Units & Measurements | Class 11 | PhysicsVerse

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Why measurement is the language of physics

The Chai Problem

Picture this. Your dadi makes chai. She says "ek chamach cheeni daalo" — one spoon of sugar. You go to your friend's house. His mother also says "ek chamach cheeni." You take the same one spoon. But the chai tastes completely different at his house.

Why?

Because his mother uses a tablespoon. Your dadi uses a teaspoon. The number is the same — one. But the SPOON is different. So the actual amount of sugar is different.

This is the entire idea of measurement in Physics. When you say "ek" — one — you are saying a number. When you say "chamach" — spoon — you are saying the unit. Together they tell you how much. Either one alone is useless.

This is why all over the world, scientists agreed on one standard set of spoons — one standard set of units. They are called SI units. So when a scientist in Mumbai says "5 metres" and a scientist in Tokyo says "5 metres", they both mean the exact same length. No confusion. No different chai taste.

The Cricket Pitch Argument

Two friends are watching IPL. Rohit says the pitch is 20 metres. Virat says no, it is 22 yards. Who is right?

Both. The pitch length is fixed. It does not change because you measure it differently. But the NUMBER you get depends on what unit you use. 22 yards is the same as 20.12 metres. Same length. Different numbers. Different units.

This gives us the most important rule of measurement:

\[Q = n \times u\]

The quantity Q equals the number n multiplied by the unit u. If you make the unit bigger, the number becomes smaller. If you make the unit smaller, the number becomes bigger. They balance each other so the actual quantity stays the same.

          Think of it like rupees and paise. 100 rupees is the same as 10,000 paise. Same money. Smaller unit, bigger number. The quantity — your money — hasn't changed.
        

Why Physics Needs This

Imagine a doctor giving you medicine. He says "take some syrup." How much? Half a spoon? Full bottle? Without a proper unit, his instruction has no meaning. You could die from too much, or stay sick from too little.

Physics is like medicine for understanding nature. Every formula, every law, every experiment depends on measurement. If measurement is loose, physics breaks. So in this chapter, we learn how to measure properly — what units to use, how to check formulas using dimensions, how many digits we should trust, and how errors creep in.

By the end of this chapter, you will never look at a number the same way again. You will see the unit hiding behind it. You will know whether to trust a measurement. You will be able to derive formulas without even knowing the physics — just using dimensions. This is the foundation. Build it strong, and the rest of Class 11 becomes easy.

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Complete theory for board, JEE & NEET

2.1

Physical Quantity

EXAM DEFINITION

A physical quantity is any quantity that can be measured, directly or indirectly, and in terms of which the laws of physics can be expressed.

Examples: length, mass, time, force, current, temperature.

A physical quantity always has TWO parts — a numerical value and a unit. Saying "the rod is 5" means nothing. Saying "the rod is 5 metres" means everything.

Fundamental Quantities

Quantities that are independent and cannot be expressed in terms of any other physical quantity. There are exactly seven of them.

Derived Quantities

Quantities formed by combining fundamental quantities through multiplication or division. Examples: velocity, force, pressure, energy.

2.2

Unit

EXAM DEFINITION

A unit is the standard amount of a physical quantity chosen to measure that quantity.

Properties a Good Unit Must Have:

Well-defined — no confusion about what it means
Convenient size — not too big, not too small for the quantity
Easily reproducible anywhere in the world
Imperishable — does not change over time
Unaffected by external conditions like temperature, pressure, humidity
Internationally accepted

2.3

Relation Between Number and Unit

If the same quantity Q is measured in two different units \(u_1\) and \(u_2\) giving numerical values \(n_1\) and \(n_2\), then:

\[Q = n_1 u_1 = n_2 u_2\]

This means \(n \propto \dfrac{1}{u}\) — number and unit size are inversely related.

          Example: Length of a pencil = 15 cm = 0.15 m.

          \(n_1 = 15\), \(u_1\) = cm; \(n_2 = 0.15\), \(u_2\) = m.

          The unit became 100 times bigger, so the number became 100 times smaller. The pencil is still the same pencil.

2.4

Systems of Units

System	Length	Mass	Time
CGS	centimetre	gram	second
FPS	foot	pound	second
MKS	metre	kilogram	second
SI	metre	kilogram	second (+ 4 more)

SI = Système International d'Unités — the international system. This is what we use in Class 11, JEE, NEET, and board exams.

The Seven SI Base Units

Quantity	Unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Luminous intensity	candela	cd
Amount of substance	mole	mol

Modern Definitions

Metre: The distance travelled by light in vacuum during a time interval of 1/299,792,458 of a second.
Kilogram: Defined by fixing Planck's constant h = 6.626 × 10⁻³⁴ J·s exactly.
Second: The duration of 9,192,631,770 vibrations of the radiation from a caesium-133 atom transitioning between two hyperfine ground state levels.
Ampere: Defined by fixing the elementary charge e = 1.602 × 10⁻¹⁹ C exactly.
Kelvin: Defined by fixing the Boltzmann constant k = 1.380 × 10⁻²³ J/K exactly.
Candela: The luminous intensity of a source emitting monochromatic radiation of frequency 540 × 10¹² Hz with radiant intensity of 1/683 W/sr.
Mole: Exactly 6.02214076 × 10²³ elementary entities (Avogadro's number fixed).

2.5

Prefixes for Powers of Ten

Power	Prefix	Symbol	Power	Prefix	Symbol
10¹	deca	da	10⁻¹	deci	d
10²	hecto	h	10⁻²	centi	c
10³	kilo	k	10⁻³	milli	m
10⁶	mega	M	10⁻⁶	micro	μ
10⁹	giga	G	10⁻⁹	nano	n
10¹²	tera	T	10⁻¹²	pico	p
10¹⁵	peta	P	10⁻¹⁵	femto	f
10¹⁸	exa	E	10⁻¹⁸	atto	a

Memory Trick — Big Prefixes Kids Make Good Tasty Pancakes Every-time → Kilo, Mega, Giga, Tera, Peta, Exa

Memory Trick — Small Prefixes My Mother Never Picks Fights Anymore → Milli, Micro, Nano, Pico, Femto, Atto

2.6

Practical Units for Small Distances

Unit	Value	Used For
1 fermi (fm)	\(10^{-15}\) m	Nuclear sizes
1 angstrom (Å)	\(10^{-10}\) m	Atomic sizes, light wavelengths
1 nanometre (nm)	\(10^{-9}\) m	Visible light wavelengths
1 micron (μm)	\(10^{-6}\) m	Bacteria, small particles

2.7

Practical Units for Large Distances

Unit	Value	Used For
1 AU	\(1.496 \times 10^{11}\) m	Mean Earth–Sun distance
1 light year (ly)	\(9.46 \times 10^{15}\) m	Distance light travels in one year
1 parsec (pc)	\(3.08 \times 10^{16}\) m = 3.26 ly	Distance at which 1 AU subtends 1 arc second

          Order: parsec > light year > AU. Parsec is the largest practical unit of distance.
        

2.8

Practical Units of Mass

Unit	Value
1 tonne (metric ton)	1000 kg
1 quintal	100 kg
1 pound (lb)	0.4536 kg
1 slug	14.57 kg
1 atomic mass unit (amu, u)	\(1.66 \times 10^{-27}\) kg = (1/12) × mass of C-12 atom
1 Chandrasekhar Limit (CSL)	1.4 × mass of Sun — largest practical unit of mass

2.9

Practical Units of Time

Unit	Details
Solar day	Time for Earth to rotate once with respect to the Sun
Sidereal day	Time for Earth to rotate once with respect to a distant star
Solar year	365.25 solar days
Lunar month	27.3 days
1 shake	\(10^{-8}\) s — smallest practical unit of time

2.10

Order of Magnitude

EXAM DEFINITION

The order of magnitude of a physical quantity is the power of 10 that is closest to its magnitude.

To find it, write the number as \(N = n \times 10^x\) where \(0.5 \lt n \leq 5\). Then \(x\) is the order of magnitude.

Number	Written as	Order
555	\(0.555 \times 10^3\)	3
0.05	\(5 \times 10^{-2}\)	−2
49	\(4.9 \times 10^1\)	1
753,000	\(0.753 \times 10^6\)	6

2.11

Dimensions of a Physical Quantity

EXAM DEFINITION

The dimensions of a physical quantity are the powers to which the fundamental quantities must be raised to represent that quantity completely.

We write dimensions inside square brackets [ ]. The seven fundamental dimensions: L, M, T, A, K, cd, mol

Dimensional Formula — Examples

\[\text{Velocity} = \frac{\text{Distance}}{\text{Time}} = \frac{[L]}{[T]} = [M^0 L T^{-1}]\]

\[F = ma \Rightarrow [M] \times [LT^{-2}] = [MLT^{-2}]\]

Important Dimensional Formulae

Quantity	Dimensional Formula	SI Unit
Area	[L²]	m²
Volume	[L³]	m³
Density	[ML⁻³]	kg m⁻³
Velocity	[LT⁻¹]	m s⁻¹
Acceleration	[LT⁻²]	m s⁻²
Force	[MLT⁻²]	newton (N)
Momentum	[MLT⁻¹]	kg m s⁻¹
Work / Energy	[ML²T⁻²]	joule (J)
Power	[ML²T⁻³]	watt (W)
Pressure	[ML⁻¹T⁻²]	pascal (Pa)
Frequency	[T⁻¹]	hertz (Hz)
Torque	[ML²T⁻²]	N m
Gravitational constant G	[M⁻¹L³T⁻²]	N m² kg⁻²
Planck's constant h	[ML²T⁻¹]	J s
Surface tension	[MT⁻²]	N m⁻¹
Coefficient of viscosity	[ML⁻¹T⁻¹]	Pa s
Electric charge	[AT]	coulomb (C)
Electric potential	[ML²T⁻³A⁻¹]	volt (V)

2.12

Principle of Homogeneity of Dimensions

EXAM DEFINITION

A physical equation is dimensionally correct only if every term on both sides of the equation has the same dimensions.

You can only add length to length, mass to mass, time to time. You cannot add a force to a velocity.

Example check: \(s = ut + \dfrac{1}{2}at^2\)

[s] = [L]

[ut] = [LT⁻¹][T] = [L] ✓

[½at²] = [LT⁻²][T²] = [L] ✓

All three terms have dimension [L]. The equation is dimensionally homogeneous.

2.13

Three Uses of Dimensional Analysis

Use 1 — Converting Units Between Systems

\[n_2 = n_1 \left[\frac{M_1}{M_2}\right]^a \left[\frac{L_1}{L_2}\right]^b \left[\frac{T_1}{T_2}\right]^c\]

Use 2 — Checking Equation Correctness

Apply the principle of homogeneity: every term on both sides must have the same dimensions.

Use 3 — Deriving Relationships

If we know what physical quantities Z depends on, assume \(Z = k \cdot A^a B^b C^c\) and compare dimensions to find the exponents.

2.14

Limitations of Dimensional Analysis

Cannot determine the value of dimensionless constants like 2, π, ½ in formulas.
Fails when a quantity depends on more than three physical quantities.
Cannot handle equations that are sums or differences of similar quantities.
Cannot derive relations involving trigonometric, logarithmic, or exponential functions.
Sometimes the choice of which quantities matter is not obvious.

2.15

Significant Figures

EXAM DEFINITION

Significant figures are the digits in a measurement that are known reliably plus the first uncertain digit.

If a length is 273.6 cm, then 2, 7, 3 are reliable and 6 is the first uncertain digit. So it has 4 significant figures.

Rules for Counting Significant Figures

All non-zero digits are significant. (13.75 → 4 sig figs)
All zeros between non-zero digits are significant. (100.05 → 5 sig figs)
Trailing zeros WITHOUT a decimal point are NOT significant. (86400 → 3 sig figs)
Trailing zeros to the right of a non-zero digit and LEFT of a decimal point ARE significant. (648700. → 6 sig figs)
All zeros to the right of a decimal point ARE significant. (161.00 cm → 5 sig figs)
Leading zeros to the left of the first non-zero digit are NOT significant. (0.0161 → 3 sig figs)
Sig figs do NOT depend on system of units. (16.4 cm = 0.164 m → both have 3 sig figs)

Sig Figs in Arithmetic

Addition / Subtraction

Final answer keeps the same number of decimal places as the term with the fewest decimal places.

Multiplication / Division

Final answer keeps the same number of significant figures as the term with the fewest significant figures.

2.16

Accuracy vs Precision

Accuracy

How close your measurement is to the true value. Depends on systematic errors.

Precision

How fine the resolution of your instrument is. Smaller least count → higher precision.

          Example: True length = 3.678 cm.

          Instrument A (LC 0.1 cm) reads 3.5 cm → MORE ACCURATE, LESS PRECISE.

          Instrument B (LC 0.01 cm) reads 3.38 cm → MORE PRECISE, LESS ACCURATE.

2.17

Errors in Measurement

EXAM DEFINITION

An error is the difference between the true value and the measured value of a quantity.

1. Systematic Errors

Always push the reading in one direction. Can be eliminated once the cause is found.

Instrumental errors — defects (zero error)
Imperfections in technique — heat loss in calorimetry
Personal errors — parallax from tilted head
External errors — scale expanding in heat

2. Random Errors

Occur irregularly, vary in both magnitude and direction. Cannot be eliminated, only reduced.

Reduced by: taking many readings and averaging.

2.18

Absolute Error and Mean Absolute Error

\[\bar{a} = \frac{a_1 + a_2 + \cdots + a_n}{n} \qquad \text{(Mean value)}\]

\[\Delta a_i = |\bar{a} - a_i| \qquad \Delta \bar{a} = \frac{\Delta a_1 + \cdots + \Delta a_n}{n}\]

Result is reported as: \(a = \bar{a} \pm \Delta \bar{a}\)

2.19

Relative Error and Percentage Error

Relative (Fractional) Error

\[\frac{\Delta \bar{a}}{\bar{a}}\]

Percentage Error

\[\frac{\Delta \bar{a}}{\bar{a}} \times 100\%\]

2.20

Propagation (Combination) of Errors

When Z depends on measured quantities A and B, errors propagate. All contributions are always added.

Rule 1 — Sum or Difference: Z = A ± B

\[\Delta Z = \Delta A + \Delta B\]

Rule 2 — Product or Quotient: Z = AB or Z = A/B

\[\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}\]

Rule 3 — Power: Z = Aⁿ

\[\frac{\Delta Z}{Z} = |n| \cdot \frac{\Delta A}{A}\]

          General rule: For \(Z = \dfrac{A^p \cdot B^q}{C^r}\):

          \(\dfrac{\Delta Z}{Z} = p\dfrac{\Delta A}{A} + q\dfrac{\Delta B}{B} + r\dfrac{\Delta C}{C}\)

2.21

Measuring Instruments — Vernier Callipers & Screw Gauge

Two instruments rule the physics lab. The metre scale gives you accuracy of 1 mm. Vernier Callipers improve this to 0.1 mm, and the Screw Gauge goes further to 0.01 mm.

A. Vernier Callipers

EXAM DEFINITION

Vernier callipers is an instrument used to measure lengths more accurately than a normal scale — typically up to 0.01 cm (0.1 mm).

Parts:

Main scale (MS) — like a normal ruler, divisions in mm or cm
Vernier scale (VS) — a small sliding scale that moves along the main scale
External jaws — for measuring outer dimensions (diameter of a cylinder)
Internal jaws — for measuring inner dimensions (inner diameter of a pipe)
Depth probe — for measuring depth of a hole

Least Count of Vernier Callipers

Typically, 10 vernier scale divisions (VSD) = 9 main scale divisions (MSD).

So: 1 VSD = (9/10) MSD = 0.9 mm

\[\text{LC} = 1 \text{ MSD} - 1 \text{ VSD} = 1 \text{ mm} - 0.9 \text{ mm} = 0.1 \text{ mm} = 0.01 \text{ cm}\]

How to Read a Vernier (Exam Method)

1 Main Scale Reading (MSR): Note the main scale division just before the zero of the vernier scale.

2 Vernier Scale Division (VSD): Find which vernier line exactly coincides with any main scale line.

3 Apply formula:

\[\text{Reading} = \text{MSR} + (\text{VSD} \times \text{LC})\]

          Example: MSR = 2.4 cm, VSD = 6, LC = 0.01 cm

          Reading = 2.4 + (6 × 0.01) = 2.4 + 0.06 = 2.46 cm

Zero Error in Vernier

Positive Zero Error

Zero of vernier is to the RIGHT of main scale zero when jaws are closed. The reading is falsely high.
Correction: Subtract zero error from observed reading.

Negative Zero Error

Zero of vernier is to the LEFT of main scale zero. The reading is falsely low.
Correction: Add the magnitude of zero error to observed reading.

\[\text{Corrected Reading} = \text{Observed Reading} - \text{Zero Error}\]

B. Screw Gauge

EXAM DEFINITION

The screw gauge (micrometer screw gauge) is used to measure small dimensions very precisely — up to 0.01 mm (0.001 cm). It works on the principle of a screw.

Parts:

Sleeve (barrel) — fixed part with main scale markings (mm and 0.5 mm lines)
Thimble — rotating cylinder with 50 divisions on its circular scale
Ratchet — applies consistent pressure; prevents over-tightening
Anvil and Spindle — the two measuring faces; the object is placed between them

Least Count of Screw Gauge

Pitch = distance moved by spindle in one full rotation = 0.5 mm (standard)

\[\text{LC} = \frac{\text{Pitch}}{\text{No. of divisions on thimble}} = \frac{0.5 \text{ mm}}{50} = 0.01 \text{ mm}\]

How to Read a Screw Gauge

1 Linear Scale Reading (LSR): Note the last visible mark on the sleeve. Each mark = 0.5 mm. Check if the 0.5 mm mark is visible past the thimble edge.

2 Thimble Scale Division (TSD): Find which circular scale line coincides with the reference (horizontal) line on the sleeve.

3 Apply formula:

\[\text{Reading} = \text{LSR} + (\text{TSD} \times \text{LC})\]

          Example: LSR = 2.5 mm, TSD = 30, LC = 0.01 mm

          Reading = 2.5 + (30 × 0.01) = 2.5 + 0.30 = 2.80 mm

Back-lash Error

When the direction of rotation is reversed, the spindle does not immediately move due to play in the screw. To avoid this, always rotate the thimble in one direction only while taking a reading.

Comparison

Feature	Vernier Callipers	Screw Gauge
Least Count	0.1 mm (0.01 cm)	0.01 mm (0.001 cm)
Range	Up to ~15 cm	Up to ~2.5 cm
Principle	Vernier principle	Screw principle
Measures	Length, outer/inner diameter, depth	Diameter of thin wires, thickness of thin sheets

          JEE/NEET Key Points:

          LC of Vernier = 1 MSD − 1 VSD = (1 − 9/10) mm = 0.1 mm

          LC of Screw Gauge = Pitch ÷ (Number of circular scale divisions)

          Back-lash error: always rotate thimble in ONE direction only.

⚠

Common Mistakes

Writing a number without a unit. "Length is 5" means nothing. Always write "5 metres".
Adding quantities with different units directly. You cannot add 5 metres and 3 seconds.
Forgetting absolute values in error rules. ΔZ is always positive — always ADD errors.
Counting leading zeros as significant figures. 0.0045 has only 2 significant figures.
Confusing accuracy (how close to true value) and precision (how fine the instrument).

💡

Board / JEE / NEET Tips

For Board Exams

Write definitions in textbook style verbatim. Show every step in dimensional analysis — nothing assumed. Always include the % sign in percentage error answers.

For JEE

Work and Torque both have dimension [ML²T⁻²] but different physical meaning — a favourite trap. Remember dimensions of Planck's constant (h), gravitational constant (G), and permittivity (ε₀). Error propagation in combined formulas is very common.

For NEET

Significant figures and unit conversion questions are easy marks — never skip them. Order of magnitude questions appear every year.

SEE IT

Four interactive simulations — learn by doing

SIMULATION 01

Cosmic Scale Explorer

Drag the slider to travel from the Planck length (10⁻³⁵ m) to the observable universe (10²⁶ m). Discover the orders of magnitude that separate the world of atoms from the world of galaxies.

10⁻³⁵ m 10⁻¹⁵ m 10⁻⁷ m 1 m 10⁷ m 10¹⁵ m 10²⁶ m

⚛️

SIMULATION 02

Smart Unit Converter

Convert between every unit you need for Class 11 — length, mass, time, and angle. Enter a value and see the conversion with the step shown.

PHYSICAL QUANTITY

FROM UNIT

VALUE

TO UNIT

→

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SIMULATION 03

Sig Fig Detective

10 rounds. A number appears — you pick how many significant figures it has. Get instant feedback with the rule explained. Build the skill that saves marks in practicals.

Score: 0 0/10

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How many significant figures does this number have?

SIMULATION 04

Vernier & Screw Gauge Reader

Master reading both instruments. Use the sliders to set any reading and see it on the instrument diagram. Then switch to Practice mode — a random reading appears and you calculate the answer.

MSR: 2.4 cm VSD: 6

MSR = 2.4 cm

VSD × LC = 6 × 0.01 = 0.06 cm

Reading = 2.46 cm

Practice Mode

SOLVE IT

10 fully solved problems — board to JEE level

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TEST IT

15 MCQs — click an option to get instant feedback

Score: 0/0 0/15 answered

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