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Units and Measurements

CBSE Class 11 Physics • Chapter 01 • Detailed Notes

Chapter Overview

This chapter covers the fundamental concepts of Units and Measurements. Ensure you understand the definitions and derivations thoroughly.

1.0 Fundamental Forces in Nature

1. Gravitational Force: Mutual attraction between masses. Weakest force but infinite range. ($F_g \propto m_1m_2/r^2$).

2. Electromagnetic Force: Force between charged particles. Stronger than gravity. Infinite range.

3. Strong Nuclear Force: Holds protons/neutrons in nucleus. Strongest force. Short range ($10^{-15}$ m). Charge independent.

4. Weak Nuclear Force: Involved in $\beta$-decay. Weak but stronger than gravity. Very short range ($10^{-16}$ m).

Relative Strength Order:

Strong Nuclear ($1$) > Electromagnetic ($10^{-2}$) > Weak Nuclear ($10^{-13}$) > Gravitational ($10^{-39}$).

1.1 Physical Quantities and Units

Physical Quantity: A quantity that can be measured and expressed in numerical value with appropriate unit.

Every physical quantity = Numerical value × Unit ($Q = nu$)

Example: Length of rod = 5 m (5 is numerical value, m is unit). Note that $n \propto 1/u$.

Classification of Physical Quantities

1. Fundamental (Base) Quantities: Quantities that cannot be expressed in terms of other quantities.

S.No.	Quantity	SI Unit	Symbol
1	Length	metre	m
2	Mass	kilogram	kg
3	Time	second	s
4	Electric Current	ampere	A
5	Temperature	kelvin	K
6	Amount of Substance	mole	mol
7	Luminous Intensity	candela	cd

Common SI Prefixes:

Power	Prefix	Symbol	Power	Prefix	Symbol
$10^{18}$	exa	E	$10^{-1}$	deci	d
$10^{15}$	peta	P	$10^{-2}$	centi	c
$10^{12}$	tera	T	$10^{-3}$	milli	m
$10^9$	giga	G	$10^{-6}$	micro	$\mu$
$10^6$	mega	M	$10^{-9}$	nano	n
$10^3$	kilo	k	$10^{-12}$	pico	p
$10^2$	hecto	h	$10^{-15}$	femto	f
$10^1$	deca	da	$10^{-18}$	atto	a

2. Derived Quantities: Quantities expressed in terms of base quantities.

Examples: Area ($L^2$), Velocity ($LT^{-1}$), Force ($MLT^{-2}$), Energy ($ML^2T^{-2}$).

1.2 Dimensional Analysis

Dimensions: Powers to which the base quantities are raised to represent a derived quantity.

Dimensional Formula: Expression showing how a physical quantity depends on base quantities.

Notation: $[Q] = [M^a L^b T^c]$.

Calculus Notation: Dimensions of differential coefficients and integrals:

$[dx/dt] = [L/T] = [LT^{-1}]$ (Velocity)
$[\int y dx] = [y][x]$ (Area under curve)

Examples of Dimensional Formulas:

Velocity: $v = \frac{\text{disp}}{\text{time}} = \frac{L}{T} \Rightarrow [M^0L^1T^{-1}]$
Acceleration: $a = \frac{\text{vel}}{\text{time}} = \frac{LT^{-1}}{T} \Rightarrow [M^0L^1T^{-2}]$
Force: $F = ma \Rightarrow [M][LT^{-2}] = [M^1L^1T^{-2}]$
Work/Energy: $W = F \cdot d \Rightarrow [MLT^{-2}][L] = [M^1L^2T^{-2}]$
Power: $P = \frac{W}{t} \Rightarrow [ML^2T^{-3}]$
Pressure: $p = \frac{F}{A} \Rightarrow [MLT^{-2}][L^{-2}] = [M^1L^{-1}T^{-2}]$

Dimensionless Quantities: Quantities with no dimensions (all powers = 0).

Pure numbers: 1, 2, $\pi$, $e$.
Ratios of same quantities: Strain ($\Delta L/L$), Angle (arc/radius), Relative Density.

Applications of Dimensional Analysis

1. Checking Dimensional Correctness

Principle of Homogeneity: In a correct equation, dimensions of all terms on both sides must be identical.

Example: Check $s = ut + \frac{1}{2}at^2$.

LHS: $[s] = [L]$

RHS: $[ut] = [LT^{-1}][T] = [L]$ and $[\frac{1}{2}at^2] = [LT^{-2}][T^2] = [L]$.

All terms have dimension $[L]$, so equation is dimensionally correct.

Worked Example 1

Q: Check dimensional correctness of: $v^2 = u^2 + 2as$

Solution:

LHS: $[v^2] = [LT^{-1}]^2 = [L^2T^{-2}]$
RHS Term 1: $[u^2] = [LT^{-1}]^2 = [L^2T^{-2}]$
RHS Term 2: $[2as] = [LT^{-2}][L] = [L^2T^{-2}]$

All terms match. Equation is dimensionally correct.

2. Deriving Relation Between Physical Quantities

Method: If quantity $Q$ depends on $a, b, c$, assume $Q = k a^x b^y c^z$. Equate dimensions to solve for $x, y, z$.

Example: Time period $T$ of pendulum depends on length $L$ and gravity $g$.

Step 1: $T = k L^x g^y$

Step 2: $[T] = [L]^x [LT^{-2}]^y = [L^{x+y} T^{-2y}]$

Step 3: Equate powers. For T: $1 = -2y \Rightarrow y = -1/2$. For L: $0 = x+y \Rightarrow x = -y = 1/2$.

Step 4: $T = k L^{1/2} g^{-1/2} = k \sqrt{\frac{L}{g}}$.

Worked Example 2

Q: Frequency ($f$) of string depends on length ($L$), tension ($T$), and mass per unit length ($\mu$). Derive formula.

Solution:

Assume $f = k L^a T^b \mu^c$. Dimensions: $[f]=[T^{-1}]$, $[T]=[MLT^{-2}]$ (Force), $[\mu]=[ML^{-1}]$.

$[T^{-1}] = [L]^a [MLT^{-2}]^b [ML^{-1}]^c = M^{b+c} L^{a+b-c} T^{-2b}$

Mass: $b+c=0 \Rightarrow c=-b$
Time: $-2b=-1 \Rightarrow b=1/2$. So $c=-1/2$.
Length: $a+b-c=0 \Rightarrow a = c-b = -1/2 - 1/2 = -1$.

Result: $f = k L^{-1} T^{1/2} \mu^{-1/2} = \frac{k}{L} \sqrt{\frac{T}{\mu}}$.

3. Converting Units

$$ n_2 = n_1 \left[\frac{M_1}{M_2}\right]^a \left[\frac{L_1}{L_2}\right]^b \left[\frac{T_1}{T_2}\right]^c $$

Worked Example 3

Q: Convert 1 Newton ($10^5$ dynes) verification.

Solution: $[F] = [MLT^{-2}]$. SI ($kg, m, s$) to CGS ($g, cm, s$).

$n_2 = 1 \left[\frac{1kg}{1g}\right]^1 \left[\frac{1m}{1cm}\right]^1 \left[\frac{1s}{1s}\right]^{-2} = 1 [1000][100][1] = 10^5$.

So, 1 N = $10^5$ dynes.

Limitations of Dimensional Analysis:

Cannot determine dimensionless constants (k).
Cannot fail for trigonometric/exponential functions.
Cannot distinguish scalars/vectors.
Cannot derive formulas with:
- More than 3 independent quantities (e.g., if $Q \propto a,b,c,d$).
- Sum/Difference of terms (e.g., $s = ut + \frac{1}{2}at^2$).

1.3 Measurement and Errors

Measurement: Comparison of unknown quantity with a known standard.

Error: Difference between true value and measured value ($E = A_m - A_t$).

Accuracy vs Precision

Accuracy: Closeness to true value.

Precision: Closeness of repeated measurements (Limit of resolution).

Example: True value = 10.0. Measurements: (10.1, 9.9) -> Accurate. (12.1, 12.0) -> Precise but not Accurate.

Types of Errors

1. Systematic Errors: Consistent direction (always + or -). Causes: Instrumental (zero error), Experimental technique, Environmental.

2. Random Errors: Irregular fluctuations. Reduced by averaging $n$ readings ($\text{Error} \propto 1/\sqrt{n}$).

3. Gross Errors: Carelessness (reading wrong scale).

Quantifying Errors

Let readings be $a_1, a_2, ..., a_n$.

Mean: $\bar{a} = \frac{\sum a_i}{n}$. (Best estimate of true value)

Absolute Error: $\Delta a_i = |a_i - \bar{a}|$.

Mean Absolute Error: $\Delta \bar{a} = \frac{\sum |\Delta a_i|}{n}$.

Relative Error: $\delta a = \frac{\Delta \bar{a}}{\bar{a}}$.

Percentage Error: $\% \text{Error} = \delta a \times 100\%$.

$$ \text{Result} = \bar{a} \pm \Delta \bar{a} $$

Worked Example 4

Q: Readings: 25.2, 25.4, 25.1, 25.3, 25.0 cm.

Solution:

Mean $\bar{a} = 25.2$ cm.

Errors: 0.0, 0.2, 0.1, 0.1, 0.2. Mean Abs Error $\Delta \bar{a} = 0.6/5 = 0.12$.

Result: $25.2 \pm 0.12$ cm.

% Error: $(0.12/25.2) \times 100 \approx 0.48\%$.

Combination of Errors

1. Sum/Difference ($Z = A \pm B$):

Max Absolute Error: $\Delta Z = \Delta A + \Delta B$. (Errors always ADD).

2. Product/Quotient ($Z = AB$ or $A/B$):

Max Relative Error: $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$.

3. Power ($Z = A^n$):

Relative Error: $\frac{\Delta Z}{Z} = n \frac{\Delta A}{A}$.

$$ Z = A^x B^y C^{-z} \Rightarrow \frac{\Delta Z}{Z} = x\frac{\Delta A}{A} +
                y\frac{\Delta B}{B} + z\frac{\Delta C}{C} $$

Worked Example 5

Q: Kinetic Energy $K = \frac{1}{2}mv^2$. $m=5.0 \pm 0.1$, $v=10.0 \pm 0.2$.

Solution: $\frac{\Delta K}{K} = \frac{\Delta m}{m} + 2\frac{\Delta v}{v}$.

$\frac{\Delta K}{K} = \frac{0.1}{5} + 2(\frac{0.2}{10}) = 0.02 + 0.04 = 0.06$.

$K = 0.5 \times 5 \times 100 = 250$ J. $\Delta K = 0.06 \times 250 = 15$.

Answer: $250 \pm 15$ J.

Significant Figures

Rules for Counting SF:

Non-zeros are significant (123 -> 3).
Zeros between non-zeros are significant (102 -> 3).
Leading zeros are NOT significant (0.005 -> 1).
Trailing zeros with decimal are significant (2.500 -> 4).
Powers of 10 (Exponential) are NOT significant ($1.2 \times 10^5 \to 2$ SF).

Rounding Off Rules:

If digit < 5, drop it. (2.34 $\to$ 2.3)
If digit > 5, increase preceding by 1. (2.36 $\to$ 2.4)
If digit = 5:
- If preceding is Even, leave unchanged. (2.45 $\to$ 2.4)
- If preceding is Odd, increase by 1. (2.35 $\to$ 2.4)

Arithmetic Operations:

Add/Sub: Result matches least decimal places. ($12.11 + 18.0 = 30.11 \to 30.1$)

Mul/Div: Result matches least significant figures. ($2.5 \times 1.25 = 3.125 \to 3.1$)

Order of Magnitude

Definition: Power of 10 closest to the magnitude of quantity. ($N = n \times 10^x$).

Rule: If $0.5 \le n < 5$, order of magnitude is $10^x$.

If $n \ge 5$, increase power by 1.

1.4 Precision Measuring Instruments

Vernier Calipers

Vernier Calipers: Instrument to measure internal/external dimensions and depths to 0.1 mm precision.

Components: Main Scale (MS), Vernier Scale (VS), Jaws, Depth bar.

Figure 1.1: Vernier Calipers Reading

Principle: 10 VSD = 9 MSD. (Using $1 \text{ MSD} = 1$ mm).

Least Count (LC): $1 \text{ MSD} - 1 \text{ VSD} = 1 - 0.9 = 0.1$ mm.

$$ LC = \frac{1 \text{ MSD}}{\text{No. of VSD divisions}} = \frac{1 \text{
                mm}}{10} = 0.1 \text{ mm} $$

Steps to Read:

MSR: Main scale reading before zero of VS.
VSR: Coinciding VS division $\times$ LC.
$\text{Total} = \text{MSR} + \text{VSR}$.

Zero Error:

Positive (+): Zero of VS is right of MS zero. (Subtract error).
Negative (-): Zero of VS is left of MS zero. (Add error).

Worked Example 8

Q: MSR = 3.2 cm. 4th VS division coincides. LC = 0.01 cm.

Solution: Total = $3.2 + (4 \times 0.01) = 3.24$ cm.

Screw Gauge (Micrometer)

Screw Gauge: Precision instrument (LC $\approx 0.01$ mm) based on screw principle.

Components: Sleeve (Main Scale), Thimble (Circular Scale), Ratchet.

Figure 1.2: Screw Gauge Reading

Principle: Based on screw motion.

Pitch: Distance moved in 1 rotation (e.g., 0.5 mm).

$$ \text{LC} = \frac{\text{Pitch}}{\text{Total Circular Divisions}} $$

Worked Example 9

Q: Pitch=1 mm, Div=100. MSR=3 mm. CSR=47. Zero Error=+0.03 mm.

Solution: LC = 1/100 = 0.01 mm.

Reading = $3 + (47 \times 0.01) = 3.47$ mm.

Corrected = $3.47 - 0.03 = \mathbf{3.44 \text{ mm}}$.

Spherometer (Brief Note)

Spherometer: Instrument to measure radius of curvature of spherical surfaces.

Principle: Based on screw principle.

$$ LC = \frac{\text{Pitch}}{\text{Number of Circular Divisions}} =
                \frac{P}{N} $$

🚀 MASTER TIP: Thoroughly practice the calculation of Least Count for both instruments. This is the most common source of error in practicals and numericals!