Units & Measurements

Chapter 1: MEASUREMENT

Science and engineering are based on measurements and comparisons. Physics is fundamentally based on the measurement of physical quantities. Thus, we need to classify things, make rules about how to measure them, and conduct experiments to establish the units for those measurements.

GPS

Without clocks of extreme accuracy, GPS would be useless as it calculates position by using the time taken by signals to travel from the satellite to a receiver.

Physical Quantity

Any quantity that can be measured using physical instruments is called a Physical Quantity.

• Eg: length, time, current, voltage, temperature, etc.
• Instruments: Scale, Clock, Ammeter, Voltmeter, Thermometer.

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Few Important Base Quantity and it’s History

Length

The standard for length has evolved significantly over time to meet the demand for higher precision.

Evolution of the Meter

1. Earth Standard (1792): The newborn Republic of France established the meter, defining it as one ten-millionth of the distance from the north pole to the equator.

2. Metal Bar Standard: For practical reasons, the Earth standard was abandoned. The meter came to be defined as the distance between two fine lines engraved near the ends of a platinum-iridium bar, kept at the International Bureau of Weights and Measures near Paris. Secondary standards were produced so every measuring device derived authority from this standard meter bar through a chain of comparisons.

3. Krypton-86 Standard (1960): A new standard based on the wavelength of light was adopted. The meter was redefined to be $1650763.73$ wavelengths of a particular orange-red light emitted by atoms of krypton-86. This awkward number was chosen so the new standard would be close to the old meter-bar standard.

4. Current Standard (1983): The demand for higher precision required a bolder step. The meter was redefined as the distance traveled by light in a vacuum during a time interval of $1/299792458$ of a second.

The Speed of Light

Because of the 1983 definition, the speed of light $c$ is now an exactly defined quantity: $c=299792458m/s$.

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Time

Time has two aspects: knowing the time of day to order events in sequence, and knowing how long an event lasts (duration). Any time standard must be able to answer two questions: “When did it happen?” and “What is its duration?”. Any phenomenon that repeats itself is a possible time standard.

Earth’s Rotation vs. Atomic Clocks

• Earth’s rotation has been used as a time standard for centuries, but there is a variation due to tidal effects caused by the Moon and large-scale winds.
• A quartz clock can be calibrated against Earth’s rotation, but it lacks the accuracy called for by modern scientific technology.
• To meet the need for a better time standard, atomic clocks have been developed. An atomic clock at the National Institute of Standards and Technology (NIST) in Boulder, Colorado, is the standard for Coordinated Universal Time (UTC) in the United States. Its time signals are sent worldwide by shortwave radio and telephone.

The Standard Second

• In 1967, the 13th General Conference on Weights and Measures adopted a standard second based on the cesium clock.
• Definition: One second is the time taken by $9192631770$ oscillations of the light emitted by a cesium-133 atom.
• Note: Atomic clocks are so consistent that two cesium clocks would have to run for 6000 years before their readings would differ by more than 1 second.

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Mass

Mass Standards

• The Standard Kilogram: The SI standard of mass is a cylinder of platinum and iridium that is kept at the International Bureau of Weights and Measures near Paris and assigned a mass of 1 kilogram. (The U.S. copy is housed in a vault at NIST and removed only once a year for checking duplicates).

• Kibble Balance: A far more accurate way of measuring mass. A standard mass is measured when the downward pull on it by gravity is balanced by an upward force from a magnetic field due to an electrical current. Its precision comes from quantum mechanical quantities that have been precisely defined or measured.

• Atomic Mass Standard: The carbon-12 atom has been assigned a mass of $12u$ by international agreement because masses of atoms can be compared with one another more precisely than with the standard kilogram.

• Conversion: $1u=1.66053886\times 10^{-27}kg$, with an uncertainty of $\pm 10$ in the last two decimal places.

Density

As a key property related to mass, density $\rho$ is the mass per unit volume:
$\rho =\frac{m}{V}$
Densities are typically listed in kilograms per cubic meter ($kg/m^3$) or grams per cubic centimetre ($g/c{m}^3$).

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Classification of Physical Quantity

Based on Directional Properties

• Scalars:
  • It only has magnitude but not any direction.
  • Ex: mass, length, time, temperature, etc.
• Vectors:
  • It has both Magnitude and Direction.
  • It follows the Vector laws of addition.
  • Ex: force, displacement, velocity, acceleration, momentum, etc.

Based on Dependency

• Fundamental / Base: A select number of physical quantities chosen by international agreement to serve as the independent foundation for all other measurements.
  • They are assigned specific, invariable standards from which other quantities are built.
• Derived: Any physical quantity defined in terms of Base Quantities and their standards.

Few Important properties of PQ & Constants

• PQs may not have Units (eg- refractive index, strain) & may not have Dimensions
  • An Unitless PQ must not have dimensions
  • A Dimensionless PQ may have Units (eg- angle 2d, 3d)
• Consatants may have dimensions.
  • Eg - Gravitational Constant, Boltzman Constant, etc and remember 1, 2, 3, 1/3, etc are constants which doesn’t have dimensions

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The Seven Base Quantities and their SI Units and Dimensions

Quantity	Symbol	SI Unit	Unit Symbol	Dimension Symbol
Length	$l,x,r$	Metre	$m$	$L$
Mass	$m$	Kilogram	$kg$	$M$
Time / Duration	$t$	Second	$s$	$T$
Electric Current	$I,i$	Ampere	$A$	$I$ or $A$
Thermodynamic Temperature	$T$	Kelvin	$K$	$\theta$ or $K$
Amount of Substance	$n$	Mole	$mol$	$mol$ or $N$
Luminous Intensity	$I_v$	Candela	$cd$	$J$ or $I_v$

Start of SI

In 1971, the 14th General Conference on Weights and Measures picked seven quantities as base quantities, thereby forming the basis of the International System of Units (aka SI, aka Metric System) + 2 supplementary units.

Charectaristics of Units

• Well defined and invariable
• Easily avialable and reproducible
• Universally accepted

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🚀 Rules for Writing Symbols of Units

Strict guidelines must be followed when documenting SI units:

1. No Dots: Do not use dots or full stops after a symbol (e.g., write cm, not c.m.). A dot is only allowed if the symbol is at the end of a sentence.
2. No Pluralization: Never add ‘s’ or ‘es’ to a symbol (e.g., $5\text{ g}$, not $5\text{ gs}$). If the full word is written out and the value is $>1$, then pluralize (e.g., $10\text{ metres}$).
3. Capitalization: Symbols named after scientists start with a capital letter (e.g., N for Newton, A for Ampere, K for Kelvin, Pa for Pascal). Other symbols are lowercase (e.g., m, kg, s).
   • Note: If writing the full name of the unit, always use lowercase even for scientists (e.g., newton, kelvin).
4. Font Style: Symbols must be printed in Roman (upright) font, never italicized, even if the surrounding text is italicized. Physical quantities (like mass $m$) can be italicized, but units (m for metre) must be upright.
5. Multiplication: Use a space or a dot to indicate multiplication of units (e.g., N m or N·m).
6. Division: Use a solidus (slash /) or negative exponents for division (e.g., $J/(m^2\cdot s)$ or $J\cdot m^{-2}\cdot s^{-1}$).
7. No Degrees for Kelvin: Never use the degree symbol with Kelvin (write 300 K, not 300 °K). Degrees are only used for Celsius (°C) and Fahrenheit (°F).
8. Avoid Abbreviations: Do not use unofficial abbreviations like sec (use s), cc (use cm³), or mps (use m/s).

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Key Derived Quantities and their SI Units

• Displacement / Distance $\to m$
• Time $\to s$
• Velocity / Speed $\to m{s}^{-1}$
  • $\left(\frac{\text{disp}}{\text{time}},\frac{\text{dist}}{\text{time}}\right)$
• Acceleration $\to m{s}^{-2}$
  • $\left(\frac{\text{velocity}}{\text{time}}\right)$
• Force $\to kgm{s}^{-2}$ (Newton)
  • $(m\cdot a)$
• Momentum $\to kgm{s}^{-1}$
  • $(P=mv)$
• Work $\to kg{m}^2{s}^{-2}$ (Joules)
  • $(F\cdot \text{disp})$
• Power $\to kg{m}^2{s}^{-3}$ (Watts)
  • $\left(\frac{\text{work}}{\text{time}}\right)$

Relations between MKS, CGS, FPS

• $1m=100cm=3.28ft$
• $1kg=1000g=2.2lb$

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Table of Prefixes for SI Units

Often a calculator uses "E" to draft "exponent of 10". Ex: $3.56E9=3.56\times 10^9$

Factor	Prefix	Symbol	Factor	Prefix	Symbol
$10^{24}$	yotta	$Y$	$10^{-1}$	deci	$d$
$10^{21}$	zetta	$Z$	$10^{-2}$	centi	$c$
$10^{18}$	exa	$E$	$10^{-3}$	milli	$m$
$10^{15}$	peta	$P$	$10^{-6}$	micro	$\mu$
$10^{12}$	tera	$T$	$10^{-9}$	nano	$n$
$10^9$	giga	$G$	$10^{-12}$	pico	$p$
$10^6$	mega	$M$	$10^{-15}$	femto	$f$
$10^3$	kilo	$k$	$10^{-18}$	atto	$a$
$10^2$	hecto	$h$	$10^{-21}$	zepto	$z$
$10^1$	deka	$da$	$10^{-24}$	yocto	$y$

$10^{-10}m=1\text{Angstrom}$ (A)

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The Two Supplementary Units of SI

(Note: $1^{\circ }=60^{\prime }=3600^{\prime \prime }$)

1. Plane Angle

It represents the amount of rotation or turning required to bring one line in coincidence with the other.

• Represented by ‘$\theta$’ and measured in Radians.
• $1rad$ is equal to the angle formed at the centre when the Arc length is equal to the Radius of the circle.
• Formula: $\theta =\frac{l}{R}$

Key Derivations:

• $\because$ when $l=R$, then $\theta =1rad$
• $\because$ when $l=2\pi R$, then $\theta =2\pi rad$
• $2\pi rad=360^{\circ }\text{or}{\pi }^c=180^{\circ }$
• $1^{\circ }=\frac{\pi }{180}rad;1rad=\frac{180^{\circ }}{\pi }$

Application: Parallax Method

*Given: A distance $R=72000km$ and a measured angle of $1^{\prime }$ to find the approximate diameter of the planet ($d$)

$\because \text{Arc length}=\text{Radius}\cdot \theta \text{ (in radians)}$
$\therefore l=R\times \left(\frac{\pi }{180\times 60}\right)=\frac{R\pi }{10800}\text{ unit}$

Approx diameter of the planet ($d$):
$=72000\times \left(\frac{\pi }{180\times 60}\right)km$
$=\frac{20\times 3.14}{3}km\text{ (apox)}$
$=\frac{62.8}{3}km\text{ (apox)}$
$=20.93km\text{ (apox)}$

2. Solid Angle

It measures how large an object appears to an observer (a specific point aka Apex). It is measured in Steradians and represented by $\Omega$.

• Steradians (sr): The 3D angle at the apex of a cone is calculated using the ratio of area and radius squared.
• Formula: $\Omega =\frac{A}{R^2}$

Area is a part of the sphere’s curved surface.

• For a Hemisphere: $A=2\pi R^2⟹\Omega =\frac{2\pi R^2}{R^2}=2\pi sr$
• For a Sphere: $A=4\pi R^2⟹\Omega =\frac{4\pi R^2}{R^2}=4\pi sr\approx 12.57sr$

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Conversion of Units:

The physical quantity (PQ) remains the same regardless of the unit used.

• General Formula: $n_1{u}_1=n_2{u}_2$.
  • $n$ = numerical value of the physical quantity.
  • $u$ = unit of the physical quantity.
• Example: Length of a pen
  • $1\text{ cm}=10\text{ mm}$
  • $100\text{ cm}=1\text{ m}$

Chain-Link Conversion Method

In this method, we multiply the original measurement by a conversion factor (a ratio of units that is equal to unity)

Few Important Conversion Factors

Length

• $1m=100cm$
• $1in=2.540cm$
• $1ft=30.48cm$
• $1yd=3ft$
• $1mi=1.609km$
• $1nmi=1852m$
• $1nmi=1.151mi$
• $1ly=9.461\times 10^{12}km$
• $1pc=3.084\times 10^{13}km$

Area & Volume

• $1m^2=10^{4}c{m}^2$
• $1acre=43560f{t}^2$
• $1ha=10000m^2$
• $1m^3=1000L$
• $1L=1000c{m}^3$
• $1gal=231i{n}^3$
• $1gal=4qt$
• $1UKgal=1.201USgal$

Mass

• $1kg=1000g$
• $1t=1000kg$
• $1slug=14.59kg$
• $1lb=453.6g$
• $1ton=2000lb$
• $1u=1.661\times 10^{-27}kg$

Time & Speed

• $1y=365.25d$
• $1d=86400s$
• $1m/s=3.6km/h$
• $1mi/h=1.609km/h$
• $1knot=1.688ft/s$

Force & Pressure

• $1N=10^{5}dyne$
• $1lb=4.448N$
• $1atm=1.013\times 10^{5}Pa$
• $1atm=14.70lb/i{n}^2$
• $1atm=76cmHg$

Energy & Power

• $1J=10^{7}erg$
• $1cal=4.1868J$
• $1eV=1.602\times 10^{-19}J$
• $1Btu=1055J$
• $1kWh=3.600\times 10^{6}J$
• $1hp=745.7W$
• $1kW=1000W$

• Conversion Examples

Conversion Task	Calculation Process	Result
$36\text{ km/hr}$ to $\text{m/s}$	$36\times \frac{1000\text{ m}}{3600\text{ s}}$	$10\text{ m/s}$
$50{\text{ mm}}^2$ to ${\text{m}}^2$	$50\times (10^{-3}\text{ m}{)}^2=50\times 10^{-6}{\text{ m}}^2$	$5\times 10^{-5}{\text{ m}}^2$
$72,000\text{ kg}\cdot \text{m/hr}$ to $\text{gm}\cdot \text{cm/s}$	$72,000\times \frac{1000\text{ gm}\times 100\text{ cm}}{3600\text{ s}}$	$2\times 10^6\text{ gm}\cdot \text{cm/s}$
$1{\text{ m}}^3$ to ${\text{cm}}^3$	$1\times (100\text{ cm}{)}^3$	$10^6{\text{ cm}}^3$
$500{\text{ kg/m}}^3$ to CGS	$500\times \frac{1000\text{ gm}}{(100\text{ cm}{)}^3}$	$0.5{\text{ gm/cm}}^3$
$10{\text{ m/s}}^2$ to FPS	$10\times 3.2{\text{ ft/s}}^2$	$32{\text{ ft/s}}^2$

• Standard SI to CGS Conversions

  • Force (Newton to Dynes): $1\text{ N}=1\text{ kg}\cdot {\text{m/s}}^2=10^5\text{ gm}\cdot {\text{cm/s}}^2=10^5\text{ dynes}$.
  • Work (Joules to Ergs): $1\text{ J}=1\text{ kg}\cdot {\text{m}}^2/{\text{s}}^2=10^7\text{ gm}\cdot {\text{cm}}^2/{\text{s}}^2=10^7\text{ ergs}$.

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Important Astronomical Units

• Light Year (ly): The distance travelled by light in one year.
  • $1\text{ ly}=9.461\times 10^{15}\text{ m}$.
• Astronomical Unit (AU): The average distance between the Sun and Earth.
  • $1\text{ AU}\approx 1.5\times 10^{11}\text{ m}$.
  • $\approx 150\text{million}km$
• Parsec (pc): A unit of length, not time or angle.
  • It is the radius of an arc whose arc length is $1\text{ AU}$ and subtends an angle of $1^{\prime \prime }$ at athe center.
  • $1\text{ pc}=3.084\times 10^{16}\text{ m}$.

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Dimensions

Powers to which fundamental PQs are raised to represent any PQ.

Important Examples

• Velocity / Speed $\to m{s}^{-1}$
  • $\left(\frac{\text{disp}}{\text{time}},\frac{\text{dist}}{\text{time}}\right)=[L{T}^{-1}]$

• Acceleration $\to m{s}^{-2}$
  • $\left(\frac{\text{velocity}}{\text{time}}\right)=[L{T}^{-2}]$
• Force $\to kgm{s}^{-2}$ (Newton)
  • $(m\cdot a)=[ML{T}^{-2}]$
• Pressure $\to N{m}^2$
  • $(\frac{\text{force}}{area})=[M{L}^{-1}{T}^{-2}]$
• Momentum $\to kgm{s}^{-1}$
  • $(P=mv)=[ML{T}^{-1}]$
• Work $\to kg{m}^2{s}^{-2}$ (Joules)
  • $(F\cdot \text{disp})=[M{L}^2{T}^{-2}]$
• Power $\to kg{m}^2{s}^{-3}$ (Watts)
  • $\left(\frac{\text{work}}{\text{time}}\right)=[M{L}^2{T}^{-3}]$
• Intensity $\to \text{kg}\cdot {\text{s}}^{-3}$ ($\text{W}/{\text{m}}^2$)
  • $\left(\frac{\text{Power}}{\text{Area}}\right)=\frac{[M{L}^2{T}^{-3}]}{[L^2]}=[M{T}^{-3}]$
• Density $\to kg{m}^{-3}$
  • $(\frac{mass}{volume})=[M{L}^{-3}]$
  • ! Relative Density is dimensionless

• Momentum / Impulse $\to Newton\cdot s$
  • $(Force\cdot Time)=[ML{T}^{-1}]$

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Few Tricky Dimensions

V1: Slightly Tricky

• Charge $\to \text{culomb(C)}$
  • $\left(I=\frac{q}{t}\right)or,q=It\therefore [AT]$
• Resistance $\to \text{ohm}$
  • $\left(P=I^{2}Ror,R=\frac{P}{I^2}\right)\therefore \frac{[M{L}^2{T}^{-3}]}{[A^2]}=[M{L}^2{T}^{-3}{A}^{-2}]$
• Voltage/Potential $\to volt(V)$
  • $(P=VI)or,V=\frac{P}{I}\therefore \frac{[M{L}^2{T}^{-3}]}{[A]}=[M{L}^2{T}^{-3}{A}^{-1}]$
• Electric Field $\to V{m}^{-1}/N{C}^{-1}$
  • $\left(F=qEor,E=\frac{F}{q}\right)\therefore \frac{[ML{T}^{-2}]}{[AT]}=[ML{T}^{-3}{A}^{-1}]$
• Magnetic Filed $\to \text{Tesla}(T)$
  the "$c$" below is the speed of light
  • $\left(E=Bcor,B=\frac{E}{c}\right)\therefore \frac{[ML{T}^{-3}{A}^{-1]}}{[L{T}^{-1}]}=[M{T}^{-2}{A}^{-1}]$

V2: Similar to Time $[T]$

• RC (Resistance x Capacitance) $\to [T]$
• L/R (Inductance / Resistance) $\to [T]$
• Frequency (1 / Time) $\to [T^{-1}]$

V3: Gradients

Tip

Whenever you see “gradient” attached to any PQ just divide the PQ’s dimension by $[L]$ aka Length

• Velocity gradient $=\frac{[L{T}^{-1}]}{[L]}=[T^{-1}]$
• Temperature gradient $=\frac{[K]}{[L]}=[K{L}^{-1}]$
• Pressure gradient $=\frac{[M{L}^{-1}{T}^{-2}]}{[L]}=[M{L}^{-2}{T}^{-2}]$
• Potential gradient $=\frac{[M{L}^2{T}^{-3}{A}^{-1}]}{[L]}=[ML{T}^{-3}{A}^{-1}]$
• Force gradient $=\frac{[ML{T}^{-2}]}{[L]}=[M{T}^{-2}]$

V4: Mix of Tricky

• Momentum / Impulse (Mass x Velocity) $\to [M]\times [L{T}^{-1}]=[ML{T}^{-1}]$
• Angular Momentum (Distance x Momentum) $\to [L]\times [ML{T}^{-1}]=[M{L}^2{T}^{-1}]$
• Moment of Inertia (Mass x Radius²) $\to [M{L}^2]$
• Energy Density (Energy / Volume) $\to \frac{[M{L}^2{T}^{-2}]}{[L^3]}=[M{L}^{-1}{T}^{-2}]$
  • & Whenever you see “density” attached to any PQ just divide the PQ’s dimension by $[L^3]$ aka volume

V5: Important Constants

• Gravitational Constant ($G$) $\to {\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$
  Derived from Newton’s Law of Gravitation: $F=\frac{G{m}_1{m}_2}{r^2}$

  • $G=\frac{F{r}^2}{m_1{m}_2}\therefore \frac{[ML{T}^{-2}][L^2]}{[M^2]}=[M^{-1}{L}^3{T}^{-2}]$

• Boltzmann Constant ($k_B$) $\to \text{J/K}$
  Derived from the relationship: $E=k_B\times \text{Temperature}$

  • $k_B=\frac{\text{Energy}}{\text{Temperature}}\therefore \frac{[M{L}^2{T}^{-2}]}{[K]}=[M{L}^2{T}^{-2}{K}^{-1}]$

• Planck’s Constant ($h$) $\to \text{J}\cdot \text{s}$
  Derived from the relationship: $E=hf$ (where $f$ is frequency)

  • $h=\frac{E}{f}\therefore \frac{[M{L}^2{T}^{-2}]}{[T^{-1}]}=[M{L}^2{T}^{-1}]$

• Coefficient of Viscosity ($\eta$) $\to \text{Pa}\cdot \text{s}$
  Derived from the Viscous Force formula (v=velocity): $F=\eta rv$

  • $\eta =\frac{F}{rv}\therefore \frac{[ML{T}^{-2}]}{[L][L{T}^{-1}]}=[M{L}^{-1}{T}^{-1}]$

V6. Advance Dimensions

• Permittivity of Free Space $\to {ϵ}_0/ϵ$
  • $\left(F=\frac{1}{4\pi {ϵ}_0}\frac{q^2}{r^2}\text{ or }{ϵ}_0=\frac{q^2}{F{r}^2}\right)\therefore \frac{[AT{]}^2}{[ML{T}^{-2}][L^2]}=[M^{-1}{L}^{-3}{T}^4{A}^2]$
• Permeability of Free Space $\to {\mu }_0/\mu$
  • $\left(c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}\text{ or }{\mu }_0=\frac{1}{{ϵ}_0{c}^2}\right)\therefore \frac{1}{[M^{-1}{L}^{-3}{T}^4{A}^2][L^2{T}^{-2}]}=[ML{T}^{-2}{A}^{-2}]$
• Speed Relationship $\to c$
  • $\left(\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}\right)\to [L{T}^{-1}]$
  • $\left(\frac{1}{{\mu }_0{ϵ}_0}\right)\to (\text{Speed}{)}^2=[L^2{T}^{-2}]$

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List of PQs with same Dimensions

1.PQs with $[L]$ (Distance)

• Distance
• Displacement
• Radius of Gyration
• Light Year
• Parsec
• Astronomical Unit (AU)
• ! Angular Displacement($\theta$) $=\frac{\text{arc length}}{\text{radius}}=[\frac{L}{L}]=[M^0{L}^0{T}^0]$

2.PQs with $[L{T}^{-1}]$ (Speed, Velocity)

• Speed
• Velocity
• Average Speed
• Average Velocity
• Terminal Velocity
• Critical Velocity
• Velocity of Light
• Escape Velocity
• Orbital Velocity
• Relative Velocity
• Instantenuous Velocity
• ! Angular Velocity $=\frac{\text{angular displacement}}{time}=[T^{-1}]$

2.PQs with $[L{T}^{-2}]$ (Accleration)

• Acceleration
• Average Accleration
• Instantenous Acceleration
• Acceleration due to Gravity
• Intensity of Gravitational Field
• Centrepetal Acceleration
• Centrfugal Acceleration
• ! Angular Acceleration $=\frac{\text{angular velocity}}{\text{time}}=\frac{T^{-1}}{T}=T^{-2}$

3.PQs with $[ML{T}^{-2}]$ (Force)

• Force
• Centrepetal Force
• Centrefugal Force
• Friction
• Gravitational Force
• Thrust
• Vuicous Force
• Spring Force
• Magnetic Force
• Non-conservative Force
• Radiation Force
• Tension
• Normal Reaction
• Weight
• Restoring Force
• Electrostatic Force
• Lorentz Force
• Buoyant Force
• ! Surface Tension and Spring Constant $=\frac{\text{force}}{\text{length}}=[M{T}^{-2}]$

4.PQs with $[M{L}^2{T}^{-2}]$ (Work, Energy)

• Work
• Energy
• Kinetic Energy
• Potential Energy
• Heat Energy
• Thermal Energy
• Vibrational Energy
• Moment of Force
• Torque or Couple
• Strain Energy

5.PQs with $[M{L}^{-1}{T}^{-2}]$ (Pressure, Modulus)

• Pressure
• Stress
• Young Modulus
• Bulk Modulus
• Shear Modulus
• Modulus of Rigidity
• Energy Density

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Principle of Homogeneity

The Principle of Homogeneity states that a physical equation is only dimensionally correct if all the terms on both sides of the equation have the same dimensions.

Core Rules

• Addition/Subtraction: You can only add or subtract quantities with the same dimensions regardless of its unit (ofc you gotta convert the units).
  • $L+L=L$
  • $L-L=L$
• LHS = RHS: The final dimensions of the Left-Hand Side must match the Right-Hand Side.
  • $L=L$
• Transcendental Functions: Arguments of logs, exponents, and trigonometric functions must be dimensionless.
• ! Always look at addition , subtraction and equal signs for finding dimension problems and fractional dimensions are plausible

🚀 Application Rules for Dimensionless Quantities

In any physical equation, pure mathematical operators act as “dimension-killers.” They only accept and output pure numbers ($M^0{L}^0{T}^0$).

• The Power Rule (Exponents)
  The “top” must be empty.
  Any value sitting in the exponent ($2^{\square }$) and the whole must be dimensionless.
• $e^a⟹[a]=[M^0{L}^0{T}^0]\text{ and }[e^a]=[M^0{L}^0{T}^0]$
• The Trig Shield (Arguments)
  Angles have units (radians), but NO dimensions.
  The value of trignometric ratios are dimensionless and pure number
  In $\sin (\theta )$, the $\theta$ is a ratio of arc-length to radius ($L/L$).
• $\sin \theta \therefore [\theta ]=[M^0{L}^0{T}^0]\text{ and }[\sin \theta ]=[M^0{L}^0{T}^0]$
• The Log Cage
  You can’t take the log of a kilogram.
  Both the base, input and output of a logarithm and the must be pure numbers.
• ${\log }_z(x)\to [x]=[M^0{L}^0{T}^0]\text{ and }[z]=[M^0{L}^0{T}^0]\text{ and }[{\log }_{z}X]=[M^0{L}^0{T}^0]$
• The Constant Crowd
  Pure numbers are just placeholders.
  Constants like $e$, $\pi$, and raw numbers ($5,1000,2.7$) never carry mass, length, or time.
• $[2.7]=[e]=[M^0{L}^0{T}^0]$

Application Example

Given: $F=a+\frac{bt}{c-v}$, find the dimension of a, b, c
$\because L\pm L=L$
$\therefore \text{The dimension of c = The dimension of v ; The dimension of a = The dimension of F }$

$$\therefore \text{The dimension of c}=[L{T}^{-1}]$$ $$\text{The dimension of a}=[ML{T}^{-2}]$$ $$\therefore \text{The dimension of b}=\frac{[ML{T}^{-2}]\times [L{T}^{-1}]}{[L]}=[M{L}^2{T}^{-4}]$$

⚠️ Limitations

• Constants: It cannot verify dimensionless constants (e.g., $2\pi$, $1/2$).
• Quantity Confusion: It cannot distinguish between different quantities with the same units (e.g., Work and Torque both use $[M{L}^2{T}^{-2}]$).

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Application of Dimensions

1. Correctness of a Formula

Whether a formula is dimensionally correct or wrong can be decided by comparing dimensions of LHS and RHS

• Dimensionally Correct: $D_{LHS}=D_{RHS}$
  • ! Dimensional correctness does not guarantee physical accuracy; the formula remains a hypothesis until validated by experimental data.
• Incorrect: $D_{LHS}\ne D_{RHS}$

2. Derivation of a Formula

We can derive a formula using dimensional analysis by assuming a power-law relationship between physical quantities.

Limitation

• We cannot determine the value of the dimensionless constant ($C$) in the formula without experimental data.
• We can not derive formula with terms like $\sin \theta ;\cos \theta ,\tan \theta ;\log ;e$
• We can not derive a formula if the constant is dimensional. (Since dimensional analysis is based on the assumption a dimensionless constant)

Example

Q1. The period of a pendulum ($T$) depends on its length ($l$) and acceleration due to gravity ($g$). Derive the formula.

Answer:

$$\begin{array}{rl}T& \propto l^a{g}^b\\ T& =C{l}^a{g}^b\\ & \\ \text{Writing dimensions on both sides:}& \\ M^0{L}^0{T}^1& =[L{]}^a[L{T}^{-2}{]}^b\\ M^0{L}^0{T}^1& =L^{a+b}{T}^{-2b}\\ & \\ \text{Comparing powers:}& \\ -2b=1& ⟹b=-\frac{1}{2}\\ a+b=0& ⟹a=\frac{1}{2}\\ & \\ \text{Substituting in (1):}& \\ T& =C{l}^{1/2}{g}^{-1/2}\\ \therefore \mathbf{T}& =\mathbf{C}\sqrt{\frac{\mathbf{l}}{\mathbf{g}}}\end{array}\text{\hspace{1em}(1)}$$

Q2. If $p$ is the pressure of a gas and $\rho$ is its density, find the dimension of velocity ($V$).

Answer:

$$\begin{array}{rl}V& \propto p^a{\rho }^b\\ V& =C{p}^a{\rho }^b\\ & \\ \text{Dimensions: }[V]=L{T}^{-1},& [p]=M{L}^{-1}{T}^{-2},[\rho ]=M{L}^{-3}\\ & \\ M^0{L}^1{T}^{-1}& =[M{L}^{-1}{T}^{-2}{]}^a[M{L}^{-3}{]}^b\\ M^0{L}^1{T}^{-1}& =M^{a+b}{L}^{-a-3b}{T}^{-2a}\\ & \\ \text{Comparing powers:}& \\ -2a=-1& ⟹a=\frac{1}{2}\\ a+b=0& ⟹b=-\frac{1}{2}\\ & \\ \text{Substituting in (1):}& \\ V& =C{p}^{1/2}{\rho }^{-1/2}\\ \therefore \mathbf{V}& =\mathbf{C}\sqrt{\frac{\mathbf{p}}{\rho }}\end{array}\text{\hspace{1em}(1)}$$

Q3. If velocity ($V$), time ($T$), and force ($F$) are chosen as base quantities, find the dimension of mass ($M$).

Answer:

$$\begin{array}{rl}M& \propto V^a{T}^b{F}^c\\ M^1{L}^0{T}^0& =[L{T}^{-1}{]}^a[T{]}^b[ML{T}^{-2}{]}^c\\ M^1{L}^0{T}^0& =M^c{L}^{a+c}{T}^{-a+b-2c}\\ & \\ \text{Comparing powers:}& \\ c& =1\\ a+c=0& ⟹a=-1\\ -a+b-2c=0& ⟹-(-1)+b-2(1)=0⟹b=1\\ & \\ \therefore [\mathbf{M}]& =[{\mathbf{F}}^1{\mathbf{T}}^1{\mathbf{V}}^{-1}]\end{array}$$

3. Conversion of Units

The numerical value of a physical quantity is inversely proportional to its unit ($n\propto \frac{1}{u}$). Thus, the product of the numerical value and the unit remains constant:
$n_1{u}_1=n_2{u}_2$
Where $u=[M^a{L}^b{T}^c]$. This allows us to convert a magnitude from one system of units to another.

Example

Q1. Convert 1 Newton (SI unit of force) to the CGS unit (dynes).

Answer:

$$\begin{array}{rl}\text{Dimensional formula of force: }[F]& =M^1{L}^1{T}^{-2}\\ \therefore a=1,b=1,c& =-2\\ & \\ \text{Using }{n}_1{u}_1& =n_2{u}_2\\ 1\times [M_1^1{L}_1^1{T}_1^{-2}]& =n_2\times [M_2^1{L}_2^1{T}_2^{-2}]\\ or,n_2& =1{\left[\frac{M_1}{M_2}\right]}^1{\left[\frac{L_1}{L_2}\right]}^1{\left[\frac{T_1}{T_2}\right]}^{-2}\end{array}$$

System Comparison:

Physical Quantity	SI System ($u_1$)	CGS System ($u_2$)
Mass ($M$)	$M_1=1\text{ kg}=10^3\text{ g}$	$M_2=1\text{ g}$
Length ($L$)	$L_1=1\text{ m}=10^2\text{ cm}$	$L_2=1\text{ cm}$
Time ($T$)	$T_1=1\text{ s}$	$T_2=1\text{ s}$

$$\begin{array}{rl}⟹n_2& =1{\left[\frac{10^3\text{ g}}{1\text{ g}}\right]}^1{\left[\frac{10^2\text{ cm}}{1\text{ cm}}\right]}^1{\left[\frac{1\text{ s}}{1\text{ s}}\right]}^{-2}\\ or,n_2& =10^3\times 10^2\times 1\\ or,n_2& =10^5\\ & \\ \therefore 1\text{ Newton}& =10^5\text{ dynes}\end{array}$$

Q2. The density of a material in CGS system is $4{\text{ g/cm}}^3$. Find its value in a new system where the unit of length is $10\text{ cm}$ and the unit of mass is $100\text{ g}$.

Answer:

$$\begin{array}{rl}\text{Dimensional formula of density: }[\rho ]& =M^1{L}^{-3}{T}^0\\ \therefore a=1,b=-3,c& =0\\ & \\ \text{Using }{n}_1{u}_1& =n_2{u}_2\\ n_2& =n_1{\left[\frac{M_1}{M_2}\right]}^a{\left[\frac{L_1}{L_2}\right]}^b\end{array}$$

Comparing the two systems:

Physical Quantity	CGS System ($u_1$)	New System ($u_2$)
Numerical Value	$n_1=4$	$n_2=?$
Mass ($M$)	$M_1=1\text{ g}$	$M_2=100\text{ g}$
Length ($L$)	$L_1=1\text{ cm}$	$L_2=10\text{ cm}$

$$\begin{array}{rl}⟹n_2& =4\times {\left[\frac{1\text{ g}}{100\text{ g}}\right]}^1\times {\left[\frac{1\text{ cm}}{10\text{ cm}}\right]}^{-3}\\ or,n_2& =4\times \left[\frac{1}{100}\right]\times [10{]}^3\\ or,n_2& =4\times \frac{1}{100}\times 1000\\ or,n_2& =40\\ & \\ \therefore \text{Value in new system }& =40\text{ units}\end{array}$$

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Error Analysis

When we perform experiments or take measurements, mistakes and uncertainties are inevitable. No instrument is perfectly precise, and no human is perfectly accurate.

Accuracy vs. Precision

• Accuracy: How close the measured value is to the True Value.
• Precision of Reading: How close the readings are to each other.
• Precision of Instrument: The resolution or the limit to which the instrument can measure (how close the measured values are to each other). Equipment with lesser least count is more precise. A highly precise instrument (e.g., measuring to $0.001cm$) might still be inaccurate if it is faulty!

Classification of Errors

Error Type	Nature	Cause	Solution
Systematic Errors	Predictable / Unidirectional	Known (Faulty calibration, environmental changes like temperature)	Apply corrections / Calibrate instruments
Random Errors	Unpredictable	Unknown (Small fluctuations in the environment or instrument)	Take multiple readings and find the average
Gross Errors	Human Error	Carelessness (e.g., writing $25$ instead of $52$)	Be attentive and careful while observing

Handling Random Errors

To reduce random errors by a factor of $n$, you must increase the number of observations by $n$ times.
Example: If the random error for $10$ readings is $x$, taking $50$ readings will reduce the error to $x/5$.

Calculation of Errors

Let the readings of an experiment be $a_1,a_2,a_3,\dots ,a_n$.

1. Mean / True Value ($\bar{a}$)
   The arithmetic mean is taken as the most accurate or “true” value.
   $\bar{a}=\frac{a_1+a_2+\cdots +a_n}{n}$

2. Absolute Error ($\Delta a$)
   The difference between the true value and the individual measured value. (Note: This can be positive or negative).
   $\Delta a_1=a_1-\bar{a};\Delta a_2=a_2-\bar{a}$

3. Mean Absolute Error ($\Delta \bar{a}$)
   The arithmetic mean of the magnitudes of the absolute errors. This represents the overall error limit.
   $\Delta \bar{a}=\frac{|\Delta a_1|+|\Delta a_2|+\cdots +|\Delta a_n|}{n}$

   • Final Reporting: $a=\bar{a}\pm \Delta \bar{a}$

4. Fractional & Percentage Error

   • Fractional Error: $\frac{\Delta \bar{a}}{\bar{a}}$
   • Percentage Error: $\frac{\Delta \bar{a}}{\bar{a}}\times 100\%$

Least Count as Error

If an error is not explicitly given in a problem, the Least Count (LC) of the measuring instrument is assumed to be the maximum possible absolute error. Cause LC of an instrument is always uncertain therefore the maximum possible error of the instrument.

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Propagation of Errors

When mathematical operations are performed on measured quantities, their individual errors propagate into the final result.

1. Addition and Subtraction

Whether adding or subtracting quantities, their absolute errors are always added.

• Given: $A=\bar{a}\pm \Delta a$ and $B=\bar{b}\pm \Delta b$
• Addition: $Z=A+B⟹Z=(\bar{a}+\bar{b})\pm (\Delta a+\Delta b)$
• Subtraction: $Z=A-B⟹Z=(\bar{a}-\bar{b})\pm (\Delta a+\Delta b)$

2. Multiplication and Division

For multiplication and division, the fractional errors are always added.

• Formula: $Z=A\cdot B\text{or}Z=\frac{A}{B}$
• Error Relation: $\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$

3. General Power Rule

If a physical quantity $X$ depends on observables $a,b,c$ raised to specific powers:
$X=\frac{a^p{b}^q}{c^r}$
Bring the powers down as multipliers and add the fractional errors:
$\frac{\Delta X}{X}=p\frac{\Delta a}{a}+q\frac{\Delta b}{b}+r\frac{\Delta c}{c}$

Important Analytical Rules

• Constants are Error-Free: Raw numbers ($2,\pi ,1/2$) do not contribute to error. For example, in $KE=\frac{1}{2}m{v}^2$, the $\frac{1}{2}$ is ignored: $\frac{\Delta K}{K}=\frac{\Delta m}{m}+2\frac{\Delta v}{v}$.
• ! The 10% Threshold: The power rule (derived via differentiation) is an approximation and is strictly valid only when percentage errors are less than 10%.
  • If an error is $>10\%$, you must calculate the exact new value using basic algebra. Example: If pendulum length increases by 44%, use $L_{new}=1.44L$ to find $T_{new}$, not the differentiation method.

Example

Q. A physical quantity $P$ is given as $P=\frac{a^2{b}^3}{c\sqrt{d}}$. The percentage error in measurement of $a,b,c$ and $d$ are 1%, 2%, 3% and 4% respectively. The percentage error in quantity $P$ will be: (JEE Main 2023)

• $\frac{\Delta P}{P}\%=2\frac{\Delta a}{a}\%+3\frac{\Delta b}{b}\%+\frac{\Delta c}{c}\%+\frac{1}{2}\frac{\Delta d}{d}\%$
• $=(2\times 1)+(3\times 2)+3+(\frac{1}{2}\times 4)=2+6+3+2=13\%$

Q. A body of mass $(5\pm 0.5)\text{ kg}$ is moving with a velocity of $(20\pm 0.4)\text{ m/s}$. Its kinetic energy will be: (JEE Main 2023)

• $K.E.=\frac{1}{2}m{v}^2=\frac{1}{2}\times 5\times (20{)}^2=1000\text{ J}$
• $\frac{\Delta K}{K}=\frac{\Delta m}{m}+2\frac{\Delta v}{v}=\frac{0.5}{5}+2\times \frac{0.4}{20}=0.1+0.04=0.14$
• $\Delta K=1000\times 0.14=140\text{ J}$
• $K.E.=(1000\pm 140)\text{ J}$

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Detailed Rules for Significant Figures

(Expanding on the core definitions)
To determine the number of significant digits in a measurement:

1. Non-Zero Digits: All non-zero digits are always significant. (e.g., $142\to 3\text{ SF}$)
2. Trapped Zeros: Zeros trapped between non-zero digits are significant. (e.g., $2.005\to 4\text{ SF}$)
3. Leading Zeros: Zeros to the left of the first non-zero digit are never significant; they only indicate the position of the decimal point. (e.g., $0.0025\to 2\text{ SF}$)
4. Trailing Zeros: Zeros at the end of a number are significant only if the number contains a decimal point.
   • $14200\to 3\text{ SF}$ (No decimal, trailing zeros are insignificant)
   • $14200.0\to 6\text{ SF}$
   • $0.00213000\to 6\text{ SF}$

Scientific Notation & Conversions

Changing the unit of measurement (e.g., $cm\to m$) must not change the number of significant figures. To avoid ambiguity, represent values in Scientific Notation ($a\times 10^b$). The order of magnitude ($10^b$) has no effect on significant figures.

Detailed Rounding Off Rules

When discarding digits to reach a specific number of significant figures, observe the digit immediately following the last desired significant figure (the “drop digit”):

• Less than 5: Leave the preceding digit unchanged. (e.g., $3.42\to 3.4$)
• Greater than 5: Increase the preceding digit by $1$. (e.g., $3.47\to 3.5$)
• Exactly 5:
  • If a non-zero digit follows the 5, increase the preceding digit by 1. (e.g., $3.451\to 3.5$)
  • If nothing (or zero) follows the 5, apply the Even/Odd Rule:
    • If the preceding digit is Even, leave it alone. (e.g., $3.450\to 3.4$)
    • If the preceding digit is Odd, increase it by 1. (e.g., $3.350\to 3.4$)

Arithmetic Operations with Significant Figures

• Addition / Subtraction (Decimal Rule): The final answer must retain the same number of decimal places as the measurement with the least number of decimal places.
• Multiplication / Division (Sig Fig Rule): The final answer must retain the same number of significant figures as the measurement with the least significant figures.

Infinite Significant Figures (Exact Numbers)

Pure numbers or exact counts possess infinite significant figures because they are perfectly precise (they have no associated measuring instrument, and thus no least count or error).

• Counting Numbers: “20 bottles”, “5 cars”, “7 days in a week”.
• Defined/Theoretical Constants: Speed of light in vacuum ($c=299792458m/s$), Avogadro’s number ($N_A$), $\pi$, or the raw integer $2$ in the formula $d=2r$.
  • ! Note: Measured constants derived from physical experiments (like the Gravitational Constant $G=6.67\times 10^{-11}N{m}^2/k{g}^2$) do have finite significant figures (in this case, 3).

Memorization Hacks for Arithmetic Operations

• SAD: Subtraction & Addition $\to$ Look at Decimals (Keep the minimum decimal places among the operands).
• MSD: Multiplication & Division $\to$ Look at Significant Digits (Keep the minimum significant digits among the operands).

The Golden Rule of Complex Calculations

When a problem involves a mix of addition, subtraction, multiplication, and division, never round off intermediate steps! Keep all intermediate digits during the calculation to avoid compounding errors, and apply the rounding rules only to the final answer.

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Order of Magnitude

The order of magnitude gives a quick, intuitive idea of a physical quantity’s scale. To determine it, you must express the number strictly in Scientific Notation: $a\times 10^b$.

• Constraint: $1\le a<10$
• Rules:
  1. If $a<5$, the order of magnitude is $\mathbf{b}$.
  2. If $a\ge 5$, the order of magnitude is $\mathbf{b}+1$.

Examples:

• $75000=7.5\times 10^4⟹$ Since $7.5\ge 5$, Order $=4+1=5$
• $0.0045=4.5\times 10^{-3}⟹$ Since $4.5<5$, Order $=-3$
• $0.00092=9.2\times 10^{-4}⟹$ Since $9.2\ge 5$, Order $=-4+1=-3$

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Advanced Dimensional Analysis (PYQ Strategies)

In competitive exams (like JEE Advanced), you will frequently encounter “new system” problems where standard base quantities (Mass, Length, Time) are arbitrarily redefined in terms of other physical constants (e.g., Planck’s constant $h$, speed of light $c$, Gravitational constant $G$).

Strategy: The Exponent Method

Whenever asked to find the dimensions of a target quantity $Z$ in terms of new base quantities $X,Y,W$:

1. Assume a power-law proportionality: $Z\propto X^a{Y}^b{W}^c$
2. Write the standard dimensional formulas (in $M,L,T,A$, etc.) for all terms on both sides of the equation.
3. Multiply the powers and equate the exponents of corresponding base dimensions ($M$ to $M$, $L$ to $L$) to form a system of linear equations.
4. Solve the linear equations for variables $a,b$, and $c$.

Example

Expressing Mass in terms of $h,c,G$

Given: Find the dimension of Mass ($M$) in a system where Planck’s constant ($h$), speed of light ($c$), and Gravitational constant ($G$) are base quantities.

$$\begin{array}{rl}M& \propto h^a{c}^b{G}^c\\ [M^1{L}^0{T}^0]& =[M{L}^2{T}^{-1}{]}^a[L{T}^{-1}{]}^b[M^{-1}{L}^3{T}^{-2}{]}^c\\ [M^1{L}^0{T}^0]& =M^{a-c}{L}^{2a+b+3c}{T}^{-a-b-2c}\end{array}$$

Forming Linear Equations:

1. For $M$: $a-c=1$
2. For $L$: $2a+b+3c=0$
3. For $T$: $-a-b-2c=0$

Solving:
From (3), we get $b=-a-2c$. Substitute this into (2):
$2a+(-a-2c)+3c=0⟹a+c=0⟹a=-c$
Substitute $a=-c$ into equation (1):
$-c-c=1⟹-2c=1⟹c=-\frac{1}{2}$
$\therefore a=\frac{1}{2}$
Now find $b$:
$b=-\left(\frac{1}{2}\right)-2\left(-\frac{1}{2}\right)=-\frac{1}{2}+1=\frac{1}{2}$

Final Relation:
$M\propto h^{1/2}{c}^{1/2}{G}^{-1/2}⟹\mathbf{M}\propto \sqrt{\frac{\mathbf{hc}}{\mathbf{G}}}$

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Vernier Caliper

A Vernier Caliper is a precision instrument used to measure internal and external distances accurately.

Parts of a Vernier Caliper

• Inner Jaw: Used to measure internal diameters (e.g., inside a tube).
• Outer Jaw: Used to measure external diameters and widths (e.g., a sphere or block).
• Main Scale: The fixed, primary measurement scale.
• Vernier Scale: The sliding, secondary scale that provides higher precision.
• Strip / Depth Probe: Used to measure the depth of holes or cylinders.

Least Count (Vernier Constant)

The Least Count (L.C.) is the smallest length that can be accurately measured by the Vernier Caliper. It is the difference between one Main Scale Division (M.S.D.) and one Vernier Scale Division (V.S.D.).

• Let $n$ Vernier Scale Divisions (V.S.D.) coincide exactly with $m$ Main Scale Divisions (M.S.D.), where normally $n\ge m$.
  $n\times \text{V.S.D.}=m\times \text{M.S.D.}$
  $\text{V.S.D.}=\frac{m}{n}\text{M.S.D.}$
• Formula for Least Count:
  $\boxed{\text{L.C.}=\text{M.S.D.}-\text{V.S.D.}=\text{M.S.D.}\left(1-\frac{m}{n}\right)}$

Reading on Vernier Calipers

To take a measurement using a Vernier Caliper, you must combine the readings from both scales.

$\boxed{\text{Total Reading}=\text{M.S.R.}+\text{V.S.R.}}$

• Main Scale Reading (M.S.R.): The reading on the Main Scale just before the zero mark of the Vernier Scale.
• Vernier Scale Reading (V.S.R.): Found by identifying which division on the Vernier Scale perfectly coincides with any mark on the Main Scale.
  $\text{V.S.R.}=\text{L.C.}\times (\text{Coinciding Vernier Division})$

Zero Error of Vernier Calipers

If the zero of the Vernier scale does not coincide with the zero of the Main scale when the jaws are fully closed (touching each other), the instrument has a Zero Error.

Correcting Zero Error

Zero error must always be subtracted (with its proper sign) from the total measured reading.
$\text{Correct Reading}=\text{Total Reading}-(\pm \text{Zero Error})$

Types of Zero Error

Positive Zero Error

When the jaws are closed, if the zero mark of the Vernier scale is to the right of the zero mark of the Main scale, the error is positive.

• Formula: $\text{Positive Zero Error}=+(\text{L.C.}\times \text{Coinciding division of vernier scale})$

Negative Zero Error

When the jaws are closed, if the zero mark of the Vernier scale is to the left of the zero mark of the Main scale, the error is negative.

• Formula: $\text{Negative Zero Error}=-(\text{L.C.}\times \text{Coinciding division of vernier scale})$

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Screw Gauge (Micrometer)

A screw gauge is an instrument used to accurately measure the diameter of a thin wire or the thickness of a sheet of metal.

Parts of a Screw Gauge

• U-frame: Holds the anvil and the spindle.
• Stud/Anvil: The fixed measuring face.
• Spindle: The movable measuring face.
• Main Scale (Pitch Scale): The linear scale engraved on the sleeve/barrel.
• Circular Scale (Head Scale): The rotating scale engraved on the thimble.
• Ratchet: Ensures uniform pressure is applied to the object being measured.

Pitch and Least Count

• Pitch (P): The distance moved by the spindle (on the main scale) due to one complete rotation of the circular scale. It is the distance between two consecutive threads.
  $\text{Pitch}=\frac{\text{Distance moved on main scale}}{\text{Number of complete rotations}}$
• Least Count (L.C.): The smallest value that can be measured by the screw gauge.
  $\boxed{\text{L.C.}=\frac{\text{Pitch}}{\text{Total number of divisions on circular scale}}}$

Reading on a Screw Gauge

To take a measurement using a screw gauge:
$\boxed{\text{Total Reading}=\text{M.S.R.}+\text{C.S.R.}}$

• Main Scale Reading (M.S.R.): The reading on the main scale just before the edge of the circular scale.
• Circular Scale Reading (C.S.R.): Found by multiplying the L.C. with the circular scale division (C.S.D.) that perfectly coincides with the base/reference line of the main scale.
  $\text{C.S.R.}=\text{L.C.}\times (\text{Coinciding Circular Division})$

Zero Error of Screw Gauge

If the zero of the circular scale does not coincide with the reference line of the main scale when the stud and spindle touch each other, there is a zero error.

Correcting Zero Error

Just like Vernier Calipers, the zero error must be subtracted (with its sign) from the total measured reading.
$\text{Correct Reading}=\text{Total Reading}-(\pm \text{Zero Error})$

Positive Zero Error

When the zero of the circular scale is below the reference line. (Error is positive, so it is subtracted from the reading).

Negative Zero Error

When the zero of the circular scale is above the reference line. (Error is negative, so subtracting it effectively adds to the reading).

Example

Q1. A screw gauge has 200 divisions on its circular scale. Its pitch is $1.0\text{ mm}$. What is its least count?
$\text{L.C.}=\frac{1.0\text{ mm}}{200}=0.005\text{ mm}=5\times 10^{-3}\text{ mm}$

Q2. A student measured the diameter of a small steel ball using a screw gauge of least count $0.001\text{ cm}$. The main scale reading is $5\text{ mm}$ and zero of circular scale division coincides with 25 divisions above the reference level. If screw gauge has a zero error of $-0.004\text{ cm}$, the correct diameter of the ball is:
$\text{M.S.R.}=5\text{ mm}=0.5\text{ cm}$
$\text{C.S.R.}=25\times 0.001\text{ cm}=0.025\text{ cm}$
$\text{Total Reading}=0.5+0.025=0.525\text{ cm}$
$\text{Correct Reading}=0.525-(-0.004)=0.529\text{ cm}$

Q3. The pitch of a screw gauge is $1\text{ mm}$ and there are 100 divisions on the circular scale. In measuring the diameter of a sphere, there are 6 divisions on the linear scale and forty divisions of the circular scale coincide with the reference line. Find the diameter of the sphere.

• $\text{Pitch}=1\text{ mm}$
• $\text{L.C.}=\frac{1\text{ mm}}{100}=0.01\text{ mm}$
• $\text{M.S.R.}=6\text{ mm}$
• $\text{C.S.R.}=40\times 0.01=0.4\text{ mm}$
• $\text{Diameter}=6+0.4=6.4\text{ mm}$

No Least Count for both Vernier and Screw

If there is no Least Count given in the qustion use $0.01$ as deafult