Unit 1: Physics and Measurement

Physics and Measurement is a scoring, formula-light unit. NEET typically asks 1–2 MCQs every year from this unit. Questions are almost always one of three flavours: (i) finding the dimensional formula of a quantity, (ii) propagating percentage errors in a composite formula, or (iii) counting significant figures / rounding. There is no calculus and very little algebra — so the unit is a near-guaranteed mark if the rules are memorised cleanly.

Because the questions are pattern-locked, the highest-yield activity is to drill the dimensional formulas table and the error-propagation rules until both are automatic.

Concept Map

• Physical quantities → fundamental (7) + derived
• Units → SI base units, supplementary units, practical units
• Dimensional analysis
  • Deduce relation between quantities
  • Check dimensional correctness
  • Convert units between systems
  • Limitations of the method
• Measurement & errors
  • Accuracy vs precision
  • Absolute, relative and percentage error
  • Combination of errors (sum, difference, product, quotient, power)
• Significant figures
  • Rules for counting
  • Rules for arithmetic
  • Rounding off
• Instruments: vernier calliper, screw gauge, least count

Topic 1: Units and the SI System

Sub-topic A: Physical Quantities

A physical quantity is anything that can be measured. Every measurement is reported as

A fundamental rule: if a measured length $L$ is written as $n_1 u_1$ in one system and $n_2 u_2$ in another, then $n_1 u_1 = n_2 u_2$, so a smaller unit gives a larger numerical value.

Sub-topic B: Fundamental and Derived Quantities

There are seven SI base quantities:

Supplementary units: radian (rad) for plane angle and steradian (sr) for solid angle. These are dimensionless.

Sub-topic C: Practical Units of Length, Mass and Time

NEET trap: 1 parsec $\approx 3.26$ light years, not 3.08 light years (which is parsec value in $10^{16}$ m). A direct factual MCQ.

Sub-topic D: SI Prefixes

Topic 2: Dimensional Analysis

Sub-topic A: Dimensional Formula and Equation

The dimensional formula of a quantity expresses it in terms of the base quantities. For velocity,

A dimensional equation is an equation that equates the symbol of the quantity to its dimensional formula, e.g. $[F] = [M L T^{-2}]$.

Sub-topic B: Standard Dimensional Formulas (memorise)

Sub-topic C: Quantities with the Same Dimensions

Identifying dimensional twins is a frequent NEET trap:

• Work, energy, torque, moment of force: $[M L^2 T^{-2}]$
• Pressure, stress, modulus of elasticity, energy density: $[M L^{-1} T^{-2}]$
• Velocity, speed of sound, escape velocity: $[L T^{-1}]$
• Angular momentum, Planck's constant: $[M L^2 T^{-1}]$
• Surface tension, surface energy/area, force gradient: $[M T^{-2}]$
• Frequency, angular velocity, velocity gradient, decay constant: $[T^{-1}]$
• Impulse, momentum: $[M L T^{-1}]$
• $RC$, $L/R$, $\sqrt{LC}$ all have dimension of time $[T]$
• $1/\sqrt{\mu_0 \varepsilon_0}$ has dimension of velocity $[L T^{-1}]$ (it equals $c$!)

Sub-topic D: Using Dimensions to Deduce a Formula

If a quantity $Q$ depends on others as $Q = k\,A^a B^b C^c$, equating dimensions on both sides gives $a, b, c$.

Example — Time period of a simple pendulum:

Assume $T \propto \ell^a g^b m^c$. Writing dimensions:

So $a + b = 0$, $-2b = 1$, $c = 0$. Solving: $b = -1/2$, $a = 1/2$, $c = 0$, giving $T = k\sqrt{\ell/g}$, with $k = 2\pi$ from experiment.

Sub-topic E: Checking Correctness of an Equation

The principle of homogeneity says every term in a physical equation must have the same dimensions. Use this to verify formulas. For example $s = ut + \tfrac{1}{2} a t^2$: $[L T^{-1}][T] = [L]$ and $[L T^{-2}][T^2] = [L]$, both equal $[L]$.

Caution: dimensional correctness is necessary but not sufficient. Equations like $s = ut + at^2$ and $s = ut + 2at^2$ are dimensionally identical but only one is right.

Sub-topic F: Conversion of Units

If a quantity has dimensions $[M^a L^b T^c]$ and numerical value $n_1$ in a system $\{M_1, L_1, T_1\}$, then in a new system $\{M_2, L_2, T_2\}$ its numerical value is

Sub-topic G: Limitations of Dimensional Analysis

• Cannot determine dimensionless constants (like the $2\pi$ above).
• Fails if the formula contains sums of terms with the same dimensions but different physics (e.g. $\tfrac{1}{2} mv^2 + mgh$).
• Cannot handle trigonometric, exponential or logarithmic functions — their arguments must be dimensionless.
• Cannot decide between scalar and vector relationships.

Topic 3: Errors in Measurement

Sub-topic A: Accuracy vs Precision

• Accuracy is how close a measurement is to the true value.
• Precision is how reproducible (close to each other) repeated measurements are.

A bullseye hit far from the centre but always in the same spot is precise but inaccurate. NEET often poses this as an assertion-reason.

Sub-topic B: Types of Errors

• Systematic errors: instrumental (zero error of vernier), imperfection (parallax), personal bias. Reduce by careful calibration.
• Random errors: unpredictable; reduce by averaging many readings.
• Gross errors: human mistakes; reread the scale.

Sub-topic C: Absolute, Relative, Percentage Error

Suppose $n$ readings $a_1, a_2, \dots, a_n$ of a quantity are taken with arithmetic mean

• Absolute error of the $i$-th reading: $\Delta a_i = \vert a_i - \bar a\vert$.
• Mean absolute error: $\Delta \bar a = \frac{1}{n} \sum \vert \Delta a_i\vert$.
• Relative error: $\Delta \bar a / \bar a$.
• Percentage error: $(\Delta \bar a / \bar a) \times 100\%$.

Sub-topic D: Propagation of Errors

For quantities $A$ and $B$ measured with absolute errors $\Delta A$ and $\Delta B$ respectively:

Sum or difference $Z = A \pm B$:

Product or quotient $Z = AB$ or $Z = A/B$:

Power $Z = A^p B^q / C^r$:

In percentage form,

Worked example — density: $\rho = m/V$ and $V = \tfrac{4}{3}\pi r^3$. If $\%\Delta m = 2\%$ and $\%\Delta r = 1\%$, then $\%\Delta V = 3\%$ and $\%\Delta \rho = 2\% + 3\% = 5\%$.

Sub-topic E: Least Count and Instrumental Error

For a vernier with $n$ vernier divisions matching $(n-1)$ main scale divisions of value $a$:

For a screw gauge with pitch $p$ and $N$ divisions on the head:

Zero error is added or subtracted to the observed reading depending on whether it is positive (zero of vernier to the right of main-scale zero) or negative (to the left).

Topic 4: Significant Figures

Sub-topic A: Counting Rules

1. All non-zero digits are significant.
2. Zeros between non-zero digits are significant. (e.g. 4007 has 4 sig figs.)
3. Leading zeros are not significant. (e.g. 0.0032 has 2 sig figs.)
4. Trailing zeros in a number with a decimal point are significant. (e.g. 4.300 has 4 sig figs.)
5. Trailing zeros in an integer without a decimal point are ambiguous; scientific notation removes ambiguity. (e.g. $4.3 \times 10^{3}$ vs $4.30 \times 10^{3}$.)

Sub-topic B: Arithmetic Rules

• Addition / subtraction: the result has as many decimal places as the input with the fewest decimal places.
• Multiplication / division: the result has as many significant figures as the input with the fewest significant figures.

Sub-topic C: Rounding Off

• If the digit to be dropped is < 5, round down. If > 5, round up.
• If it is exactly 5, round to the nearest even digit (banker's rounding).

Topic 5: Measurement Strategies (NCERT-style)

For lengths much larger than the metre stick (planetary distances) one uses the parallax method:

with $b$ the baseline and $\theta$ the parallax angle (in radians). For very small lengths (atomic sizes) one uses electron-microscope imaging. Masses of subatomic particles are inferred from $E = mc^2$ and reactor calibration.

NEET Pattern MCQ Tips

NEET tests this unit through these specific question shapes — recognise them on sight:

1. Direct dimension recall: "Which has the same dimensions as Planck's constant?" → angular momentum.
2. Error propagation: percentage error in a derived quantity given errors in measured quantities — apply the powers rule.
3. Significant figures: count sig figs in something like 0.003200, or add 23.27 + 1.5 + 0.026 and report the sum.
4. Assertion-Reason: typical Reason: "A dimensionally correct equation must be physically correct." The reason is false.
5. Conversion: convert a quantity between SI and CGS — usually power-of-10 manipulation.
6. Least count: deduce LC of vernier or screw gauge from numerical data.

Common Confusions and Traps

• "Light year" is a distance, not a time. (Trap!)
• The radian and steradian are not dimensionless because they are angles — actually, they are dimensionless: angle = arc/radius cancels $L/L$.
• Surface tension has dimensions $[M T^{-2}]$, same as spring constant per unit length and same as surface energy per unit area.
• Stefan's constant $\sigma$ involves $T^4$ — many students drop the $\Theta^{-4}$ factor.
• The combination $\sqrt{LC}$ has dimensions of time, not frequency.
• Gravitational constant $G$ has $M^{-1}$ in its formula — easy to miss the minus sign.
• $\varepsilon_0$ involves $A^2$, $\mu_0$ involves $A^{-2}$ — their product is $1/c^2$.

Quick Revision Card

• 7 SI base quantities; 2 supplementary (rad, sr).
• $[F] = [M L T^{-2}]$, $[E] = [M L^2 T^{-2}]$, $[P] = [M L^2 T^{-3}]$.
• Energy, work, torque all share $[M L^2 T^{-2}]$.
• Pressure, stress, modulus all share $[M L^{-1} T^{-2}]$.
• For $Z = A^p B^q / C^r$: $\%\Delta Z = p\%\Delta A + q\%\Delta B + r\%\Delta C$.
• Sum/difference of measurements → add absolute errors.
• Dimensional correctness is necessary, not sufficient.
• Vernier LC = $a/n$; screw gauge LC = pitch/head divisions.
• $1\ \text{ly} \approx 9.46 \times 10^{15}\ \text{m}$; $1\ \text{pc} \approx 3.26\ \text{ly}$.
• $1/\sqrt{\mu_0 \varepsilon_0} = c$ — dimensions of speed.

Worked NEET Examples

Example 1: Dimensions of energy density

Energy density $u$ is energy per unit volume.

• $[E] = [M L^2 T^{-2}]$.
• $[V] = [L^3]$.

So $[u] = [M L^2 T^{-2}]/[L^3] = [M L^{-1} T^{-2}]$.

This matches the dimensions of pressure — a useful cross-check often tested in NEET.

Example 2: Period of a pendulum on Jupiter

The dimensional formula $T = k\sqrt{\ell/g}$ tells us $T \propto 1/\sqrt g$. On Jupiter, $g_J \approx 2.5 g_E$. A pendulum that takes 2 s on Earth takes $2/\sqrt{2.5} \approx 1.26$ s on Jupiter.

Example 3: Percentage error in $g$ via pendulum

If $T = 2\pi\sqrt{\ell/g}$, then $g = 4\pi^2\ell/T^2$, so

If $\ell = 50.0$ cm is measured with $\Delta\ell = 0.1$ cm and $T = 1.42$ s with $\Delta T = 0.01$ s,

Example 4: Force-Length-Time as base quantities

If $F, L, T$ are the chosen base quantities, then since $F = ML T^{-2}$, mass has $[M] = [F L^{-1} T^2]$. Energy is $E = FL$, so $[E] = [FL]$. Power is $E/T$, so $[P] = [F L T^{-1}]$.

Example 5: Mean and percentage error from a data table

Five measurements of a length: 5.62, 5.58, 5.65, 5.60, 5.63 cm.

• Mean $\bar a = 5.616$ cm.
• Absolute deviations: 0.004, 0.036, 0.034, 0.016, 0.014.
• Mean absolute error $\approx 0.021$ cm.
• Relative error $\approx 0.0037$.
• Percentage error $\approx 0.37\%$.

Reported: $\bar a = (5.62 \pm 0.02)$ cm.

Additional Quick Examples — Identifying Dimensions

Identify the dimensions of the following constants by their defining equations:

• $h$ from $E = h\nu$: $[h] = [E]/[\nu] = [ML^2T^{-2}]/[T^{-1}] = [M L^2 T^{-1}]$.
• $\hbar = h/(2\pi)$ — same dimensions as $h$.
• $G$ from $F = Gm_1m_2/r^2$: $[G] = [F][L^2]/[M^2] = [M^{-1}L^3T^{-2}]$.
• Stefan's constant $\sigma$ from $P = \sigma A T^4$: $[\sigma] = [\text{power}]/[\text{area}][\Theta^4] = [M T^{-3}\Theta^{-4}]$.
• Coefficient of viscosity $\eta$ from $F = \eta A (dv/dy)$: $[\eta] = [F]/([L^2][T^{-1}]) = [M L^{-1} T^{-1}]$.
• Surface tension from $T = F/L$: $[T] = [M T^{-2}]$.
• Universal gas constant $R$ from $PV = nRT$: $[R] = [ML^{-1}T^{-2}][L^3]/([\text{mol}][K]) = [M L^2 T^{-2}\Theta^{-1}\text{mol}^{-1}]$.
• Boltzmann $k_B$: same as $R$ but without the $\text{mol}^{-1}$.
• Permittivity $\varepsilon_0$ from $F = q_1 q_2/(4\pi\varepsilon_0 r^2)$: $[\varepsilon_0] = [q^2]/([F][r^2]) = [A^2 T^2]/([M L T^{-2}][L^2]) = [M^{-1} L^{-3} T^4 A^2]$.

Significant Figures — Tabulated Cases

Measurement of Small and Large Lengths

• Echo method: sound or radar bounces off object; $d = vt/2$ where $t$ is the round-trip time.
• Triangulation/parallax for stars (within 100 parsecs).
• Spectroscopic parallax for farther stars (uses luminosity).
• Atomic dimensions: scanning tunnelling microscope or X-ray diffraction.
• Nuclear dimensions: scattering experiments (Rutherford-type).

NCERT-Mandated Constants Worth Memorising

Formula Sheet