Physics and Measurement Notes – Complete Chapter

Physical Quantities & SI Units

Types of Physical Quantities

Fundamental Quantities — have their own dimension; cannot be expressed in terms of other quantities
Derived Quantities — expressed in terms of fundamental quantities; e.g. velocity, acceleration, force, work
Supplementary Quantities — have units but no dimension; e.g. plane angle (radian), solid angle (steradian)

S.No.	Fundamental Quantity	SI Unit	Symbol
1	Length	Metre	m
2	Mass	Kilogram	kg
3	Time	Second	s
4	Temperature	Kelvin	K
5	Amount of Substance	Mole	mol
6	Electric Current	Ampere	A
7	Luminous Intensity	Candela	cd

Special Units of Length

Astronomical Unit (AU) — average distance between Earth and Sun ≈ 1.5 × 10¹¹ m
Light Year — distance travelled by light in one year ≈ 9.46 × 10¹⁵ m
Parsec — distance at which 1 AU subtends an angle of 1 arc-second ≈ 3.08 × 10¹⁶ m

Unit Conversion — Key Formula

Since numerical value × unit = constant:

\[ n_1 u_1 = n_2 u_2 \]

where n = numerical value, u = unit. If unit increases → numerical value decreases.

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

where a, b, c are powers of M, L, T in the dimensional formula.

Dimensional Analysis & Principle of Homogeneity

Principle of Homogeneity

Physical quantities of the same dimension can be added, subtracted, and equated
Physical quantities of different dimensions can only be multiplied or divided — NOT added/subtracted/equated
e.g. Z = A + B is valid only if [Z] = [A] = [B]

e.g. v = at + b → [v] = [at] → [b] = LT⁻¹, [a] = LT⁻²

Both terms on RHS must have same dimension as LHS (velocity = LT⁻¹)

Applications of Dimensional Analysis

① Check correctness of a physical equation — both sides must have same dimensions
② Derive formula — e.g. time period of simple pendulum T depends on length l and g; use dimensional method to derive T = k√(l/g)
③ Convert units — from one system to another using n₁u₁ = n₂u₂
④ Find new unit — e.g. if unit of length = 2 m and unit of acceleration = 8 m/s², find unit of time in new system

⚡ Trick — Deriving Time in New Unit System

Given: new unit of length = 2 m, new unit of acceleration = 8 m/s²

[T] = [L]^{1/2}[a]^{-1/2} \quad \Rightarrow \quad T_{new} = \sqrt{\frac{L_{new}}{a_{new}}} = \sqrt{\frac{2}{8}} = \frac{1}{2} \text{ s}

Worked Examples

Q: In P = P₀e^−at², find dimension of a.

Exponent must be dimensionless → [at²] = 1 → [a][T²] = 1 → [a] = T⁻²

Ans: [a] = T⁻² ✓

Q: In y = A sin(ωt − kx), find dimensions of ω and k.

Angle must be dimensionless → [ωt] = 1 → [ω] = T⁻¹ | [kx] = 1 → [k] = L⁻¹

Ans: [ω] = T⁻¹, [k] = L⁻¹ ✓

Q: v = at + b/(t+c). Find dimension of b.

[v] = LT⁻¹; [b/(t+c)] = LT⁻¹ → [b] = LT⁻¹ × T = L (since [t+c] = T)

Ans: [b] = L ✓

Limitations of Dimensional Analysis

Cannot Do the Following

Cannot derive dimensionless proportional constants — e.g. constant k in T = k√(l/g)
Cannot derive dimensionless functions like sin θ, cos θ, tan θ, e^x etc.
Cannot distinguish between two quantities with the same dimensional formula — e.g. work and torque both have [ML²T⁻²]
Cannot derive formula if a quantity depends on two quantities of the same dimension
Works only when the quantity depends on a maximum of 3 physical quantities
Cannot derive formula which has sum or difference of physical quantities — e.g. v = u + at cannot be derived this way

Important Dimensional Formulae

Basic Quantities

Mass → [M] | Length → [L] | Time → [T] | Current → [A] | Temperature → [K]
Velocity → [LT⁻¹] | Acceleration → [LT⁻²]
Force → [MLT⁻²] | Momentum → [MLT⁻¹] | Impulse → [MLT⁻¹]
Energy / Work / Torque → [ML²T⁻²] | Power → [ML²T⁻³]
Pressure / Stress / Modulus of Elasticity → [ML⁻¹T⁻²]
Surface Tension / Spring Constant / Force Gradient → [MT⁻²]

Physical Quantity	Dim. Formula	Same Formula Quantities
Work / Energy / Torque	[ML²T⁻²]	Work, KE, PE, Heat, Torque
Power	[ML²T⁻³]	Power, VI, I²R, V²/R
Force / Weight	[MLT⁻²]	Force, weight, tension, thrust, friction
Momentum / Impulse	[MLT⁻¹]	Linear momentum, impulse
Angular Momentum / Planck's h	[ML²T⁻¹]	L = Iω, h = E/f
Time	[T]	Time period, RC, √LC, L/R
Angular velocity / Frequency	[T⁻¹]	ω, f, velocity gradient, decay constant

Extended Formulae — Derivation Table

Physical Quantity	Derivation	Dim. Formula
Gravitational Constant G	F = Gm₁m₂/r² → G = Fr²/m²	[M⁻¹L³T⁻²]
Planck's Constant h	E = hf → h = E/f	[ML²T⁻¹]
Boltzmann Constant k_B	E = k_BT → k_B = E/T	[ML²T⁻²K⁻¹]
Capacitance C	E = ½CV² → C = 2E/V²	[M⁻¹L⁻²T⁴A²]
Electric Resistance R	H = I²Rt → R = H/(I²t)	[ML²T⁻³A⁻²]
Self/Mutual Inductance L	E = ½LI² → L = 2E/I²	[ML²T⁻²A⁻²]
Magnetic Field B	F = BIl → B = F/(Il)	[MT⁻²A⁻¹]
Magnetic Flux φ	φ = BA	[ML²T⁻²A⁻¹]
Electric Field E	F = qE → E = F/q	[MLT⁻³A⁻¹]
Electric Potential V	V = W/q	[ML²T⁻³A⁻¹]
Permittivity ε₀	F = q²/(4πε₀r²)	[M⁻¹L⁻³T⁴A²]
Permeability μ₀	F = μ₀I₁I₂l/(2πr)	[MLT⁻²A⁻²]
Coefficient of Viscosity η	F = 6πηrv	[ML⁻¹T⁻¹]
Surface Tension S	F = Sl → S = F/l	[MT⁻²]
Thermal Conductivity K	dQ/dt = KA·dT/dx	[MLT⁻³K⁻¹]
Specific Heat Capacity s	Q = msΔT	[L²T⁻²K⁻¹]
Latent Heat L	Q = mL	[L²T⁻²]
Magnetic Moment	U = MB → M = U/B	[L²A] = [M⁰L²T⁰A¹]

Dimensionless Quantities (No Unit or Supplementary Unit)

Strain, Angle (radian), Solid Angle (steradian), Refractive Index, Relative Density
Poisson's Ratio, Reynolds Number, Coefficient of Friction, Electric Susceptibility
Relative Permittivity / Permeability, all trigonometric functions (sin, cos, tan), e^x, log x

Charge & Mass Density Formulae

Linear charge density λ = Q/l → [ATL⁻¹]
Areal charge density σ = Q/A → [ATL⁻²]
Volume charge density ρ = Q/V → [ATL⁻³]
Linear mass density λ_m = M/l → [ML⁻¹]
Area mass density σ_m = M/A → [ML⁻²]
Volume density (density) ρ = M/V → [ML⁻³]

Unit–Dimension Relationships — Key Statements

Important True / False — Must Remember

A unitless quantity may have dimension → FALSE
A dimensionless quantity may be unitless → TRUE (e.g. coefficient of friction)
A quantity having unit must have dimension → FALSE (angle has unit = radian but is dimensionless)
A quantity having dimension may not have unit → FALSE (dimension implies unit always)

Physical Equation vs Dimensional Equation

Equation physically correct → may or may not be dimensionally correct
Equation physically incorrect → must be dimensionally incorrect
Equation dimensionally correct → may or may not be physically correct
Equation dimensionally incorrect → must be physically incorrect

Types of Physical Constants

Dimensionless & Constant — pure numbers like π, 1, 2
Dimensional & Constant — G (gravitational constant), ε₀ (permittivity)
Dimensionless & Variable — coefficient of friction μ, angle θ
Dimensional & Variable — velocity, time, length, force

Significant Figures

Definition

Number of certain digits + one uncertain digit in a measurement. Indicates the precision of the instrument used.

Rules for Counting Significant Figures

Rule 1: All non-zero digits are significant → 44 m = 2 s.f.
Rule 2: All zeros between non-zero digits are significant → 405 m = 3 s.f.
Rule 3: Zeros on the left side of first non-zero digit are NOT significant → 0.043 = 2 s.f.
Rule 4: Trailing zeros after decimal point are significant → 0.06900 = 4 s.f.
Rule 5: Trailing zeros without decimal are NOT significant → 2000 g = 1 s.f., 40 cm = 1 s.f.
Rule 6: Exact numbers have infinite significant figures → 25 pens = ∞ s.f.
Rule 7: Powers of 10 are NOT counted → 2.303 × 10²³ = 4 s.f.

Number	S.F.	Reason
44 m	2	Both non-zero digits count
40 cm	1	Trailing zero without decimal not significant
405 m	3	Zero between non-zeros is significant
2000 g	1	Trailing zeros without decimal not significant
0.043 cm	2	Leading zeros not significant
30.043	5	All digits including captured zero
0.06900	4	Trailing zeros after decimal are significant
2.303 × 10²³	4	Power of 10 not counted

Arithmetic Operations with Significant Figures

Addition / Subtraction → final result to minimum decimal places of any operand
Multiplication / Division → final result to minimum significant figures of any operand
e.g. 4.2 × 1 = 4.2 → 1 s.f. (from "1") → final = 4

⚡ Rounding Off Rules

Digit to be removed < 5 → previous digit unchanged → 43.463 → 43.46
Digit to be removed > 5 → previous digit +1 → 84.328 → 84.33
Digit to be removed = 5 → previous digit unchanged if even → 43.465 → 43.46
Digit to be removed = 5 → previous digit +1 if odd → 43.415 → 43.42

Q: Number of significant figures in 0.06900?

Leading zeros not significant; trailing zeros after decimal ARE significant → 6, 9, 0, 0 = 4 s.f.

Ans: 4 significant figures ✓

Q: Result of 2.42 + 3.1 + 4.7 to correct significant figures?

Addition → min decimal places = 1 (from 3.1 and 4.7); 2.42 + 3.1 + 4.7 = 10.22 → round to 1 decimal = 10.2

Ans: 10.2 ✓

Accuracy and Precision

Definitions

Accuracy — closeness of measured value to the true value
Precision — extent (fineness) of measurement; depends on the least count of the instrument; smaller LC = more precise

Example — True value = 6.571 m

Measurements: (a) 6.52 m (b) 6.513 m (c) 6.61 m (d) 6.684 m

Most Accurate = (b) 6.513 m — closest to true value 6.571

Most Precise = (b) 6.513 m — maximum decimal places

Note: If true value is not given → most precise = most accurate

Error Analysis

Types of Errors

Systematic Error — consistently in same direction; due to faulty instrument or improper method; can be corrected
Random Error — due to unpredictable fluctuations; reduced by taking multiple readings and averaging
Gross Error / Blunder — due to carelessness of experimenter; e.g. reading wrong scale

Absolute Error: δa = |x_true − x_measured|

\overline{\delta a} = \frac{\delta a_1 + \delta a_2 + \cdots + \delta a_n}{n} \quad \text{(Mean Absolute Error)}

Relative Error = δa̅ / a_mean

Percentage Error = (δa̅ / a_mean) × 100%

Combination of Errors — Golden Rule: Errors Always Add

Sum: Y = A + B → δY = δA + δB
Difference: Y = A − B → δY = δA + δB
Product: Y = AB → δY/Y = δA/A + δB/B
Division: Y = A/B → δY/Y = δA/A + δB/B
Power: Y = A^pB^qC^r → δY/Y = p(δA/A) + q(δB/B) + r(δC/C)

⚡ Special Case — Parallel Resistance Error

If R_e = R₁R₂/(R₁+R₂), let Z = R₁ + R₂

\frac{\delta R_e}{R_e} = \frac{\delta R_1}{R_1} + \frac{\delta R_2}{R_2} + \frac{\delta R_1 + \delta R_2}{R_1 + R_2}

Q: X = A²B³. Errors in A and B are 2% and 1% respectively. Find % error in X.

δX/X × 100 = 2(δA/A × 100) + 3(δB/B × 100) = 2(2%) + 3(1%) = 4% + 3% = 7%

Ans: 7% ✓

Q: R = V/I, V = (100 ± 5) V, I = (10 ± 0.2) A. Find % error in R.

δR/R × 100 = δV/V × 100 + δI/I × 100 = 5/100 × 100 + 0.2/10 × 100 = 5% + 2% = 7%

Ans: 7% ✓

Measuring Instruments

Instrument	Least Count
Metre Scale	1 mm = 0.1 cm
Vernier Calliper	0.1 mm = 0.01 cm
Screw Gauge	0.01 mm = 0.001 cm
Stop Watch	0.1 s or 0.01 s

Vernier Calliper

Vernier Scale Principle

Has a Main Scale (MSD) and a Vernier Scale (VSD)
If n VSD coincides with (n−1) MSD: (n−1) MSD = n VSD
1 VSD = (n−1)/n MSD

\text{Least Count} = 1\,\text{MSD} - 1\,\text{VSD} = \frac{1}{n}\,\text{MSD}

Total Reading = MSR + (VSR × LC)

MSR = Main Scale Reading | VSR = number of coinciding Vernier division

⚡ Standard Example

If 10 VSD = 9 MSD (where 1 MSD = 1 mm):

LC = 1 MSD − 1 VSD = 1 − 9/10 = 0.1 mm = 0.01 cm

If MSR = 2.3 cm and 7th division coincides: Total = 2.3 + 7 × 0.01 = 2.37 cm

Screw Gauge

Screw Gauge Principle

Works on principle of screw and nut
Pitch = distance moved by screw in one complete rotation
Circular Scale = scale on the thimble, divided into equal divisions

Pitch = Distance moved on main scale / Number of rotations

Least Count = Pitch / Total divisions on circular scale

Total Reading = Linear scale reading + (Circular scale reading × LC)

Standard: Pitch = 1 mm, Circular divisions = 100 → LC = 0.01 mm

Zero Error in Instruments

Positive Zero Error — zero of Vernier is to the right of main scale zero when jaws closed → subtract from reading
Negative Zero Error — zero of Vernier is to the left of main scale zero → add to reading
Corrected Reading = Observed reading − Zero error (with sign)

Q: In a Vernier calliper, 20 VSD = 19 MSD (1 MSD = 1 mm). Find the least count.

1 VSD = 19/20 MSD; LC = 1 − 19/20 = 1/20 mm = 0.05 mm = 0.005 cm

Ans: LC = 0.05 mm ✓

Q: Screw gauge has pitch 0.5 mm and 50 divisions on circular scale. Linear = 2 mm, circular = 30. Find diameter.

LC = 0.5/50 = 0.01 mm; Total = 2 + 30 × 0.01 = 2 + 0.30 = 2.30 mm

Ans: Diameter = 2.30 mm ✓

Q: Vernier calliper has zero error = +0.02 cm. Observed reading = 3.14 cm. Find correct reading.

Corrected = Observed − Zero error = 3.14 − 0.02 = 3.12 cm

Ans: 3.12 cm ✓

Q: T = 2π√(l/g). Errors in l and g are 1% and 2% respectively. Find % error in T.

δT/T × 100 = ½(δl/l × 100) + ½(δg/g × 100) = ½(1) + ½(2) = 0.5 + 1.0 = 1.5%

Ans: 1.5% ✓

Q: Which is more precise — instrument with LC 0.001 cm or 0.01 cm?

Smaller least count = higher precision

Ans: Instrument with LC = 0.001 cm is more precise ✓

Physics & Measurement Notes