Units and MeasurementsNEET Physics · Class 11 · NCERT Chapter 1

Introduction

Every numerical answer in NEET Physics begins with a measurement. Whether you are computing a force, an electric field, a wavelength or a half-life, you need to know what you are measuring, the unit you are measuring it in, and how confident you are in the value. Units and Measurements is the chapter that gives you that vocabulary.

For NEET, the chapter is tested in three concrete ways: vernier calliper and screw gauge readings, dimensional analysis, and error propagation. Each appears every two to three years. This page covers all three thoroughly, with worked NEET problems at the end and live calculators on the Interactive Learning page.

The International System of Units (SI)

The SI is built on seven base quantities, each with its own base unit. Every other physical quantity is derived from these.

Seven SI base units

Common derived units

SI prefixes you must know

Significant figures

Significant figures (sig figs) are the digits in a measurement that carry real information. They tell the reader how precise the measurement is. The five rules below cover every NEET question on the topic.

1. All non-zero digits are significant. 234 has 3 sig figs.
2. Zeros between non-zero digits are significant. 2003 has 4 sig figs.
3. Leading zeros are not significant. 0.0042 has 2 sig figs.
4. Trailing zeros after a decimal point are significant. 4.20 has 3 sig figs.
5. Trailing zeros in a whole number are ambiguous unless written in scientific notation. 1500 can be 2, 3 or 4 sig figs; $1.500\times 10^3$ is unambiguously 4.

Rule of thumb in arithmetic: when you add or subtract, the result keeps as many decimal places as the term with the fewest. When you multiply or divide, the result keeps as many significant figuresas the term with the fewest.

Rounding off rules

NCERT and NEET follow the standard rounding rules. Round the digit you keep based on the digit just after it.

• If the next digit is less than 5, round down. 5.673 → 5.67
• If the next digit is more than 5, round up. 5.677 → 5.68
• If the next digit is exactly 5 with no non-zero digits after it, round to the nearest even number (banker's rounding). 5.675 → 5.68 but 5.685 → 5.68.
• If the digit being kept is non-zero followed by 5 plus other non-zero digits, round up. 5.6751 → 5.68

Dimensions of physical quantities

The dimensions of a physical quantity are the powers to which the seven base quantities must be raised to represent that quantity. For NEET you mostly need three: length $L$, mass $M$, and time $T$. We write the dimensional formula in square brackets.

Notice that Planck's constant and angular momentum have the same dimensions. NEET loves this fact.

Dimensional analysis and applications

Dimensional analysis is a quick check on every formula you write. It has three uses you will see in NEET.

1. Checking dimensional homogeneity

Every term in a physics equation must have the same dimensions. If the LHS is a force, every term on the RHS must be a force. Take $v^2=u^2+2as$:

All three terms have the same dimensions. The equation is dimensionally consistent.

2. Deriving relationships

Suppose the time period $T$ of a simple pendulum depends on its length $\ell$, mass $m$, and acceleration due to gravity $g$. We can write $T=k{\ell }^a{m}^b{g}^c$ where $k$ is a dimensionless constant. Comparing dimensions on both sides:

Equating powers: $b=0$, $a+c=0$, $-2c=1$. Solving gives $c=-1/2$, $a=1/2$, $b=0$. So $T\propto \sqrt{\ell /g}$ — the standard pendulum result.

3. Converting units

To convert $g=9.8{\text{m/s}}^2$ into cm/s², multiply by $\frac{100\text{cm}}{1\text{m}}$:

Measurement of length

Length goes from atomic radii ($\sim 10^{-10}\text{m}$) all the way to galactic distances ($\sim 10^{22}\text{m}$). Different scales need different instruments. NEET focuses on three.

Vernier calliper

A vernier calliper measures length to the nearest $0.01\text{cm}$ (or finer). It uses a sliding vernier scale alongside the main scale. The smallest division it can measure is the least count:

If the main scale has 1 mm divisions and the vernier scale has 10 divisions matching 9 mm, then $1\text{VSD}=0.9\text{mm}$ and $\text{LC}=1-0.9=0.1\text{mm}=0.01\text{cm}$.

The full reading is:

where MSR is the main scale reading just before the zero of the vernier, and VSR is the number of the vernier division that lines up exactly with a main scale division.

Screw gauge (micrometer)

A screw gauge measures lengths to the nearest $0.001\text{cm}$. It works on the principle that a screw advances by a fixed pitch for every rotation.

For a typical screw gauge, pitch $=0.5\text{mm}$ and the circular scale has 50 divisions, giving $\text{LC}=0.5/50=0.01\text{mm}$. The reading formula is the same shape as the vernier calliper: $\text{Reading}=\text{MSR}+(\text{CSR}\times \text{LC})$ with CSR = circular scale reading.

Parallax method

For very large distances we use parallax. Look at a distant object from two points $A$ and $B$ separated by a known baseline $b$. The angle $\theta$ between the two sight lines is the parallax angle, and the distance is:

This is how astronomers measure the distance to nearby stars using the diameter of Earth's orbit as the baseline.

Measurement of mass

Masses on Earth are measured with a beam balance against standard masses, accurate to about $\sim 10^{-3}\text{g}$. For atomic masses we use the unified atomic mass unit:

It is defined as one-twelfth the mass of an unbound carbon-12 atom in its ground state. Particle masses in nuclear physics are expressed in MeV/c² via Einstein's relation $E=m{c}^2$.

Measurement of time

Time is measured by anything that repeats: a swinging pendulum, a quartz crystal, an atomic transition. The modern second is defined by the caesium-133 atomic clock, which is accurate to one part in $10^{13}$. For NEET, the pendulum result you need is:

This formula appears every year in some form, often inside an error-propagation question.

Errors in measurement

No measurement is exact. The deviation from the true value is the error. Errors come in two flavours.

Systematic errors

Repeatable, one-sided errors — they bias every reading in the same direction. Causes include zero error in the instrument, a poorly calibrated scale, parallax, and human habits. Systematic errors do not shrink with more readings; you must correct or remove them.

Random errors

Unpredictable scatter around the true value — they shift the reading equally in either direction. Caused by small uncontrollable fluctuations. Random errors do shrink with more readings, like $1/\sqrt{N}$.

Accuracy vs precision

Accuracy is how close a measurement is to the true value. Precision is how close repeated readings are to each other. You can be precise but inaccurate (a stuck instrument), or accurate but imprecise (a noisy instrument that averages out correctly), or both, or neither.

Mean and mean absolute error

Take $N$ readings $a_1,a_2,\dots ,a_N$ of a quantity. The mean is the best estimate of the true value:

The absolute error of the $i$th reading is $|\Delta a_i|=|a_i-\bar{a}|$. The mean absolute error is:

We then quote the result as $\bar{a}\pm \Delta \bar{a}$. The relative error is $\Delta \bar{a}/\bar{a}$ and the percentage error is $(\Delta \bar{a}/\bar{a})\times 100$.

Combination of errors

When measured quantities combine in a formula, their errors combine too. For NEET, four rules cover almost every problem.

Rule 1 — Sum or difference

If $Z=A\pm B$, then:

Absolute errors add up. This is the worst-case bound.

Rule 2 — Product

If $Z=AB$, then:

Relative errors add up.

Rule 3 — Quotient

If $Z=A/B$, the relative-error formula is the same as the product rule:

Rule 4 — Powers

If $Z=A^p{B}^q/C^r$, then:

Each exponent multiplies its relative error. Take $g=\frac{4{\pi }^2\ell }{T^2}$ as a NEET classic: the percentage error in $g$ is $\frac{\Delta \ell }{\ell }\times 100+2\frac{\Delta T}{T}\times 100$.

Worked NEET problems

Summary cheat sheet

• SI base units (7): m, kg, s, A, K, mol, cd. Memorise the table.
• Vernier: $\text{LC}=1\text{MSD}-1\text{VSD}$, reading $=\text{MSR}+(\text{VSR}\times \text{LC})$.
• Screw gauge: $\text{LC}=\text{Pitch}/N$, reading $=\text{MSR}+(\text{CSR}\times \text{LC})$.
• Sig figs: Addition keeps fewest decimal places. Multiplication keeps fewest sig figs.
• Error rules: Sum/difference → absolute errors add. Product/quotient → relative errors add. Powers → relative errors add weighted by the exponent.
• Common dimension twins: angular momentum and Planck's constant share $[M{L}^2{T}^{-1}]$. Energy and torque share $[M{L}^2{T}^{-2}]$ (but they are different physical quantities — torque is a vector, energy is a scalar).

Next: try the 12 interactive widgets for vernier, screw gauge, error propagation and dimensional analysis, or work through the 30+ NEET PYQs with full solutions. To time yourself, take the free 10-question mock test.