Units, Dimensions and Significant Figures for JEE – Full Basics in One Shot

Units, Dimensions and Significant Figures for JEE – Complete Guide with Solved Examples

Physics is the science of measurement, and every JEE aspirant must master the basics of units, dimensions, and significant figures before diving into mechanics or electromagnetism. This chapter may seem simple, but JEE regularly tests your understanding through dimensional analysis, error calculations, and numerical precision questions. In this post, we'll cover everything you need: fundamental and derived quantities, dimensional formulas, dimensional analysis tricks, and significant figure rules with plenty of solved examples.​

What Are Physical Quantities?

A physical quantity is anything that can be measured and expressed as a number with a unit. For example, the length of a table is 2 meters, the mass of a book is 500 grams, and the time taken for a reaction is 3.5 seconds. Every physical quantity has two parts: a magnitude (numerical value) and a unit.​

Physical quantities are classified into two types:

Fundamental quantities: These are independent quantities that cannot be expressed in terms of other quantities. The SI system defines seven fundamental quantities:​

• Length (meter, m)

• Mass (kilogram, kg)

• Time (second, s)

• Electric current (ampere, A)

• Temperature (kelvin, K)

• Amount of substance (mole, mol)

• Luminous intensity (candela, cd)

Derived quantities: These are expressed in terms of fundamental quantities. Examples include:​

• Velocity = distance/time (m/s)

• Force = mass × acceleration (kg·m/s²)

• Pressure = force/area (kg/m·s²)

• Energy = force × distance (kg·m²/s²)

Dimensions and Dimensional Formulas

The dimension of a physical quantity shows which fundamental quantities it depends on and to what power. We use square brackets to denote dimensions. The seven fundamental dimensions are:​

• Mass [M]

• Length [L]

• Time [T]

• Electric current [I]

• Temperature [K]

• Amount of substance [mol]

• Luminous intensity [cd]

For mechanics problems in JEE, we mainly use [M], [L], and [T].

Dimensional Formulas of Important Quantities

Quantity	Formula	Dimensional Formula
Velocity	distance/time	[M⁰L¹T⁻¹]
Acceleration	velocity/time	[M⁰L¹T⁻²]
Force	mass × acceleration	[M¹L¹T⁻²]
Work/Energy	force × distance	[M¹L²T⁻²]
Power	work/time	[M¹L²T⁻³]
Pressure	force/area	[M¹L⁻¹T⁻²]
Momentum	mass × velocity	[M¹L¹T⁻¹]
Impulse	force × time	[M¹L¹T⁻¹]
Density	mass/volume	[M¹L⁻³T⁰]

Important note: Some different quantities can have the same dimensions. For example, work, energy, and torque all have dimensions [M¹L²T⁻²], but they are physically different. Similarly, momentum and impulse both have [M¹L¹T⁻¹].

Three Major Uses of Dimensional Analysis

1. Checking the correctness of equations:

The principle of homogeneity states that in any correct physical equation, dimensions on both sides must be identical.​

Example 1: Check if the equation ${𝑣}^2={𝑢}^2+2𝑎𝑠$ is dimensionally correct.

Left side: $[{𝑣}^2]=[{𝐿}^1{𝑇}^{-1}{]}^2=[{𝐿}^2{𝑇}^{-2}]$

Right side: $[{𝑢}^2]=[{𝐿}^2{𝑇}^{-2}]$, and $[2𝑎𝑠]=[{𝐿}^1{𝑇}^{-2}][{𝐿}^1]=[{𝐿}^2{𝑇}^{-2}]$

Since both sides have the same dimensions, the equation is dimensionally correct.​

2. Converting units from one system to another:

If a quantity has dimensions $[{𝑀}^{𝑎}{𝐿}^{𝑏}{𝑇}^{𝑐}]$, and you know its value in one system, you can convert it to another system using:
${𝑛}_2{𝑢}_2={𝑛}_1{𝑢}_1{\left(\frac{{𝑀}_1}{{𝑀}_2}\right)}^{𝑎}{\left(\frac{{𝐿}_1}{{𝐿}_2}\right)}^{𝑏}{\left(\frac{{𝑇}_1}{{𝑇}_2}\right)}^{𝑐}$

Example 2: Convert 1 newton into dynes.

Force has dimensions [M¹L¹T⁻²].

• 1 N = 1 kg·m/s²

• 1 dyne = 1 g·cm/s²

$1\text{ N}=1\times {\left(\frac{1000}{1}\right)}^1{\left(\frac{100}{1}\right)}^1{\left(\frac{1}{1}\right)}^{-2}={10}^5\text{ dynes}$

3. Deriving relationships between physical quantities:

If you know which quantities a formula depends on, you can find the relationship (up to a dimensionless constant).​

Example 3: The time period $𝑇$ of a simple pendulum depends on length $𝑙$, mass $𝑚$, and acceleration due to gravity $𝑔$. Find the relation.

Assume $𝑇=𝑘\cdot {𝑙}^{𝑎}\cdot {𝑚}^{𝑏}\cdot {𝑔}^{𝑐}$, where $𝑘$ is dimensionless.

Dimensions: $[𝑇]=[𝐿{]}^{𝑎}[𝑀{]}^{𝑏}[𝐿{𝑇}^{-2}{]}^{𝑐}$

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

Comparing powers:

• For M: $𝑏=0$

• For L: $𝑎+𝑐=0$

• For T: $-2𝑐=1\Rightarrow 𝑐=-1\mathrm{/}2$

So $𝑎=1\mathrm{/}2$, and $𝑇=𝑘\sqrt{𝑙\mathrm{/}𝑔}$. The actual formula is $𝑇=2𝜋\sqrt{𝑙\mathrm{/}𝑔}$, where $2𝜋$ is the dimensionless constant dimensional analysis cannot find [].

Significant Figures – The Basics

When you measure something, the precision of your result depends on your instrument [][]. Significant figures tell us how many digits in a number are reliable.

Rules for Counting Significant Figures

1. All non-zero digits are significant [][].
   Example: 1234 has 4 significant figures.

2. Zeros between non-zero digits are significant [][].
   Example: 1002 has 4 significant figures.

3. Leading zeros (before the first non-zero digit) are NOT significant [][].
   Example: 0.0034 has 2 significant figures (3 and 4).

4. Trailing zeros after a decimal point ARE significant [][].
   Example: 2.300 has 4 significant figures.

5. Trailing zeros in a whole number without a decimal are ambiguous [].
   Example: 2500 could have 2, 3, or 4 significant figures depending on measurement. In JEE, assume 2 unless stated otherwise.

6. Exact numbers have infinite significant figures [].
   Example: 12 eggs in a dozen, $𝜋$ in formulas.

Practice Problems on Counting Significant Figures

• 0.00456: 3 significant figures (4, 5, 6)

• 2.030: 4 significant figures (2, 0, 3, 0)

• 500: 1 significant figure (unless context says otherwise)

• 500.0: 4 significant figures

• 6.022 × 10²³: 4 significant figures (6, 0, 2, 2)

Significant Figures in Calculations

Rule 1: Addition and Subtraction

The result should have as many decimal places as the number with the fewest decimal places in the data [][].

Example 4:
$12.11+18.0+1.012=31.122$

The number 18.0 has only 1 decimal place, so the answer is rounded to 1 decimal place: 31.1 [].

Rule 2: Multiplication and Division

The result should have as many significant figures as the number with the fewest significant figures in the data [][].

Example 5:
$2.5\times 3.42=8.55$

2.5 has 2 significant figures, 3.42 has 3. So the answer is rounded to 2 significant figures: 8.6 [].

Example 6:
$4.56\mathrm{/}1.4=\mathrm{3.257...}$

1.4 has 2 significant figures, so answer is 3.3 [].

Rule 3: Rounding Off

• If the digit after the last significant figure is less than 5, keep the last digit unchanged [][].

• If it is 5 or more, increase the last digit by 1 [][].

Example 7:
Round 2.346 to 3 significant figures: 2.35
Round 2.344 to 3 significant figures: 2.34

Common JEE Mistakes and Quick Tips

• Mistake 1: Treating dimensions and units as the same thing. Dimensions are qualitative ([M¹L¹T⁻²]), units are quantitative (newton, dyne) [].

• Mistake 2: Forgetting that dimensionless quantities (like strain, refractive index, angle) have no dimensions but can still have units (radian, steradian) [][].

• Mistake 3: Ignoring significant figures in numerical questions. JEE expects you to report answers with the correct number of significant figures based on given data [][].

• Mistake 4: Mixing up leading and trailing zeros. Remember: leading zeros never count, trailing zeros after a decimal always count [][].

• Quick Tip: Always write very large or very small numbers in scientific notation (e.g., $3.0\times {10}^8$) to avoid ambiguity in significant figures [].

• Quick Tip for dimensional analysis: If an equation has exponential, logarithmic, or trigonometric functions, the argument inside must be dimensionless []. For example, in ${𝑒}^{-𝑘𝑡}$, $𝑘𝑡$ must be dimensionless, so $𝑘$ has dimensions [T⁻¹].

---