NEET Physics — Chapter 1

Units & Measurements

Every physical measurement rests on a foundation of units and standards. This chapter covers the SI system and its seven base units, how to express any physical quantity using dimensional formulas, the three applications of dimensional analysis, how to count significant figures and propagate errors through calculations, and how precision instruments like the vernier caliper and screw gauge work. Mastering this chapter gives you the language in which all of physics is written — and it accounts for 1–2 direct marks in NEET every year.

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1. Physical Quantities & Units

A physical quantity is anything that can be measured and expressed as a number with a unit. Physics divides all quantities into two broad categories:

Fundamental (Base) quantities — independent, cannot be derived from others.
Derived quantities — defined in terms of fundamental quantities (e.g., speed = length/time).

The International System of Units (SI) defines seven base quantities:

Base Quantity	Symbol	SI Unit	Unit Symbol
Length	L	metre	m
Mass	M	kilogram	kg
Time	T	second	s
Electric Current	I	ampere	A
Thermodynamic Temperature	θ	kelvin	K
Amount of Substance	N	mole	mol
Luminous Intensity	J	candela	cd

Supplementary units (not base, but used alongside SI):

Radian (rad) — plane angle
Steradian (sr) — solid angle

NEET tip: The SI system has exactly seven base units. Radian and steradian are supplementary — not base units. Newton, joule, watt, pascal are all derived units. This distinction is a favourite NEET MCQ point.

Examples of derived quantities and their units:

Speed — m/s (derived from length and time)
Force — kg·m/s² = newton (N)
Energy — kg·m²/s² = joule (J)
Pressure — N/m² = pascal (Pa)

2. Dimensional Formulas of Key Quantities

The dimensional formula of a quantity expresses it as a product of powers of the base dimensions M (mass), L (length), and T (time). Write the definition of the quantity and substitute dimensions step by step.

Mechanical quantities:

\text{Velocity} = \frac{\text{displacement}}{\text{time}} \Rightarrow [LT^{-1}]

\text{Acceleration} = \frac{\text{velocity}}{\text{time}} \Rightarrow [LT^{-2}]

\text{Force} = m \times a \Rightarrow [MLT^{-2}]

\text{Work / Energy} = F \times d \Rightarrow [ML^2T^{-2}]

\text{Power} = \frac{W}{t} \Rightarrow [ML^2T^{-3}]

\text{Pressure} = \frac{F}{A} \Rightarrow [ML^{-1}T^{-2}]

\text{Momentum} = m \times v \Rightarrow [MLT^{-1}]

\text{Impulse} = F \times t \Rightarrow [MLT^{-1}]

\text{Torque} = F \times r \Rightarrow [ML^2T^{-2}]

\text{Angular momentum} = m \times v \times r \Rightarrow [ML^2T^{-1}]

Surface and fluid quantities:

\text{Surface tension} = \frac{F}{L} \Rightarrow [MT^{-2}]

\text{Coefficient of viscosity} = \frac{\text{shear stress}}{\text{velocity gradient}} \Rightarrow [ML^{-1}T^{-1}]

Important physical constants:

G\ (\text{gravitational constant}) \Rightarrow [M^{-1}L^3T^{-2}]

h\ (\text{Planck's constant}) \Rightarrow [ML^2T^{-1}]

k_B\ (\text{Boltzmann constant}) \Rightarrow [ML^2T^{-2}K^{-1}]

R\ (\text{gas constant}) \Rightarrow [ML^2T^{-2}K^{-1}\text{mol}^{-1}]

Caution: Torque and Work/Energy both have the same dimensional formula

[ML^2T^{-2}]

, yet they are completely different physical quantities. Similarly, Momentum and Impulse share

[MLT^{-1}]

. Having the same dimensions does NOT mean quantities are equal or interchangeable.

3. Dimensional Analysis — Applications & Limitations

Dimensional analysis uses the principle of homogeneity: every term in a physically valid equation must have the same dimensional formula. This gives three key applications.

Application 1 — Checking an equation

Verify $v^2 = u^2 + 2as$ :

[v^2] = [L^2T^{-2}],\quad [u^2] = [L^2T^{-2}],\quad [2as] = [LT^{-2}][L] = [L^2T^{-2}] \checkmark

All terms match — the equation is dimensionally consistent.

Caution: Dimensional consistency is necessary but not sufficient. The equation

s = ut + at^2

is dimensionally correct but physically wrong (missing the factor

\frac{1}{2}

). Pure numbers like 2,

\pi

\frac{1}{2}

are invisible to dimensional analysis.

Application 2 — Unit conversion

If a quantity has dimensions $[M^a L^b T^c]$ , then:

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

Example: 1 J in CGS. Energy $\Rightarrow [ML^2T^{-2}]$ . $n_2 = 1 \times (1000)^1 \times (100)^2 \times (1)^{-2} = 10^7$ erg.

Application 3 — Deriving relationships

Time period of a simple pendulum: assume $T = k\,L^a m^b g^c$ . Matching dimensions:

M^0L^0T^1 = M^b L^{a+c} T^{-2c}

\Rightarrow b=0,\quad c = -\tfrac{1}{2},\quad a = \tfrac{1}{2}

\therefore\; T \propto \sqrt{\frac{L}{g}}

Limitations of dimensional analysis:

Cannot find the value of dimensionless constants ( $\pi$ , 2, $\frac{1}{2}$ , etc.)
Cannot handle equations involving exponential, logarithmic, or trigonometric functions
Cannot be applied when a quantity depends on two different quantities that have the same dimensions
Cannot distinguish between scalars and vectors of the same dimension

NEET tip: NEET frequently asks you to check whether a given equation is dimensionally correct, or to find the dimensions of an unknown constant in a formula. Always isolate the unknown and express it in terms of the other quantities.

4. Measurement & Significant Figures

Every measurement is approximate. Significant figures (sig figs) convey how precisely a quantity is known. The number of sig figs is determined by the measuring instrument's least count.

Rules for counting significant figures:

All non-zero digits are significant: 2345 has 4 sig figs.
Zeros between non-zero digits are significant: 1007 has 4 sig figs.
Leading zeros are NOT significant: 0.0034 has 2 sig figs.
Trailing zeros after a decimal point ARE significant: 3.600 has 4 sig figs.
Trailing zeros in a whole number are ambiguous: 3400 may have 2, 3, or 4 sig figs — use scientific notation to be clear.

Arithmetic with significant figures:

Addition/Subtraction — result has the same number of decimal places as the term with fewest decimal places. Example: $3.14 + 1.1 = 4.2$ (not 4.24).
Multiplication/Division — result has the same number of significant figures as the term with fewest sig figs. Example: $2.5 \times 3.14 = 7.9$ (2 sig figs).

Rounding rules:

If the digit to be dropped is less than 5 → round down (retain preceding digit).
If the digit to be dropped is more than 5 → round up.
If the digit to be dropped is exactly 5 → round to the nearest even digit (banker's rounding).

Pro tip: In NEET, significant figures questions are straightforward. The most common trap is with zeros — remember that trailing zeros after a decimal are significant, but leading zeros are not. When in doubt, convert to scientific notation.

5. Errors in Measurement

No measurement is perfect. The difference between the measured value and the true (accepted) value is the error. Understanding types of errors is essential for NEET.

Types of errors:

Systematic error — consistent, same direction every time; caused by faulty instruments, zero error, wrong calibration. Can be corrected once identified.
Random error — irregular, different each time; caused by environmental fluctuations, observer variability. Reduced by taking multiple readings and averaging.
Gross error — blunders due to careless reading or recording; eliminated by careful observation.

Definitions (for $n$ measurements $a_1, a_2, \ldots, a_n$ ):

\text{Mean value: }\bar{a} = \frac{a_1 + a_2 + \cdots + a_n}{n}

\text{Absolute error of }i\text{-th reading: }\Delta a_i = |\bar{a} - a_i|

\text{Mean absolute error: }\overline{\Delta a} = \frac{\Delta a_1 + \Delta a_2 + \cdots + \Delta a_n}{n}

\text{Relative (fractional) error} = \frac{\overline{\Delta a}}{\bar{a}}

\text{Percentage error} = \frac{\overline{\Delta a}}{\bar{a}} \times 100\%

The final result is reported as: $a = \bar{a} \pm \overline{\Delta a}$

NEET tip: The mean absolute error is always positive (we use absolute values of individual errors). The relative error has no unit — it is a pure ratio. Percentage error = relative error × 100.

6. Error Propagation — Combining Errors

When a derived quantity depends on measured quantities, errors propagate (combine). The rules depend on whether quantities are added, multiplied, or raised to powers.

Rule 1 — Addition and Subtraction

If $Z = A + B$ or $Z = A - B$ , then absolute errors add:

\Delta Z = \Delta A + \Delta B

Caution: For both addition AND subtraction, absolute errors always add — never subtract. This means subtraction of two nearly equal quantities causes a large relative error, making it a poor experimental strategy.

Rule 2 — Multiplication and Division

If $Z = A \times B$ or $Z = A / B$ , then relative errors add:

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

Rule 3 — Powers

If $Z = A^p \cdot B^q \cdot C^r$ , then:

\frac{\Delta Z}{Z} = p\,\frac{\Delta A}{A} + q\,\frac{\Delta B}{B} + r\,\frac{\Delta C}{C}

The power multiplies the relative error of that quantity. A quantity raised to a high power contributes a large relative error to the result.

Worked example: Kinetic energy $K = \frac{1}{2}mv^2$ . If mass has 2% error and speed has 3% error:

\frac{\Delta K}{K} = \frac{\Delta m}{m} + 2\cdot\frac{\Delta v}{v} = 2\% + 2 \times 3\% = 8\%

Pro tip: To minimise error in a derived quantity, measure the quantity that appears with the highest power most carefully. In

K = \frac{1}{2}mv^2

, a 1% error in

v

causes a 2% error in

K

— twice as damaging as a 1% error in

m

7. Instruments & Least Count

The least count (LC) of an instrument is the smallest measurement it can reliably make. It determines the precision of a reading and the uncertainty to assign to each measurement.

Vernier Caliper

A vernier scale slides alongside the main scale to improve precision. If $n$ vernier scale divisions (VSD) equal $(n-1)$ main scale divisions (MSD):

\text{Least Count} = 1\text{ MSD} - 1\text{ VSD} = \frac{1\text{ MSD}}{n}

Typical vernier calipers have 10 VSD = 9 MSD, so LC = 0.1 mm = 0.01 cm.

Reading a vernier: Total reading = Main scale reading + (vernier coincidence × LC)

Screw Gauge (Micrometer)

\text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}}

Typical: pitch = 0.5 mm, 50 circular scale divisions → LC = 0.5/50 = 0.01 mm = 0.001 cm.

Reading a screw gauge: Total = Linear scale reading + (circular scale reading × LC)

Zero Error:

Positive zero error — zero of circular scale is below the reference line when jaws are closed. The instrument over-reads. Correct reading = observed − zero error.
Negative zero error — zero of circular scale is above the reference line. The instrument under-reads. Correct reading = observed + |zero error|.

Pro tip: Always apply zero correction: Correct reading = Observed reading − Zero error. If zero error is negative (e.g., −0.03 mm), subtracting a negative adds to the reading. Practise this sign convention — NEET MCQs test it directly.

8. Important Conversions & SI Prefixes

NEET questions often use non-SI units. Knowing these conversions saves precious time in the exam.

SI Prefixes (must memorise):

Prefix	Symbol	Factor
giga	G	$10^9$
mega	M	$10^6$
kilo	k	$10^3$
hecto	h	$10^2$
deca	da	$10^1$
deci	d	$10^{-1}$
centi	c	$10^{-2}$
milli	m	$10^{-3}$
micro	μ	$10^{-6}$
nano	n	$10^{-9}$
pico	p	$10^{-12}$

Key physical conversions:

Quantity	Value in SI
1 electron-volt (eV)	$1.6 \times 10^{-19}$ J
1 atomic mass unit (amu or u)	$1.66 \times 10^{-27}$ kg
1 light year (ly)	$9.46 \times 10^{15}$ m
1 parsec (pc)	$3.08 \times 10^{16}$ m $\approx 3.26$ ly
1 astronomical unit (AU)	$1.5 \times 10^{11}$ m
1 angstrom (Å)	$10^{-10}$ m
1 fermi (fm)	$10^{-15}$ m
1 litre (L)	$10^{-3}$ m³
1 bar	$10^5$ Pa
1 atmosphere (atm)	$1.013 \times 10^5$ Pa
1 calorie (cal)	4.186 J

NEET tip: The angstrom (Å =

10^{-10}

m) is commonly used for atomic sizes and wavelengths of visible light. The fermi (

10^{-15}

m) is used for nuclear sizes. These appear in Modern Physics and Optics questions.

9. NEET Exam Traps & Common Mistakes

NEET MCQs on Units & Measurements reliably target certain conceptual pitfalls. Knowing these traps in advance gives you a significant edge.

Trap 1 — Dimensionless quantities

Several physical quantities appear to have dimensions but are actually dimensionless:

Angle (radian) — ratio of arc length to radius: $[L/L] = [M^0L^0T^0]$
Strain — ratio of change in length to original length: $[L/L]$
Refractive index — ratio of speeds: $[LT^{-1}/LT^{-1}]$
Relative density (specific gravity) — ratio of densities
Coefficient of friction — ratio of forces
Reynolds number, dielectric constant, fine structure constant $\alpha$

Caution: "Dimensionless" does NOT mean "unitless" in the sense of being trivial. Radian is a unit of angle — it just has no dimensional formula. NEET sometimes asks if a quantity has dimensions or not.

Trap 2 — Quantities sharing the same dimensional formula

Dimensional Formula	Quantities
$[ML^2T^{-2}]$	Work, Energy, Torque, Heat
$[MLT^{-1}]$	Momentum, Impulse
$[ML^{-1}T^{-2}]$	Pressure, Stress, Young's modulus, Bulk modulus
$[ML^2T^{-1}]$	Angular momentum, Planck's constant $h$
$[MT^{-2}]$	Surface tension, Spring constant (force/length)
$[ML^{-1}T^{-1}]$	Coefficient of viscosity, Momentum/Area

Trap 3 — Zero error sign convention

Students frequently get the sign wrong. The universal rule is:

\text{True reading} = \text{Observed reading} - \text{Zero error}

Positive zero error (+3 divisions): True = Observed − 3 div → smaller value.
Negative zero error (−3 divisions): True = Observed − (−3) = Observed + 3 → larger value.

Trap 4 — Significant figures in addition vs multiplication

Addition/subtraction uses decimal places; multiplication/division uses significant figures. Mixing these rules is the most common calculation error.

Trap 5 — Dimensional analysis cannot detect wrong numerical constants

$s = 2ut + at^2$ is dimensionally correct but physically wrong. NEET may ask you to identify the error — dimensional analysis alone cannot reveal it.

Pro tip: When a NEET question asks "which pair has the same dimensions?", use the table above. When asked "which is dimensionless?", check if it is a ratio of like quantities. These two question types account for the majority of Units & Measurements marks.

10. Quick Revision — Dimensional Formulas of 20 Quantities

Use this table for rapid last-minute revision before NEET. Verify each formula by tracing back to the definition — do not just memorise.

S.No.	Physical Quantity	Formula/Definition	Dimensional Formula
1	Velocity	displacement/time	$[LT^{-1}]$
2	Acceleration	velocity/time	$[LT^{-2}]$
3	Force	$ma$	$[MLT^{-2}]$
4	Work / Energy	$Fd$	$[ML^2T^{-2}]$
5	Power	Work/time	$[ML^2T^{-3}]$
6	Pressure / Stress	Force/Area	$[ML^{-1}T^{-2}]$
7	Momentum / Impulse	$mv$ / $Ft$	$[MLT^{-1}]$
8	Torque	$F imes r$	$[ML^2T^{-2}]$
9	Angular Momentum	$mvr$	$[ML^2T^{-1}]$
10	Surface Tension	Force/Length	$[MT^{-2}]$
11	Coefficient of Viscosity	stress/velocity gradient	$[ML^{-1}T^{-1}]$
12	Gravitational Constant (G)	$Fr^2/(m_1m_2)$	$[M^{-1}L^3T^{-2}]$
13	Planck's Constant (h)	$E/ u $</td><td class="py-1">$ [ML^2T^{-1}]$
14	Boltzmann Constant ( $k_B$ )	Energy/Temperature	$[ML^2T^{-2}K^{-1}]$
15	Density	mass/volume	$[ML^{-3}]$
16	Moment of Inertia	$mr^2$	$[ML^2]$
17	Angular Velocity	angle/time	$[T^{-1}]$
18	Frequency	cycles/time	$[T^{-1}]$
19	Electric Charge	current × time	$[AT]$
20	Resistance	Voltage/Current	$[ML^2T^{-3}A^{-2}]$

NEET tip: Note that Angular velocity and Frequency both have the same dimensional formula

[T^{-1}]

, yet they are different — one is in rad/s, the other in Hz (cycles/s). Also note that Torque (row 8) and Work/Energy (row 4) share

[ML^2T^{-2}]

— a frequent trap question.

Pro tip: Derive, don't memorise. For any unfamiliar quantity in NEET, write its definition as a formula, substitute known dimensions, and simplify. This approach never fails, even for unusual constants you haven't seen before.

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