or extremely large, we use Scientific Notation (powers of 10) and certain Special Units to make measurements and calculations more convenient.

Special Units for Small (Microscopic) Lengths

For atomic and nuclear scales, the metre is too large a unit. We use the following:

• Fermi ($f$): Used specifically in nuclear physics to measure the size of a nucleus.

  $1 \text{ fermi} = 1 \text{ f} = 10^{-15} \text{ m}$

• Angstrom (Å): Used in atomic physics to measure the size of atoms and the wavelength of light.

  $1 \text{ angstrom} = 1 \text{ Å} = 10^{-10} \text{ m}$

Special Units for Large (Astronomical) Lengths

For distances between planets and stars, the following units are used:

1. Astronomical Unit (AU)

It is the average distance of the centre of the Sun from the centre of the Earth.

$1 \text{ AU} = 1.496 \times 10^{11} \text{ m}$

2. Light Year (ly)

It is the distance that light travels in vacuum in one year with a velocity of $3 \times 10^8 \text{ m s}^{-1}$.

Derivation:

$1 \text{ light year} = \text{Speed of light} \times \text{Time (1 year in seconds)}$

$1 \text{ ly} = (3 \times 10^8 \text{ m/s}) \times (365.25 \times 24 \times 60 \times 60 \text{ s})$

$1 \text{ ly} = 9.46 \times 10^{15} \text{ m}$

3. Parsec (pc)

Parsec (Parallactic Second) is the distance at which the average radius of the Earth’s orbit ($1 \text{ AU}$) subtends an angle of 1 arc second.

Derivation:

Using the formula $\theta = \frac{l}{r} \implies r = \frac{l}{\theta}$

Where $l = 1 \text{ AU}$ and $\theta = 1''$ (converted to radians).

$1 \text{ pc} = \frac{1.496 \times 10^{11} \text{ m}}{4.85 \times 10^{-6} \text{ rad}}$

$1 \text{ pc} \approx 3.08 \times 10^{16} \text{ m}$

Comparison Table of Units

Unit Name	Fermi ($f$)	Angstrom (Å)	Astron. Unit (AU)	Light Year (ly)	Parsec (pc)
Value in Metres	$10^{-15} \text{ m}$	$10^{-10} \text{ m}$	$1.496 \times 10^{11} \text{ m}$	$9.46 \times 10^{15} \text{ m}$	$3.08 \times 10^{16} \text{ m}$

Range and Order of Lengths

The following table illustrates the wide spectrum of sizes and distances that are studied in Physics. Note that the ratio of the longest and shortest lengths in our universe is approximately $10^{41}$.

Size of object or distance	Length (m)
Size of a proton	$10^{-15}$
Size of atomic nucleus	$10^{-14}$
Size of hydrogen atom	$10^{-10}$
Length of typical virus	$10^{-8}$
Wavelength of light	$10^{-7}$
Size of red blood corpuscle	$10^{-5}$
Thickness of a paper	$10^{-4}$
Height of Mount Everest above sea level	$10^{4}$
Radius of the Earth	$10^{7}$
Distance of Moon from the Earth	$10^{8}$
Distance of the Sun from the Earth	$10^{11}$
Distance of Pluto from the Sun	$10^{13}$
Size of our galaxy	$10^{21}$
Distance to Andromeda galaxy	$10^{22}$
Distance to the boundary of observable universe	$10^{26}$

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Measurement of Mass

Mass is a fundamental and basic property of matter. Unlike weight, mass does not depend on the temperature, pressure, or the location of the object in space. For instance, the mass of an object remains the same whether it is on the Earth, the Moon, or in deep space.

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SI Unit of Mass

The kilogram ($kg$) is the internationally accepted base unit of mass. Historically, it was defined by a physical prototype—a cylinder made of platinum-iridium alloy kept at the International Bureau of Weights and Measures in France. However, physical objects can accumulate dust or lose atoms over time, leading to slight variations in measurement.

The Modern Definition (Post-2018)

Mass is a basic property of any object that does not change with location. The standard unit to measure mass is the Kilogram ($kg$).

Before 2018, the kilogram was defined by a physical metal cylinder kept in France. However, physical objects can change over time (due to dust or wear). To make the measurement permanent and universal, scientists now use a constant of nature called the Planck Constant ($h$).

Key Constant Values

• Planck Constant ($h$): $6.62607015 \times 10^{-34} \text{ kg m}^2 \text{ s}^{-1}$
• Speed of Light ($c$): $299,792,458 \text{ m s}^{-1}$

Derivation of Mass

The new definition links mass to energy. We use two famous equations from physics to understand how mass is calculated from the Planck constant:

1. Quantum Energy Equation: $E = h\nu$ (where $\nu$ is frequency)
2. Mass-Energy Equation: $E = mc^2$

Since both represent energy ($E$), we can put them together:

$mc^2 = h\nu$

To find the mass ($m$), we rearrange the formula:

$m = \frac{h\nu}{c^2}$

Conclusion: This formula shows that if we know the fixed value of $h$ and the speed of light $c$, we can define the kilogram ($m$) using the Second (linked to frequency $\nu$) and the Metre (linked to $c$).

Dimensional Consistency

To check if the units are correct, we look at the dimensions:

• The unit of $h$ is $\text{kg m}^2 \text{ s}^{-1}$.
• The unit of $c^2$ is $\text{m}^2 \text{ s}^{-2}$.
• The unit of frequency ($\nu$) is $\text{s}^{-1}$.

When we put them in the formula $m = \frac{h\nu}{c^2}$:

$\text{Mass Unit} = \frac{(\text{kg m}^2 \text{ s}^{-1}) \times (\text{s}^{-1})}{\text{m}^2 \text{ s}^{-2}}$

The $\text{m}^2$ and $\text{s}^{-2}$ cancel out, leaving only kilogram ($kg$).

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Unified Atomic Mass Unit ($u$)

When dealing with atoms and molecules, using the kilogram is impractical because the numbers are too small. For example, the mass of a hydrogen atom is approximately $0.00000000000000000000000000167 \text{ kg}$. To simplify this, scientists established the Unified Atomic Mass Unit.

Definition

One unified atomic mass unit ($1 \text{ u}$) is defined as exactly one-twelfth ($\frac{1}{12}$) of the mass of an unexcited atom of the Carbon-12 isotope ($_{6}^{12}C$), including the mass of its electrons.

Mathematical Expression and Derivation

The relationship is given by the formula:

$1 \text{ u} = \frac{\text{Mass of one atom of } ^{12}C}{12}$

To find the value in kilograms, we use Avogadro's number ($N_A \approx 6.022 \times 10^{23} \text{ mol}^{-1}$):

Mass of $1 \text{ mole}$ of $^{12}C = 12 \text{ g} = 0.012 \text{ kg}$

Mass of $1 \text{ atom}$ of $^{12}C = \frac{0.012}{6.022 \times 10^{23}} \text{ kg}$

$1 \text{ u} = \frac{1}{12} \times \left( \frac{0.012}{6.022 \times 10^{23}} \right) \text{ kg}$

Result: $1 \text{ u} \approx 1.66 \times 10^{-27} \text{ kg}$

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Methods of Measuring Mass

The technique used to measure mass depends entirely on the scale of the object being measured. We categorize these into three main methods:

1. Common Balance (Macroscopic Scale)

For everyday objects like vegetables, grains, or metals, we use a Common Balance (Beam Balance). This is frequently seen in Indian grocery shops and markets. It works by comparing the unknown mass with a known standard mass using the principle of moments.

2. Gravitational Method (Large Scale)

To measure the mass of planets, moons, and stars, we cannot use a balance. Instead, we use Newton’s Law of Gravitation. By observing the orbital period ($T$) and the radius of the orbit ($R$) of a satellite or planet, the mass ($M$) of the central body can be calculated.

The gravitational force provides the centripetal force:

$G \frac{Mm}{R^2} = \frac{mv^2}{R}$

From this, the mass $M$ of the heavy celestial body is derived as:

$M = \frac{v^2 R}{G}$

3. Mass Spectrograph (Atomic Scale)

For sub-atomic particles and atoms, a Mass Spectrograph is used. In this device, a charged particle is made to move through uniform electric and magnetic fields. The particle follows a curved trajectory.

• The radius ($r$) of this trajectory is directly proportional to the mass ($m$) of the particle.
• By measuring the deflection or radius, the precise mass of the particle is determined.

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Comparison of Measurement Techniques

Scale	Example Object	Instrument/Method
Small (Atomic)	Protons, Electrons, Atoms	Mass Spectrograph
Medium (Everyday)	Rice, Gold, Books	Common Balance / Electronic Scale
Large (Celestial)	Earth, Sun, Jupiter	Gravitational Method (Newton's Laws)

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Range and Order of Masses

The range of masses encountered in the universe is incredibly vast, spanning over $85$ orders of magnitude. The table below lists the typical masses of various objects:

Object	Mass ($kg$)
Electron	$10^{-30}$
Proton	$10^{-27}$
Uranium atom	$10^{-25}$
Red blood cell	$10^{-13}$
Dust particle	$10^{-9}$
Rain drop	$10^{-6}$
Mosquito	$10^{-5}$
Grape	$10^{-3}$
Human	$10^{2}$
Automobile	$10^{3}$
Boeing 747 aircraft	$10^{8}$
Moon	$10^{23}$
Earth	$10^{25}$
Sun	$10^{30}$
Milky way galaxy	$10^{41}$
Observable Universe	$10^{55}$

Measurement of Time

To measure any time interval, we require a clock. In modern science, time measurement has reached extraordinary levels of precision by moving from mechanical vibrations to atomic vibrations. The current international standard for time is based on the Atomic Clock.

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The Atomic Standard of Time (Caesium Clock)

The Caesium Atomic Clock uses the vibrations of atoms as a "pendulum." In a standard wristwatch, a quartz crystal vibrates at $32,768$ times per second. In contrast, a Caesium atom vibrates billions of times per second, making it significantly more stable and accurate.

Mechanism of Vibration

The "vibrations" in an atomic clock actually refer to the transitions between two energy states of an atom. Specifically, we use the Hyperfine Levels of the ground state of the Caesium-133 atom. When the atom absorbs a specific frequency of microwave radiation, it jumps between these levels. The frequency of this radiation is so constant that it serves as the ultimate "tick" of the universal clock.

Standard Definition of a Second

The Second ($s$) is defined by taking the fixed numerical value of the Caesium frequency, $\Delta\nu_{Cs}$, which is the unperturbed ground-state hyperfine transition frequency of the Caesium-133 atom.

The definition is mathematically stated as:

$\Delta\nu_{Cs} = 9,192,631,770 \text{ Hz}$

This means that exactly $9,192,631,770$ cycles of this radiation must pass for one second to be completed.

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Time Maintenance and IST in India

In India, the National Physical Laboratory (NPL) in New Delhi is the custodian of the nation's time. It maintains the Indian Standard Time (IST) using an ensemble of four Caesium atomic clocks.

• Precision: The clocks at NPL are so precise that the uncertainty is merely $1$ part in $10^{15}$. This is written as an uncertainty of $\pm 1 \times 10^{-15}$.
• Stability and Drift: These clocks are exceptionally stable. Over the duration of an entire year, they do not gain or lose more than $32 \ \mu s$ ($32 \text{ millionths of a second}$).
• Global Linking: IST is linked to the International Atomic Time (TAI) and Coordinated Universal Time (UTC) maintained by the BIPM in France.

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Linking Time to Length

Because time can be measured with such immense accuracy ($1$ part in $10^{15}$), whereas other quantities like length cannot, the scientific community decided to define Length in terms of Time. This was done by fixing the Speed of Light ($c$).

The Metre is now defined as the distance light travels in vacuum in a time interval of:

$t = \frac{1}{299,792,458} \text{ seconds}$

This ensures that as long as our atomic clocks are accurate, our definition of the metre remains perfectly constant everywhere in the universe.

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Range and Order of Time Intervals

The events in our universe occur over a massive range of time, from the fleeting life of an unstable particle to the vast age of the universe itself. The ratio between the longest and shortest time intervals is approximately $10^{41}$.

Event	Time Interval ($s$)
Life-span of most unstable particle	$10^{-24}$
Time required for light to cross a nuclear distance	$10^{-22}$
Period of x-rays	$10^{-19}$
Period of atomic vibrations	$10^{-15}$
Period of light wave	$10^{-15}$
Life time of an excited state of an atom	$10^{-8}$
Period of radio wave	$10^{-6}$
Period of a sound wave	$10^{-3}$
Wink of eye	$10^{-1}$
Time between successive human heart beats	$10^{0}$
Travel time for light from Moon to the Earth	$10^{0}$
Travel time for light from the Sun to the Earth	$10^{2}$
Time period of a satellite	$10^{4}$
Rotation period of the Earth	$10^{5}$
Rotation and revolution periods of the Moon	$10^{6}$
Revolution period of the Earth	$10^{7}$
Travel time for light from nearest star	$10^{8}$
Average human life-span	$10^{9}$
Age of Egyptian pyramids	$10^{11}$
Time since dinosaurs became extinct	$10^{15}$
Age of the universe	$10^{17}$

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Cosmic Ratios and Curious Coincidences

When we examine the tables for the range of Length, Time, and Mass, a fascinating and mysterious pattern emerges. Scientists have discovered that certain ratios between the largest and smallest scales of our universe result in the same numerical order of magnitude: $10^{41}$.

Analysis of the "Large Number" Coincidence

By comparing the extremes of the universe, we can derive the following mathematical ratios based on the data provided in the previous sections:

1. Ratio of Lengths ($R_L$)

The ratio of the longest length (distance to the boundary of the observable universe) to the shortest length (size of a proton) is calculated as:

$R_L = \frac{10^{26} \text{ m}}{10^{-15} \text{ m}} = 10^{41}$

2. Ratio of Time Intervals ($R_T$)

The ratio of the longest time interval (age of the universe) to the shortest time interval (life-span of the most unstable particle) is calculated as:

$R_T = \frac{10^{17} \text{ s}}{10^{-24} \text{ s}} \approx 10^{41}$

3. Ratio of Masses ($R_M$)

The ratio of the largest mass (observable universe) to the smallest mass (electron) is approximately the square of this "magic number":

$R_M = \frac{10^{55} \text{ kg}}{10^{-30} \text{ kg}} \approx 10^{85}$

Interestingly, $(10^{41})^2 = 10^{82}$, which is very close to the order of magnitude of the mass ratio of the universe.

Summary of Cosmic Ratios

The following table summarizes these coincidences which have led many physicists, including Paul Dirac, to speculate about deeper laws of nature.

Physical Quantity	Calculated Ratio	Order of Magnitude
Length (Largest/Smallest)	$10^{26} / 10^{-15}$	$10^{41}$
Time (Longest/Shortest)	$10^{17} / 10^{-24}$	$10^{41}$
Mass (Largest/Smallest)	$10^{55} / 10^{-30}$	$\approx (10^{41})^2$

Whether these ratios are purely accidental or point toward a deeper physical symmetry in the fabric of the universe remains one of the greatest mysteries in modern physics.

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Accuracy, Precision of Instruments and Errors in Measurement

Measurement is the cornerstone of all experimental science and technology. However, the result of every measurement by any measuring instrument contains some degree of uncertainty. This uncertainty is technically called an error. To understand the quality of a measurement, we must analyze two fundamental concepts: Accuracy and Precision.

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Accuracy and Precision

While often used interchangeably in daily life, these terms have distinct meanings in scientific measurement:

1. Accuracy: It is a measure of how close the measured value is to the true value of the quantity. Accuracy depends on the calibration of the instrument and the removal of systematic biases.

2. Precision: It tells us to what resolution or limit the quantity is measured. Precision is determined by the least count of the instrument. A measurement can be very precise but inaccurate if the instrument is not correctly calibrated.

Illustrative Example

Suppose the true value of a certain length is exactly $3.678 \text{ cm}$.

• Experiment A: Using an instrument with resolution $0.1 \text{ cm}$, the measured value is $3.5 \text{ cm}$.
• Experiment B: Using an instrument with higher resolution $0.01 \text{ cm}$, the value is $3.38 \text{ cm}$.

In this case, the first measurement is more accurate (closer to $3.678$) but less precise. The second measurement is more precise (higher resolution) but less accurate.

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Classification of Errors

Errors in measurement are generally classified into two broad categories based on their nature and source: Systematic Errors and Random Errors.

1. Systematic Errors

These are errors that tend to be in one direction, either consistently positive or consistently negative. Since the cause is often known, these errors can be minimized or corrected. Major sources include:

• Instrumental Errors: These arise from inherent flaws in the instrument's design or calibration. For example, if a thermometer in an Indian research lab reads $104 \ ^\circ C$ instead of $100 \ ^\circ C$ at the boiling point of water, it has a calibration error. Similarly, "Zero Error" occurs when the zero mark of a scale does not coincide with the reference point.
• Imperfection in Experimental Technique: Sometimes the procedure itself introduces error. For instance, measuring body temperature by placing a thermometer under the armpit always gives a value lower than the actual internal body temperature. External factors like changes in humidity or wind velocity can also systematically bias results.
• Personal Errors: These arise from the observer's individual habits or lack of care. A common example is Parallax Error, where an observer holds their head slightly to the left or right while reading a needle position on a scale, leading to a consistently wrong reading.

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2. Random Errors

Random errors occur irregularly and vary in both sign and size. They are unpredictable and arise due to:

• Random fluctuations in experimental conditions like temperature, voltage supply, or mechanical vibrations.
• Unbiased personal errors made by the observer while taking readings.

Because these errors are random, they can be minimized by taking a large number of observations and calculating their Arithmetic Mean. The likelihood of overestimating the value becomes equal to the likelihood of underestimating it, causing the errors to cancel out on average.

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3. Least Count Error

The Least Count is the smallest value that can be measured by a measuring instrument. The uncertainty associated with this limit is called the Least Count Error.

• It is a type of random error but has a limited size.
• For a metre scale, the least count is $1 \text{ mm} = 0.1 \text{ cm}$.
• For Vernier Callipers, it is typically $0.01 \text{ cm}$.
• For a Screw Gauge, it is typically $0.001 \text{ cm}$.

We can reduce this error by using instruments of higher precision or by improving experimental techniques.

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Mathematical Analysis of Errors

In scientific experiments, since the "true value" of a physical quantity is often unknown, we use statistical methods to estimate it and determine the reliability of our measurements. This process involves calculating the average of multiple readings and analyzing the deviations from that average.

1. Arithmetic Mean ($a_{mean}$)

When we perform an experiment multiple times, random errors cause the readings to vary. By taking the arithmetic mean, we assume that overestimations and underestimations will cancel each other out, providing us with a value closest to the true value.

If $a_1, a_2, a_3, ..., a_n$ are the values obtained in $n$ measurements, then the arithmetic mean is calculated as:

$a_{mean} = \frac{a_1 + a_2 + a_3 + ... + a_n}{n}$

In summation notation, this is expressed as:

$a_{mean} = \sum\limits_{i=1}^{n} \frac{a_i}{n}$

2. Absolute Error ($|\Delta a|$)

The Absolute Error is the magnitude of the difference between the true value (taken as the arithmetic mean) and the individual measured value. It tells us how much a specific reading "deviated" from the average.

The absolute errors for individual readings are calculated as follows:

$\Delta a_1 = a_1 - a_{mean}$

$\Delta a_2 = a_2 - a_{mean}$

... and so on up to $\Delta a_n = a_n - a_{mean}$.

Note: While the calculated $\Delta a_i$ can be positive or negative, the absolute error is always considered as the magnitude, denoted by $|\Delta a_i|$, which is always positive.

3. Mean Absolute Error ($\Delta a_{mean}$)

To find the overall uncertainty in our set of measurements, we take the arithmetic mean of all the individual absolute errors. This is called the Mean Absolute Error.

The formula for Mean Absolute Error is:

$\Delta a_{mean} = \frac{|\Delta a_1| + |\Delta a_2| + |\Delta a_3| + ... + |\Delta a_n|}{n}$

Or in summation form:

$\Delta a_{mean} = \sum\limits_{i=1}^{n} \frac{|\Delta a_i|}{n}$

The final result of a measurement is usually reported as: $a = a_{mean} \pm \Delta a_{mean}$. This implies that the true value is likely to lie between $(a_{mean} + \Delta a_{mean})$ and $(a_{mean} - \Delta a_{mean})$.

4. Relative Error and Percentage Error

Absolute error does not always give a clear picture of the "seriousness" of the error. For example, an error of $1 \text{ cm}$ is huge if you are measuring a pen, but negligible if you are measuring a national highway. To solve this, we use Relative and Percentage errors.

Relative Error

It is the ratio of the mean absolute error to the mean value of the quantity measured.

$\text{Relative Error} = \frac{\Delta a_{mean}}{a_{mean}}$

Percentage Error ($\delta a$)

When the relative error is expressed in percent, it is called the Percentage Error.

$\delta a = \left( \frac{\Delta a_{mean}}{a_{mean}} \right) \times 100 \%$

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Summary Table of Error Analysis

Term	Mathematical Definition	Purpose
Arithmetic Mean	$\sum \frac{a_i}{n}$	Acts as the 'True Value'
Absolute Error	$|a_i - a_{mean}|$	Individual deviation
Mean Absolute Error	$\sum \frac{|\Delta a_i|}{n}$	Average uncertainty
Relative Error	$\frac{\Delta a_{mean}}{a_{mean}}$	Normalized error
Percentage Error	$\text{Relative Error} \times 100$	Standardized error in %

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Combination of Errors

When we perform an experiment, the final result often depends on several different measured quantities. Since each measurement has its own uncertainty, these errors "propagate" or combine to affect the final calculated value. Understanding how these errors combine is essential for reporting accurate scientific results.

1. Error of a Sum or a Difference

The fundamental principle in error analysis is that we always calculate the Maximum Possible Error. This ensures that the true value of the quantity is guaranteed to lie within the range we specify. This is why, even when two quantities are subtracted, their absolute errors are added, never subtracted.

A. Error of a Sum

Suppose two physical quantities $A$ and $B$ have measured values $(A \pm \Delta A)$ and $(B \pm \Delta B)$. Let $Z$ be their sum:

$Z = A + B$

Derivation:

1. Including the errors, the value of $Z$ becomes:

   $Z \pm \Delta Z = (A \pm \Delta A) + (B \pm \Delta B)$

2. Rearranging the terms on the right-hand side:

   $Z \pm \Delta Z = (A + B) \pm \Delta A \pm \Delta B$

3. Since we know that $Z = A + B$, the term $(A + B)$ cancels out from both sides:

   $\pm \Delta Z = \pm \Delta A \pm \Delta B$

4. The four possible values for $\Delta Z$ are $(+\Delta A + \Delta B)$, $(+\Delta A - \Delta B)$, $(-\Delta A + \Delta B)$, and $(-\Delta A - \Delta B)$.
5. The maximum absolute error is therefore:

   $\Delta Z = \Delta A + \Delta B$

B. Error of a Difference

Suppose we want to find the error in the difference of two quantities $Z = A - B$.

Derivation:

1. Including the errors, the value of $Z$ becomes:

   $Z \pm \Delta Z = (A \pm \Delta A) - (B \pm \Delta B)$

2. Opening the brackets:

   $Z \pm \Delta Z = (A - B) \pm \Delta A \mp \Delta B$

3. Since $Z = A - B$, the equation simplifies to:

   $\pm \Delta Z = \pm \Delta A \mp \Delta B$

4. To find the most pessimistic or maximum possible error, we take the sum of the magnitudes of the errors. We do not subtract the errors because a smaller error would lead us to be overconfident in our result.
5. Thus, the maximum absolute error is:

   $\Delta Z = \Delta A + \Delta B$

Summary Rule for Sum and Difference

Rule: When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Mathematical Operation	Measured Result ($Z$)	Maximum Absolute Error ($\Delta Z$)
Addition	$Z = A + B$	$\Delta A + \Delta B$
Subtraction	$Z = A - B$	$\Delta A + \Delta B$

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2. Error of a Product or a Quotient

When physical quantities are related by multiplication or division, the errors do not combine as simple additions of absolute values. Instead, it is the Relative Error (or Fractional Error) that propagates. This section explains how uncertainties are calculated for derived quantities like area, density, or speed.

1. Error of a Product

Suppose a physical quantity $Z$ is the product of two independent measured quantities $A$ and $B$. Let the measured values be $(A \pm \Delta A)$ and $(B \pm \Delta B)$, where $\Delta A$ and $\Delta B$ are their respective absolute errors.

Mathematical Derivation

We start with the relationship:

$Z = AB$

Including the uncertainties, the measured value of $Z$ becomes:

$Z \pm \Delta Z = (A \pm \Delta A)(B \pm \Delta B)$

Expanding the brackets on the right-hand side:

$Z \pm \Delta Z = AB \pm A\Delta B \pm B\Delta A \pm \Delta A\Delta B$

To find the relative error, we divide the left side by $Z$ and the right side by $AB$ (since $Z = AB$):

$\frac{Z \pm \Delta Z}{Z} = \frac{AB \pm A\Delta B \pm B\Delta A \pm \Delta A\Delta B}{AB}$

$1 \pm \frac{\Delta Z}{Z} = 1 \pm \frac{\Delta B}{B} \pm \frac{\Delta A}{A} \pm \frac{\Delta A\Delta B}{AB}$

In scientific measurements, the absolute errors $\Delta A$ and $\Delta B$ are very small compared to the actual values $A$ and $B$. Therefore, their product $\Delta A\Delta B$ is negligibly small and can be ignored. After canceling $1$ from both sides, we get:

$\pm \frac{\Delta Z}{Z} = \pm \frac{\Delta A}{A} \pm \frac{\Delta B}{B}$

For the Maximum Relative Error, we sum the individual relative errors:

$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$

2. Error of a Quotient

The same rule applies when quantities are divided. If $Z = A/B$, the maximum relative error is still the sum of the relative errors of $A$ and $B$.

Logic using Logarithmic Differentiation

A more sophisticated way to derive this for any power or quotient is by using natural logarithms:

$ln \ Z = ln \ A - ln \ B$

Differentiating both sides:

$\frac{dZ}{Z} = \frac{dA}{A} - \frac{dB}{B}$

To obtain the maximum possible error, we take the absolute values and add them:

$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$

The General Rule

Rule: When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers or dividers.

Relationship	Result ($Z$)	Maximum Relative Error ($\frac{\Delta Z}{Z}$)
Product	$Z = A \times B$	$\frac{\Delta A}{A} + \frac{\Delta B}{B}$
Quotient	$Z = A / B$	$\frac{\Delta A}{A} + \frac{\Delta B}{B}$

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3. Error in Case of a Power

When a physical quantity is raised to a mathematical power, the relative error in that quantity is multiplied by the value of the power. This is a very important rule because it shows that quantities with higher exponents contribute more significantly to the total uncertainty of the result.

1. Derivation for a Simple Power

Suppose a physical quantity $Z$ depends on a measured quantity $A$ such that:

$Z = A^2$

We can write this expression as a product of the same quantity:

$Z = A \times A$

Applying the Product Rule of errors (where relative errors are added):

$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta A}{A}$

$\frac{\Delta Z}{Z} = 2 \frac{\Delta A}{A}$

Hence, the relative error in $A^2$ is twice the relative error in $A$.

2. General Dimensional Equation Rule

In a more complex scenario where a quantity $Z$ depends on multiple variables raised to different powers, such as:

$Z = \frac{A^p B^q}{C^r}$

The maximum relative error in $Z$ is obtained by summing the relative errors of each component multiplied by their respective powers. Note that even if a quantity is in the denominator (like $C^r$), its error is added because we always seek the maximum possible uncertainty.

The General Formula:

$\frac{\Delta Z}{Z} = p \left( \frac{\Delta A}{A} \right) + q \left( \frac{\Delta B}{B} \right) + r \left( \frac{\Delta C}{C} \right)$

Rule: The relative error in a physical quantity raised to the power $k$ is the $k$ times the relative error in the individual quantity.

Summary of Power Rule Multipliers

Quantity Form	Power ($k$)	Relative Error Contribution
$A^2$	$2$	$2 \left( \frac{\Delta A}{A} \right)$
$A^3$	$3$	$3 \left( \frac{\Delta A}{A} \right)$
$\sqrt{A}$	$1/2$	$\frac{1}{2} \left( \frac{\Delta A}{A} \right)$
$\frac{1}{A}$	$1$	$1 \left( \frac{\Delta A}{A} \right)$
$A^{3/2}$	$1.5$	$1.5 \left( \frac{\Delta A}{A} \right)$

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Visualization of Power Impact

Quantities with higher powers in a formula contribute more to the final error. For instance, in the formula for $g$ above, an error in time ($T$) has double the impact of an error in length ($L$) because $T$ is squared.

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Summary Table of Error Rules

The following table summarizes how absolute and relative errors combine during various arithmetic operations.

Mathematical Operation	Result ($Z$)	Error Rule (Maximum Error)
Addition	$Z = A + B$	$\Delta Z = \Delta A + \Delta B$ (Absolute errors add)
Subtraction	$Z = A - B$	$\Delta Z = \Delta A + \Delta B$ (Absolute errors add)
Multiplication	$Z = A \times B$	$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$ (Relative errors add)
Division	$Z = A / B$	$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$ (Relative errors add)
Power	$Z = A^k$	$\frac{\Delta Z}{Z} = k \left( \frac{\Delta A}{A} \right)$ (Relative error multiplied by power)

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Practical Challenges in Measurement

One of the most fascinating challenges in measurement is dealing with the fractal nature of objects. A common example is the measurement of international boundaries and coastlines.

• The France-Belgium Discrepancy: It is an interesting fact that the length of the common international boundary between France and Belgium mentioned in the official documents of the two countries differs substantially.
• The Coastline Paradox: Imagine measuring the coastline of Gujarat or Andhra Pradesh. Roads and rivers have mild bends, but coastlines are extremely irregular.
• Impact of Scale: If you measure a coastline with a $100 \text{ m}$ tape, you skip many small crevices. If you use a $1 \text{ m}$ ruler, you account for those crevices, and the total measured length increases.
• Fractals and Chaos: This phenomenon belongs to the area of Fractals, a complex field in theoretical physics and mathematics where the measured length depends on the scale of the "measuring stick."

Least Count of Measuring Instruments

The Least Count is the smallest value that can be measured by a measuring instrument. All readings or measured values are reliable only up to this limit. Using instruments with a smaller least count increases the precision of the measurement.

Standard Least Counts in Laboratory

Instrument	Least Count (in cm)
Metre Scale	$0.1 \text{ cm}$
Vernier Callipers	$0.01 \text{ cm}$
Screw Gauge	$0.001 \text{ cm}$
Spherometer	$0.001 \text{ cm}$

Detailed Analysis of Error Propagation

In many experiments, like finding the acceleration due to gravity ($g$) using a simple pendulum, we combine multiple measurements (Length and Time). The error in each measurement contributes to the total error in the result.

Significant Figures

Dimensions of Physical Quantities

Dimensional Formulae and Dimensional Equations

Dimensional Analysis and Its Applications

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Additional Exercises

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