Units and Measurements JEE Notes PDF, Download Now

Units and Measurements is the first chapter of JEE Physics and the one every other chapter quietly depends on. It decides how you write a quantity, how you check a formula, how many digits you keep in an answer, and how much error your final result carries. These Units and Measurements JEE notes cover the full chapter, including SI units, dimensional analysis, significant figures, error propagation, unit conversion, and Vernier caliper and screw gauge numericals, along with important JEE questions for fast revision and exam-focused practice.

Units and Measurements JEE Notes: Important Concepts

Every measurement in physics needs two parts: a number and a unit. "5 metres" is meaningful; "5" alone is not. To keep measurements comparable worldwide, science uses the SI system (Système International), so that 1 metre means the same thing in every lab.

Physical Quantity

A physical quantity is anything that can be measured and expressed in units: length, mass, time, speed, force.

The Seven SI Base Units

The SI system fixes exactly seven base units. Every other unit in physics is a combination of these seven.

JEE tip: Almost every mechanics formula uses only the MKS trio: metre, kilogram and second. Get comfortable with these three and most of the chapter follows.

SI Prefixes

Very large and very small quantities are written with prefixes standing for powers of 10.

Dimensional Analysis and Unit Conversion

Dimensional analysis is the highest-yield topic in this chapter for JEE Main. Every physical quantity can be written as a combination of base quantities: mass (M), length (L), time (T), and where needed current (A) and temperature (K).

Dimensions are the powers to which the base quantities are raised to represent a quantity:

$$[Q] = [M^{a}L^{b}T^{c}]$$, where a, b, c are the dimensional exponents.

Worked example: dimensions of force $$\text{Force}=\text{mass}\times\text{acceleration}=\text{mass}\times\frac{\text{velocity}}{\text{time}}=\text{mass}\times\frac{\text{length}}{\text{time}^{2}}$$ $$\Rightarrow [F]=[M]\,[LT^{-2}]=[MLT^{-2}]$$ So force carries 1 power of mass, 1 of length and −2 of time.

Key Dimensional Formulas

The Three Uses of Dimensional Analysis

1. Checking whether a formula is correct. Both sides of a valid equation must carry identical dimensions.

Is $$v=u+at$$ dimensionally correct?

• LHS: $$[v]=[LT^{-1}]$$
• RHS: $$[u]+[a][t]=[LT^{-1}]+[LT^{-2}][T]=[LT^{-1}]+[LT^{-1}]$$ ✓

2. Deriving a relation between quantities. If you know what a quantity depends on, dimensional consistency gives you the powers.

Time period T of a pendulum depends on length l and gravity g:

• Let $$T=kl^{a}g^{b}\Rightarrow [T]=[L^{a}][L^{b}T^{-2b}]$$
• Compare $$L^{0}T^{1}=L^{a+b}T^{-2b}\Rightarrow a+b=0$$ and $$-2b=1$$
• So $$b=-\frac{1}{2}$$, $$a=\frac{1}{2}$$ ⇒ $$T=k\sqrt{\frac{l}{g}}$$
• The constant $$k=2\pi$$ cannot be obtained from dimensions.

3. Converting a quantity between unit systems. For a quantity of dimensions $$[M^{a}L^{b}T^{c}]$$:

$$n_{2}=n_{1}\left(\frac{M_{1}}{M_{2}}\right)^{a}\left(\frac{L_{1}}{L_{2}}\right)^{b}\left(\frac{T_{1}}{T_{2}}\right)^{c}$$ ($$1=\text{old system}$$, $$2=\text{new system}$$)

Worked example: convert 1 joule to CGS Energy has dimensions $$[ML^{2}T^{-2}]$$. $$n_{2}=1\times\left(\frac{1\,\mathrm{kg}}{1\,\mathrm{g}}\right)^{1}\times\left(\frac{1\,\mathrm{m}}{1\,\mathrm{cm}}\right)^{2}\times\left(\frac{1\,\mathrm{s}}{1\,\mathrm{s}}\right)^{-2}=1\times1000\times100^{2}\times1=10^{7}$$ ⇒ $$1\,\mathrm{J}=10^{7}\,\mathrm{erg}$$

Simple Unit Conversion

The physical quantity itself does not change when the unit changes; only the number adjusts:

$$n_{1}u_{1}=n_{2}u_{2}$$

Smaller unit ⇒ larger number, and vice versa.

Convert $$36\,\mathrm{km\,h^{-1}}$$ to $$\mathrm{m\,s^{-1}}$$: $$36\times\frac{1000\,\mathrm{m}}{3600\,\mathrm{s}}=10\,\mathrm{m\,s^{-1}}$$ Shortcut: $$\mathrm{km\,h^{-1}}\rightarrow\mathrm{m\,s^{-1}}:\times\frac{5}{18}$$. $$\mathrm{m\,s^{-1}}\rightarrow\mathrm{km\,h^{-1}}:\times\frac{18}{5}$$.

What Dimensional Analysis Cannot Do

• It cannot find dimensionless constants ($$\frac{1}{2}$$, $$2\pi$$, and similar).
• It cannot separate two quantities that share dimensions: work and torque are both $$[ML^{2}T^{-2}]$$.
• It fails for logarithmic, trigonometric and exponential terms (their arguments must be dimensionless).

Significant Figures and Errors in Measurement

An answer is only as precise as the instrument that produced it. Significant figures count the digits that are actually reliable: all digits known with certainty, plus one estimated digit.

Rules for Counting Significant Figures

Rounding Rules in Calculations

• Addition / subtraction: the answer keeps as many decimal places as the term with the fewest decimal places.
• Multiplication / division: the answer keeps as many significant figures as the term with the fewest significant figures.

Examples:

• $$4.56\times1.4=6.384\rightarrow6.4$$ (2 sig figs, limited by 1.4)
• $$12.11+0.3=12.41\rightarrow12.4$$ (1 decimal place, limited by 0.3)

Types of Error

Since the true value is usually unknown, we take several readings and treat the mean as the best estimate; each reading's error is its deviation from that mean.

Worked example: five readings of 2.63, 2.56, 2.42, 2.71, 2.80 m

• Mean: $$\bar{a}=\frac{2.63+2.56+2.42+2.71+2.80}{5}=2.624\,\mathrm{m}$$
• Absolute errors: $$0.006,\ 0.064,\ 0.204,\ 0.086,\ 0.176\ \mathrm{m}$$
• Mean absolute error: $$\overline{\Delta a}=\frac{0.006+0.064+0.204+0.086+0.176}{5}=0.1072\,\mathrm{m}\approx0.107\,\mathrm{m}$$
• Percentage error: $$\frac{0.1072}{2.624}\times100\%\approx4.09\%$$

Propagation of Errors

When measured values are combined, their errors combine too. The rule depends on the operation.

Addition or subtraction: absolute errors add. If $$Z=A\pm B$$, then $$\Delta Z=\Delta A+\Delta B$$

Multiplication, division or powers: relative errors add, weighted by the power. If $$Z=\frac{A^{p}B^{q}}{C^{r}}$$, then

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

Worked example $$Z=\frac{A^{2}B}{C^{1/2}}$$, with $$\frac{\Delta A}{A}=2\%$$, $$\frac{\Delta B}{B}=3\%$$, $$\frac{\Delta C}{C}=4\%$$. $$\frac{\Delta Z}{Z}=(2\times2\%)+(1\times3\%)+\left(\frac{1}{2}\times4\%\right)=4\%+3\%+2\%=9\%$$

JEE tip: The power becomes a multiplier of the percentage error, so the quantity raised to the highest power dominates the final error. When a question asks which measurement must be made most carefully, the answer is the one with the highest power in the formula.

Measuring Instruments: Vernier Caliper and Screw Gauge

Two instruments are directly examinable in JEE, and both follow the same three-step pattern: find the least count, take the observed reading, then correct for zero error.

Least Count (LC) is the smallest measurement an instrument can resolve.

Vernier Caliper

A Vernier caliper has a fixed main scale (usually in mm) and a sliding Vernier scale, which lets you read fractions of one main scale division.

• $$LC=\left|1\,MSD-1\,VSD\right|$$. For a standard direct Vernier where $$n\,VSD=(n-1)\,MSD$$, $$LC=\frac{1\,MSD}{n}$$, where $$n=\text{number of Vernier divisions}$$. Typically $$LC=\frac{1\,\mathrm{mm}}{10}=0.1\,\mathrm{mm}=0.01\,\mathrm{cm}$$
• $$\text{Reading}=MSR+(VSR\times LC)-\text{zero error}$$
  • $$MSR=\text{main-scale reading just before the Vernier zero}$$
  • $$VSR=\text{coinciding Vernier division number}$$

Worked example: $$MSR=3.2\,\mathrm{cm}$$, $$VSR=6$$, $$LC=0.01\,\mathrm{cm}$$, $$\text{zero error}=+0.02\,\mathrm{cm}$$ $$\text{Reading}=3.2+(6\times0.01)-0.02=3.24\,\mathrm{cm}$$

Screw Gauge (Micrometer)

A screw gauge measures down to 0.01 mm using a screw mechanism, with a main scale on the barrel and a circular scale on the rotating thimble. Pitch is the distance the screw advances in one full rotation (usually 0.5 mm or 1 mm).

• $$LC=\frac{\text{Pitch}}{\text{number of circular-scale divisions}}$$ Typically $$LC=\frac{0.5\,\mathrm{mm}}{50}=0.01\,\mathrm{mm}=0.001\,\mathrm{cm}$$
• $$\text{Reading}=MSR+(CSR\times LC)-\text{zero error}$$
  • $$MSR=\text{main-scale reading on the sleeve/barrel}$$
  • $$CSR=\text{circular-scale division aligned with the reference line}$$

Worked example: $$\text{Pitch}=0.5\,\mathrm{mm}$$, $$50\ \text{circular divisions}$$, $$MSR=3\,\mathrm{mm}$$, $$CSR=27$$, $$\text{zero error}=-0.03\,\mathrm{mm}$$

• $$LC=\frac{0.5}{50}=0.01\,\mathrm{mm}$$
• $$\text{Observed reading}=3+(27\times0.01)=3.27\,\mathrm{mm}$$
• $$\text{Corrected reading}=3.27-(-0.03)=3.30\,\mathrm{mm}$$

Zero Error and Its Correction

With the jaws (or faces) fully closed, the two zeros should coincide. If they don't, the instrument carries a zero error.

One formula covers both cases, provided you keep the sign:

$$\text{Corrected reading}=\text{observed reading}-\text{zero error}$$

Worked example: full Vernier problem A Vernier caliper has 10 Vernier divisions and $$1\,MSD=1\,\mathrm{mm}$$. With the jaws closed, the 4th Vernier division coincides with a main scale line. While measuring an object, $$MSR=2.3\,\mathrm{cm}$$ and the 7th Vernier division coincides.

1. $$LC=\frac{1\,\mathrm{mm}}{10}=0.1\,\mathrm{mm}=0.01\,\mathrm{cm}$$
2. Zero error: the Vernier zero sits ahead of the main scale zero, so it is positive. $$\text{Zero error}=4\times0.01=+0.04\,\mathrm{cm}$$
3. $$\text{Observed reading}=2.3+(7\times0.01)=2.37\,\mathrm{cm}$$
4. $$\text{Corrected reading}=2.37-0.04=2.33\,\mathrm{cm}$$

Important Formulas and Results at a Glance

JEE Important Points, Common Mistakes and Quick Revision

Points JEE Repeatedly Tests

• Dimensional correctness does not guarantee physical correctness, since a missing $$\frac{1}{2}$$ or $$2\pi$$ survives the dimensional check untouched.
• Quantities sharing dimensions cannot be told apart dimensionally: work and torque are both $$[ML^{2}T^{-2}]$$.
• Arguments of sin, cos, log and exponential functions are always dimensionless, which is a favourite way of framing "find the dimensions of a and b" questions.
• In error propagation, the term with the highest power decides where precision matters most.
• Vernier and screw gauge numericals almost always hide a zero error.

Common Mistakes to Avoid

1. Forgetting the zero error correction, the single biggest source of lost marks in instrument numericals.
2. Mishandling a negative zero error: subtracting a negative value means you add its magnitude.
3. Mixing the two rounding rules: decimal places for addition/subtraction, significant figures for multiplication/division.
4. Ignoring the power multiplier in $$\frac{\Delta Z}{Z}$$, or averaging percentage errors instead of adding them.
5. Mixing unit systems mid-problem: convert everything to MKS before substituting.
6. Miscounting significant figures in numbers like 0.0052 (2, not 4) and 3.200 (4, not 2).
7. Using dimensional analysis to derive a constant. It can only give the form of the relation.

Quick Revision Notes for Units and Measurements

• 7 SI base units: metre, kilogram, second, ampere, kelvin, mole, candela.
• Fundamental quantities stand alone; derived quantities are built from them.
• $$[Q] = [M^{a}L^{b}T^{c}]$$. Dimensional analysis checks formulas, derives relations and converts units.
• Dimensional analysis fails on numeric constants, on identical-dimension pairs, and on log/trig/exponential terms.
• Sig figs: non-zero digits and sandwiched zeros count; leading zeros never do; trailing zeros count only after a decimal point.
• Errors add as absolute values in sums and differences, and as relative values (weighted by powers) in products and quotients.
• Vernier $$LC=\left|1\,MSD-1\,VSD\right|$$, typically $$0.01\,\mathrm{cm}$$. Screw gauge $$LC=\frac{\text{pitch}}{\text{number of circular divisions}}$$, typically $$0.01\,\mathrm{mm}$$.
• $$\text{Corrected reading}=\text{observed reading}-\text{zero error}$$, sign included.
• Standard results worth memorising: $$1\,\mathrm{J}=10^{7}\,\mathrm{erg}$$; $$\mathrm{km\,h^{-1}}\rightarrow\mathrm{m\,s^{-1}}:\times\frac{5}{18}$$; $$T=2\pi\sqrt{\frac{l}{g}}$$.

This chapter usually contributes one or two straightforward questions in JEE Main, and every mark here is a formula-and-care mark rather than a concept-depth mark. Revise the tables above using the JEE formula sheet, practise five to ten Vernier and screw gauge numericals until zero error becomes automatic, and this becomes one of the most reliably scoring topics on the paper.