Units and Measurements — NEET notes

Units & Systems of Measurement

Every measurement needs a unit. SI defines 7 base units from which all other units are derived. Knowing which quantities are fundamental vs derived, and how CGS/FPS/SI differ, is essential NEET knowledge.

Physical Quantity = Numerical Value Unit

Why We Need Standard Units

A measurement is meaningless without a standard reference. Units provide that standard so everyone gets the same number for the same quantity.

When you say a rod is '5 long' — 5 what? Metres? Feet? Cubits? A unit is a standard reference quantity. Every measurement is a comparison: you're saying 'this length is 5 times the standard length (metre).' Without a universally agreed standard, science would be chaos.

• A physical quantity = numerical value unit (e.g., length = 5 m)
• If you change the unit, the numerical value changes inversely: 5 m = 500 cm
• The magnitude of a physical quantity is independent of the unit chosen: n_1 u_1 = n_2 u_2
• Larger unit → smaller numerical value; smaller unit → larger numerical value

n_1 u_1 = n_2 u_2 (magnitude is invariant)

Fundamental & Derived Quantities

Only 7 independent physical quantities form the base of SI. Every other quantity (speed, force, energy...) is derived from combinations of these 7.

A fundamental (base) quantity cannot be expressed in terms of other quantities — it is defined independently. A derived quantity is expressed as a combination of base quantities through equations.

• There are exactly 7 fundamental quantities in SI — memorize all 7 with their units and symbols
• The kilogram is the ONLY base unit with a prefix ('kilo') in its name
• Two supplementary quantities were defined earlier: plane angle (radian, rad) and solid angle (steradian, sr) — now treated as derived
• Derived examples: velocity ( m/s ), acceleration ( m/s^2 ), force ( kg /s^2 = N )

SI, CGS & FPS Systems

Different countries historically used different unit systems. The SI system (Systeme International) is now universally adopted for scientific work.

Three major systems evolved historically based on different choices for the base units of length, mass, and time.

• SI = internationally accepted system with 7 base units — used in ALL scientific work
• CGS = centimetre-gram-second — still used in some theoretical physics
• FPS = foot-pound-second (British system) — used in everyday life in the US
• SI was established by the General Conference on Weights and Measures (CGPM) in 1971
• 1 N = 10^5 dyne , 1 J = 10^7 erg — classic conversion factors for NEET

Measurement of Length

Length ranges from subatomic ( 10^ -15 m) to intergalactic ( 10^ 26 m) scales. For everyday objects we use rulers; for astronomical distances we need clever tricks like parallax, and special units like AU, light year, and parsec.

D = b

Range of Lengths in Nature

The lengths encountered in physics span an astonishing range. NCERT gives specific values for sizes of various objects — and NEET tests them.

The size of a proton is 10^ -15 m and the observable universe stretches to 10^ 26 m — a range of 41 orders of magnitude. No single instrument can cover this range, which is why different measurement techniques exist for different scales.

• 1 fermi (fm) = 10^ -15 m — used for nuclear sizes
• 1 angstrom ( ) = 10^ -10 m — used for atomic/molecular sizes
• 1 micron ( m) = 10^ -6 m — used for biological cells
• These are NOT SI units but are widely used in their respective fields

Parallax Method

Parallax uses the apparent shift of an object when viewed from two different positions to calculate its distance. This is how astronomers measure star distances.

Hold your thumb at arm's length and close one eye, then the other. Your thumb appears to shift against the background. This shift is called parallax. The closer the thumb, the larger the shift.

In astronomy, we observe a star from two positions of Earth's orbit (6 months apart). The two observation points are separated by 2 1 AU . The apparent angular shift of the star against distant background stars is 2 , where is the parallax angle.

D = b

Here b is the baseline (distance between the two observation points), is the parallax angle (in radians), and D is the distance to the object. This works because for very distant objects, is tiny, so .

• Basis (b): the distance between the two observation points
• Parallax angle ( ): half the total angular shift observed
• Formula valid only when D b (so is small, measured in radians)
• For nearby objects: basis can be the distance between your two eyes ( 6 cm)
• For stars: basis = diameter of Earth's orbit 3 10^ 11 m

Quick recall

A star shows a parallax of 1 arcsecond when observed from Earth's orbit. How far away is it?

Exactly 1 parsec — that's the definition of a parsec. 1 pc = 3.08 10^ 16 m.

AU, Light Year & Parsec

For astronomical distances, metres are absurdly small. Three special units handle cosmic scales: the AU, the light year, and the parsec.

Saying the nearest star is ' 4 10^ 16 metres away' is technically correct but practically useless. Astronomers use larger, more intuitive units.

• 1 parsec 3.26 light years — the parsec is the larger unit
• Parsec comes from parallax of one arcsecond
• The nearest star (Proxima Centauri) is 1.3 pc 4.2 ly away
• These are the MOST frequently tested values from this chapter in NEET

1 ly = (3 10^8)(365.25 24 3600) 9.46 10^ 15 m

Significant Figures

Significant figures tell you how precisely a quantity was measured. Every digit that carries real measurement information is significant. Knowing the rules for counting, rounding, and arithmetic with sig figs is a NEET must.

Result has as many sig figs as the least precise input

Counting Significant Figures

The rules for identifying significant figures revolve around one key question: is this digit giving real measurement information, or is it just a placeholder?

Significant figures are all the digits in a measurement that are known reliably, plus one uncertain (estimated) digit. They tell you the precision of a measurement — more sig figs means higher precision.

• Rule 1: All non-zero digits are significant. 246 has 3 sig figs
• Rule 2: Zeros between non-zero digits are significant. 1.008 has 4 sig figs
• Rule 3: Leading zeros (before first non-zero digit) are NOT significant. 0.0042 has 2 sig figs
• Rule 4: Trailing zeros after decimal point ARE significant. 4.700 has 4 sig figs
• Rule 5: Trailing zeros in a number WITHOUT a decimal point are ambiguous. 2300 could be 2, 3, or 4 sig figs — use scientific notation to be clear

Rounding Off

When a calculation gives more digits than your data justifies, you must round off. The rules seem simple but have a tricky edge case that NEET loves.

After performing calculations, the result often has more digits than justified by the precision of the input data. You must round off to the correct number of significant figures.

• If the digit to be dropped is < 5, drop it (round down). 7.82 7.8
• If the digit to be dropped is > 5, increase preceding digit by 1 (round up). 7.86 7.9
• If the digit to be dropped is exactly 5: round to the nearest even digit. 7.85 7.8 but 7.75 7.8
• This 'round to even' rule (banker's rounding) prevents systematic bias in large datasets

Arithmetic with Significant Figures

When you add/subtract, you match decimal places. When you multiply/divide, you match significant figures. Mixing up these two rules is the #1 student error.

The precision of a calculated result depends on the precision of the inputs. But the rule is different for addition/subtraction vs multiplication/division.

• In addition: 236.1 + 5.03 + 0.214 = 241.344 241.3 (1 decimal place, limited by 236.1 )
• In multiplication: 3.4 2.123 = 7.2182 7.2 (2 sig figs, limited by 3.4 )
• Exact numbers (like counting numbers or defined constants) have infinite sig figs and never limit the result
• Example: circumference = 2 r . The '2' is exact. = 3.14159... has infinite sig figs. Only r limits the result

Errors in Measurement

No measurement is perfect. Understanding the types of errors (systematic vs random), how to quantify them (absolute, relative, percentage), and how errors propagate through calculations is one of the highest-yield NEET topics in this chapter.

Percentage Error = a a_ mean 100 %

Types of Errors

Errors fall into three categories based on their origin. Systematic errors have a consistent direction; random errors fluctuate; gross errors are human blunders.

Every measurement has uncertainty. The difference between the measured value and the true value is called error. Understanding error types is crucial because different types require different remedies.

• Systematic errors — consistent, reproducible, always in the same direction (too high or too low)
• Random errors — unpredictable fluctuations, equally likely to be positive or negative
• Gross errors — human blunders: misreading a scale, recording wrong numbers, using wrong formula

Absolute, Relative & Percentage Error

Three ways to express the same error. Absolute error has units; relative error is dimensionless; percentage error is relative error times 100.

After taking multiple readings a_1, a_2, , a_n , the best estimate of the true value is the arithmetic mean a_ mean = 1 n _ i=1 ^ n a_i .

Absolute error: a_i = |a_i - a_ mean |

Mean absolute error: a = 1 n _ i=1 ^ n | a_i|

Relative error = a a_ mean Percentage error = a a_ mean 100 %

• Absolute error has the same units as the measurement. L = 0.2 cm
• Relative error is dimensionless (a pure ratio). L / L = 0.004
• Percentage error = relative error 100 . = 0.4 %
• The result is reported as: a = a_ mean a
• A measurement with small percentage error has high accuracy

Combination of Errors

When a derived quantity depends on measured quantities, errors combine according to specific rules. This is the HIGHEST-YIELD topic in this chapter for NEET numericals.

If a result Z depends on measured quantities A and B , the error in Z depends on HOW A and B combine. The rules differ for addition/subtraction vs multiplication/division.

Z = A + B Z = A + B (absolute errors add)

Z = A - B Z = A + B (errors still ADD, never subtract!)

Z = A^p B^q / C^r Z Z = |p| A A + |q| B B + |r| C C

• Errors always ADD in the worst case — never subtract (maximum possible error principle)
• For Z = A B : Z = A + B (absolute errors add)
• For Z = AB or Z = A/B : Z Z = A A + B B (relative errors add)
• For Z = kA^p B^q C^r : Z Z = |p| A A + |q| B B + |r| C C
• The quantity with the highest power contributes the most error — measure it most carefully!

Quick recall

If Z = A^2 B^ 1/3 C and percentage errors in A , B , C are 1%, 3%, 4% respectively, what is the percentage error in Z ?

Z Z 100 = 2(1 %) + 1 3 (3 %) + 1 2 (4 %) = 2 % + 1 % + 2 % = 5 %

Dimensional Analysis

Every physical quantity has a dimensional formula built from [M], [L], and [T] (and occasionally [A], [K], [mol], [cd]). Dimensional analysis lets you check if an equation makes physical sense, convert units between systems, and even derive approximate formulas.

[Force] = [M L T^ -2 ]

Dimensions & Dimensional Formulae

The dimension of a quantity tells you what fundamental quantities it is made of and in what powers. It is the identity card of the quantity.

The dimension of a physical quantity is the power to which the fundamental quantities must be raised to represent it. We use square brackets to denote dimensions: [M] for mass, [L] for length, [T] for time, [A] for current, [K] for temperature, [mol] for amount, and [cd] for luminous intensity.

• Quantities with NO dimensions: angle (radian), refractive index, strain, relative density, all trigonometric ratios
• Dimensionless ≠ unitless: Angle has no dimensions but has a unit (radian). Pure numbers are both dimensionless and unitless
• Work and torque have the same dimensional formula [ML^2T^ -2 ] but are physically different — dimensions don't capture everything
• Two quantities can be added/subtracted ONLY if they have the same dimensions (principle of homogeneity)

Checking Dimensional Consistency

An equation is dimensionally consistent if every term on both sides has the same dimensions. This is a necessary (but not sufficient) condition for correctness.

The Principle of Homogeneity states: every term in a valid physical equation must have the same dimensions. You can't add metres to seconds — just like you can't add apples to oranges.

To check an equation, find the dimensions of each term separately. If they all match, the equation is dimensionally consistent. If any term differs, the equation is definitely wrong.

• If an equation is dimensionally inconsistent → it is definitely wrong
• If an equation is dimensionally consistent → it may or may not be correct
• Each term in a sum/equation must have identical dimensions
• Arguments of , , , e^x must be dimensionless
• Transcendental functions ( , , , e^x ) produce dimensionless outputs

Quick recall

Check if v^2 = u^2 + 2as is dimensionally consistent.

[v^2] = [LT^ -1 ]^2 = [L^2T^ -2 ] . [u^2] = [L^2T^ -2 ] . [2as] = [LT^ -2 ][L] = [L^2T^ -2 ] . All terms match → dimensionally consistent.

Applications of Dimensional Analysis

Beyond checking equations, dimensional analysis can derive relations between quantities and convert units between systems. But it has clear limitations — it can't find dimensionless constants or handle functions like sin and log.

n_2 = n_1 ( M_1 M_2 )^a ( L_1 L_2 )^b ( T_1 T_2 )^c

Deriving Relations

If you know a quantity depends on certain variables, dimensional analysis can determine the powers of those variables. This is how many famous formulas were first guessed.

Suppose you know that the time period T of a simple pendulum depends on length l , mass m , and acceleration due to gravity g . Without solving any differential equations, dimensional analysis can tell you HOW they combine.

Method: Assume T = k l^a m^b g^c where k is a dimensionless constant. Write dimensions on both sides and equate powers of [M] , [L] , and [T] .

[T] = [L]^a [M]^b [LT^ -2 ]^c = [M^b L^ a+c T^ -2c ]

• Equating powers: [M]: b = 0 , [L]: a + c = 0 , [T]: -2c = 1
• Solving: c = - 1 2 , a = 1 2 , b = 0 → T does NOT depend on mass!
• Result: T = k l/g (the actual constant k = 2 can't be found dimensionally)
• This method works when the relation is a power-law (product of variables raised to powers)

Quick recall

Derive the dependence of the velocity of sound v on pressure P and density of the medium.

Let v = k P^a ^b . [LT^ -1 ] = [ML^ -1 T^ -2 ]^a [ML^ -3 ]^b . Equating: [M]: a+b=0 , [L]: -a-3b=1 , [T]: -2a=-1 . So a= 1 2 , b=- 1 2 . Therefore v = k P/ .

Converting Units Between Systems

To convert a quantity from one system (say CGS) to another (SI), use the dimensional formula to find the conversion factor systematically.

Since the magnitude of a physical quantity is invariant ( n_1 u_1 = n_2 u_2 ), we can use dimensional analysis to convert between unit systems. If the dimensional formula is [M^a L^b T^c] , then:

n_2 = n_1 ( M_1 M_2 )^a ( L_1 L_2 )^b ( T_1 T_2 )^c

Here, subscript 1 is the 'from' system and subscript 2 is the 'to' system.

• This method works for ANY unit system conversion — not just CGS↔SI
• You need to know the dimensional formula of the quantity being converted
• The conversion factor for each base unit: 1 kg = 10^3 g , 1 m = 10^2 cm
• Time is usually the same (seconds) in both CGS and SI, so the T factor is often 1

Limitations of Dimensional Analysis

Dimensional analysis is powerful but has clear boundaries. Knowing what it CAN'T do is just as important as knowing what it can — and NEET tests both.

Despite its elegance, dimensional analysis has fundamental limitations that you must know.

• Cannot determine dimensionless constants — the 2 in T = 2 l/g , the 1 2 in KE = 1 2 mv^2
• Cannot derive relations involving transcendental functions — , , , e^x are dimensionless and invisible to dimensional analysis
• Cannot distinguish between quantities with the same dimensions — work vs torque, energy vs moment of force
• Fails when a quantity depends on more than 3 mechanical variables — you get 3 equations ( M , L , T ) but more than 3 unknowns
• Cannot determine whether a quantity depends on another — if the dependence is through a dimensionless combination, dimensional analysis misses it entirely

Accuracy, Precision & Instruments

Accuracy is how close you are to the true value; precision is how consistent your readings are. Instruments like vernier calipers and screw gauges achieve high precision through clever mechanical design. Knowing least count calculations is a NEET staple.

Least Count = 1 MSD - 1 VSD

Accuracy vs Precision

These two words are used interchangeably in everyday language but mean very different things in physics. NEET tests this distinction explicitly.

Accuracy tells you how close a measurement is to the true value. Precision tells you how reproducible the measurement is — how closely repeated readings agree with each other.

• A cheap digital scale showing 72.345 kg every time (but true weight is 70 kg) is precise but not accurate
• A bathroom scale showing 69, 70, 71, 70 kg (true weight 70 kg) is accurate but not very precise
• The number of significant figures in a measurement indicates its precision, not accuracy
• More decimal places = higher precision. But precision without accuracy is useless

Least Count

Every instrument has a limit to how finely it can measure. This limit is the least count — and it determines the maximum precision of the instrument.

The least count (LC) of an instrument is the smallest measurement it can make. For a ruler with mm markings, LC = 1 mm. For a vernier caliper, LC is much smaller — typically 0.01 mm or 0.1 mm.

• Least count = smallest division on the main scale, OR the difference between 1 MSD and 1 VSD for vernier instruments
• Smaller least count → higher precision → more significant figures in the measurement
• The least count error is the maximum uncertainty in a single reading: LC
• LC error is always present — it's an inherent limitation of the instrument, not a mistake

Vernier Caliper

The vernier caliper uses a sliding vernier scale alongside the main scale. The clever trick: the vernier has divisions slightly smaller than the main scale, and the mismatch gives sub-millimetre readings.

A vernier caliper has two scales: the main scale (MS) with 1 mm divisions, and the vernier scale (VS) where n vernier divisions coincide with (n-1) main scale divisions. Typically, 10 VSD = 9 MSD.

LC = 1 MSD - 1 VSD = 1 mm - 0.9 mm = 0.1 mm = 0.01 cm

Reading = MSR + (VSR LC) - Zero Error

• MSR = Main Scale Reading — the mark on the main scale just BEFORE (left of) the zero of the vernier
• VSR = Vernier Scale Reading — the vernier division that BEST ALIGNS with any main scale division
• Zero error — if the zero of the VS doesn't coincide with zero of MS when jaws are closed
• Positive zero error (zero of VS is to the RIGHT of zero of MS) → subtract correction
• Negative zero error (zero of VS is to the LEFT of zero of MS) → add correction

Quick recall

MSR = 3.2 cm, VSR = 6, LC = 0.01 cm, zero error = +0.03 cm. What is the reading?

Reading = 3.2 + (6 0.01) - 0.03 = 3.2 + 0.06 - 0.03 = 3.23 cm.

Screw Gauge (Micrometer)

A screw gauge uses a screw mechanism to achieve 0.01 mm precision — 10 times better than a vernier caliper. The circular scale magnifies tiny linear movements into easily readable angular divisions.

A screw gauge converts rotational motion of a thimble into linear motion of a spindle using a precision screw. One full rotation advances the spindle by one pitch (usually 0.5 mm or 1 mm). The thimble has 50 or 100 divisions, giving each division a value of pitch/number of divisions.

LC = Pitch Number of circular scale divisions = 0.5 mm 50 = 0.01 mm

Reading = PSR + (CSR LC) - Zero Error

• PSR = Pitch Scale Reading (also called Linear Scale Reading) — the main scale reading visible before the thimble edge
• CSR = Circular Scale Reading — the circular division aligned with the reference line
• Pitch = distance moved by spindle for one complete rotation of the thimble
• Typical LC = 0.01 mm = 10^ -5 m — suitable for measuring wire diameters
• Zero error concept is identical to vernier caliper — positive if zero passes reference line, negative if it doesn't reach it

Quick recall

A screw gauge has pitch 0.5 mm and 50 circular divisions. PSR = 3 mm, CSR = 27. Zero error = -0.03 mm. Find the reading.

LC = 0.5/50 = 0.01 mm. Reading = 3 + (27 0.01) - (-0.03) = 3 + 0.27 + 0.03 = 3.30 mm.