Class 11 Physics: Units and Measurement | Physics Q&A

CBSE Class 11 › Unit I: Physical World

Units and Measurement

NCERT Chapter 1 • SI Units, Significant Figures & Dimensional Analysis

NCERT 2025–26 Unit I • ~4 Marks JEE Main • 1 Question

Illustration of a stopwatch, digital caliper, and prototype kilogram on graph paper with physics symbols. — Precision in measurement is the foundation of all physics.

1. Fundamental & Derived Units

Measurement involves comparing a physical quantity with an internationally accepted reference standard called a unit.

Fundamental (Base) Units: The units for the base quantities which are independent of each other (e.g., Length, Mass, Time).
Derived Units: Units expressed as combinations of base units (e.g., Velocity = Length/Time).

A complete set of these units is known as the System of Units.

Problem 1: Which of the following is a derived unit?
(a) Kilogram (b) Second (c) Newton (d) Candela

Show Answer

Answer: (c) Newton. It is derived from kg·m·s⁻².

Problem 2: Why can’t we use foot or pound as standard units in scientific work?

Show Answer

Because they are not universally reproducible or based on invariant constants. SI units are defined via fundamental constants (e.g., speed of light), ensuring global consistency.

Problem 3: Convert a pressure of 1 Pascal (Pa) into CGS units (dyne/cm²).

Show Answer

1 Pa = 1 N/m² = (10⁵ dyne) / (10² cm)²
= 10⁵ / 10⁴ dyne/cm²
= 10 dyne/cm².

2. The International System (SI)

The standard scheme used globally is the Système Internationale d’ Unites (SI). It consists of 7 Base Units and 2 Supplementary Units.

Base Quantity	Unit Name	Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Electric Current	Ampere	A
Temperature	Kelvin	K
Amount of Substance	Mole	mol
Luminous Intensity	Candela	cd

Modern SI Definitions (Since 2019):

Base units are now defined by fixing exact values of universal constants—e.g., the metre is defined using the speed of light (c = 299,792,458 m/s), and the kilogram via the Planck constant (h).

Supplementary Units (Plane & Solid Angle)

Besides the base units, SI defines two dimensionless units:

Plane Angle ( $d\theta$ ): Ratio of arc length to radius. Unit: Radian (rad).
Solid Angle ( $d\Omega$ ): Ratio of intercepted area to square of radius. Unit: Steradian (sr).

Diagram comparing 2D plane angle (radian) and 3D solid angle (steradian) geometries. — Visualizing Plane Angle ( $d\theta$ ) vs. Solid Angle ( $d\Omega$ ).

d\theta = \dfrac{ds}{r} \quad \text{(radian)}

d\Omega = \dfrac{dA}{r^2} \quad \text{(steradian)}

Problem 4: Are radian and steradian dimensionless? Justify.

Show Answer

Yes. Both are ratios of same-dimensional quantities (length/length and area/area²), so their dimensions cancel out.

3. Significant Figures

The reported result of a measurement is a number that includes all reliable digits plus the first uncertain digit. These are called Significant Figures.

Key Rule:

A change of units does NOT change the number of significant figures.
Example: $2.308$ cm and $0.02308$ m both have 4 significant figures.

Dartboard illustration showing the difference between high precision (grouped shots) and high accuracy (bulls-eye). — Accuracy vs. Precision: Significant figures help convey precision.

Rules for Counting Significant Figures

All non-zero digits are significant.
Zeros between two non-zero digits are significant (e.g., 2005 has 4).
Leading zeros (left of the first non-zero) are NOT significant (e.g., 0.007 has 1).
Trailing zeros without a decimal are NOT significant (e.g., 12300 has 3).
Trailing zeros WITH a decimal are significant (e.g., 3.500 has 4).

Scientific Notation: The Unambiguous Way

To eliminate confusion about trailing zeros, always use scientific notation:

Example: 4.700 m = 4.700 × 10³ mm → clearly has 4 significant figures.
The power of 10 does not affect significant digit count.

The exponent in a × 10^b gives the order of magnitude (e.g., Earth’s diameter ≈ 10⁷ m → order = 7).

Problem 5: How many significant figures in (a) 0.00450 kg, (b) 1200 m, (c) 1.200 × 10⁴ g?

Show Answer

(a) 3 (Leading zeros ignored, trailing zero counts), (b) 2 (Trailing zeros no decimal), (c) 4 (Scientific notation base counts).

Problem 6: Express 0.0006032 in scientific notation and state its significant figures.

Show Answer

6.032 × 10⁻⁴ → 4 significant figures.

4. Errors in Measurement

No measurement is perfect. The difference between the measured value and the true value is the error. We must also account for error propagation in calculations.

A. Arithmetic Operations with Significant Figures

Multiplication & Division: The final result should retain as many significant figures as the number with the least significant figures.

Example: Density calculation.
Mass = 4.237 g (4 sig figs)
Volume = 2.51 cm³ (3 sig figs)
Density = $4.237 / 2.51 = 1.688...$ $\rightarrow$ Round to 1.69 g cm⁻³ (3 sig figs).

Addition & Subtraction: The final result should retain as many decimal places as the number with the least decimal places.

Example: Summing masses.
$436.32$ (2 decimals) + $227.2$ (1 decimal) + $0.301$ (3 decimals) = $663.821$ .
Round to 663.8 g (1 decimal place).

B. Rounding Rules

If the digit to be dropped is 5 and no further digits follow:
• Round to the nearest even number for the preceding digit.
Examples:
– 2.745 → 2.74 (4 is even)
– 2.735 → 2.74 (3 is odd → round up)

Uncertainty Tip:

1. Addition/Subtraction: Absolute errors add up. ( $\Delta Z = \Delta A + \Delta B$ )
2. Multiplication/Division: Relative (%) errors add up. ( $\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}$ ).
3. Powers ( $Z = A^n$ ): Relative error is multiplied by $n$ . ( $\dfrac{\Delta Z}{Z} = n \dfrac{\Delta A}{A}$ ).

Problem 7: The length and breadth of a rectangle are $(5.7 \pm 0.1)$ cm and $(3.4 \pm 0.2)$ cm. Calculate area with error limits.

Show Answer

Area $A = l \times b = 5.7 \times 3.4 = 19.38$ cm².
Relative Error $\dfrac{\Delta A}{A} = \dfrac{\Delta l}{l} + \dfrac{\Delta b}{b} = \dfrac{0.1}{5.7} + \dfrac{0.2}{3.4}$
$= 0.0175 + 0.0588 = 0.0763$ .
Absolute Error $\Delta A = 0.0763 \times 19.38 = 1.48$ .
Rounding to significant figures (since errors are 1 sig fig approx):
Area = (19.4 ± 1.5) cm².

Problem 8: If percentage errors in measurement of Mass and Speed are 2% and 3% respectively, what is the error in Kinetic Energy?

Show Answer

$K = \dfrac{1}{2}mv^2$ .
% Error in $K$ = (% Error in $m$ ) + 2 × (% Error in $v$ )
$= 2\% + 2(3\%) = 2\% + 6\% =$ 8%.

Problem 9: Round the following to 3 significant digits: (a) 2.345 (b) 2.335

Show Answer

(a) 2.34 (Preceding digit 4 is even, drop 5).
(b) 2.34 (Preceding digit 3 is odd, round up).

5. Dimensional Analysis

The dimensions of a physical quantity are the powers to which the base quantities are raised to represent that quantity.

We use square brackets: Length $[L]$ , Mass $[M]$ , Time $[T]$ .

Flowchart connecting physical quantities to fundamental dimensions Mass [M], Length [L], and Time [T]. — Breaking down reality into basic dimensions.

Common Dimensions:
Velocity: $[LT^{-1}]$
Force: $[MLT^{-2}]$
Energy: $[ML^2T^{-2}]$

Note: Pure numbers, ratios of same quantities (e.g., angle = arc/radius), and mathematical functions (sin, log, exp) are dimensionless.

Principle of Homogeneity

Physical quantities can only be added or subtracted if they have the same dimensions. This allows us to check the consistency of equations.

Application: Deducing Relation for Pendulum Period 3 Marks

Step 1: Assumption
Let time period $T$ depend on length $l$ , mass $m$ , and gravity $g$ .
$T = k \, l^x \, g^y \, m^z$ .

Step 2: Dimensional Equation
Write dimensions for both sides:
$[T] = [L]^x [LT^{-2}]^y [M]^z$
$[M^0 L^0 T^1] = M^z L^{x+y} T^{-2y}$ .

Step 3: Equating Powers
$z = 0$ (Independent of mass)
$-2y = 1 \Rightarrow y = -1/2$
$x + y = 0 \Rightarrow x = 1/2$ .

Step 4: Final Formula
$T = k \, l^{1/2} \, g^{-1/2}$
$T = k \sqrt{\dfrac{l}{g}}$ .

Problem 10: Check dimensional consistency of $v = u + at$ .

Show Answer

All terms have dimension [LT⁻¹]. Equation is dimensionally consistent.

Problem 11: A dimensionless quantity is made from $e, \varepsilon_0, h, c$ . Find exponents in $e^\alpha \varepsilon_0^\beta h^\gamma c^\delta$ .

Show Answer

Using dimensional analysis: $(\alpha, \beta, \gamma, \delta) = (2, -1, -1, -1)$ . This gives the fine structure constant $\alpha = \dfrac{e^2}{4\pi\varepsilon_0 \hbar c}$ .

Problem 12: The centripetal force $F$ depends on mass $m$ , velocity $v$ , and radius $r$ . Derive the formula.

Show Answer

Let $F = k m^a v^b r^c$ .
$[MLT^{-2}] = [M]^a [LT^{-1}]^b [L]^c = [M^a L^{b+c} T^{-b}]$ .
Equating powers: $a=1$ , $-b=-2 \Rightarrow b=2$ , $b+c=1 \Rightarrow 2+c=1 \Rightarrow c=-1$ .
Formula: $F = k \dfrac{mv^2}{r}$ .

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