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Physics · CDS

PC01 — Units, Dimensions & Measurements

📖 PC01 · CDS General Science — Physics 🎯 CDS Level : High Priority

Every Physics problem begins here. Units tell us how much; dimensions tell us what kind. CDS consistently places 1–2 questions from this chapter — they are direct, factual, and completely scoring once you know the key dimensional formulae and conversion rules.

📌 CDS pattern in this chapter:
(1) Identify the dimensional formula of a given quantity (Force, Energy, Pressure, etc.);
(2) Which pair of quantities shares the same dimensional formula;
(3) Check whether a given equation is dimensionally correct;
(4) SI unit of a derived quantity;
(5) Least count of Vernier caliper / Screw gauge.

Topics at a Glance

① SI Units

7 base quantities; derived units

② Dimensional Analysis

Formulae; checking equations; conversion

③ Significant Figures

Rules for counting; rounding off

④ Errors in Measurement

Absolute, relative, percentage error

⑤ Vernier Caliper

Least count; zero error; reading

⑥ Screw Gauge

Pitch; least count; reading

1. International System of Units (SI)

1.1

7 Fundamental Quantities & Their SI Units

All physical measurements trace back to these seven base quantities

#	Fundamental 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

💡 Mnemonic for 7 base quantities: My Kind Sister Taught A Little Child — Mass, K (temperature), Substance (amount), Time, Ampere, Length, Candela.

⚡ Important Derived Units (Most CDS-tested)

Force: Newton (N) = kg m s⁻² Energy / Work: Joule (J) = kg m² s⁻² = N·m Power: Watt (W) = J/s = kg m² s⁻³ Pressure: Pascal (Pa) = N/m² = kg m⁻¹ s⁻² Frequency: Hertz (Hz) = s⁻¹ Electric Charge: Coulomb (C) = A·s Voltage: Volt (V) = J/C = kg m² s⁻³ A⁻¹ Resistance: Ohm (Ω) = V/A = kg m² s⁻³ A⁻² Magnetic Field: Tesla (T) = kg s⁻² A⁻¹

Joule and Newton-metre have the same unit but different physical meanings (energy vs torque). This is a repeated CDS trap.

2. Dimensional Analysis

2.1

Dimensional Formulae & Their Applications

Express every quantity in terms of M (mass), L (length), T (time) — and A (current) when needed

⚡ Master Table of Dimensional Formulae

Velocity: [LT⁻¹] Acceleration: [LT⁻²] Force: [MLT⁻²] Momentum: [MLT⁻¹] Work / Energy: [ML²T⁻²] Power: [ML²T⁻³] Pressure: [ML⁻¹T⁻²] Torque: [ML²T⁻²] Angular momentum: [ML²T⁻¹] Planck's const (h): [ML²T⁻¹] Surface tension: [MT⁻²] Viscosity: [ML⁻¹T⁻¹] Gravitational G: [M⁻¹L³T⁻²] Density: [ML⁻³] Frequency: [T⁻¹] Latent heat: [L²T⁻²] Strain: dimensionless Refractive index: dimensionless Specific heat (c): [L²T⁻²K⁻¹]

Same-dimension pairs (key CDS question):
Work and Torque → both [ML²T⁻²] (same dimension, different meaning)
Angular momentum and Planck's constant h → both [ML²T⁻¹]
Impulse and Momentum → both [MLT⁻¹]

Fig. 1 — Three applications of dimensional analysis. The critical limitation: it cannot determine pure numbers like ½ or 2π that appear in physical equations.

Worked Example — Dimensional Check

Q: Is the equation s = ut + ½at² dimensionally correct?

LHS: [s] = L

RHS: [ut] = LT⁻¹ × T = L | [½at²] = LT⁻² × T² = L

All three terms = [L]. ✓ Equation is dimensionally correct. (Note: ½ is dimensionless — dimensional analysis cannot verify it, but it confirms the structure.)

3. Significant Figures & Rounding Off

3.1

Rules for Counting Significant Figures

Every measured value has a precision — significant figures express it

📌 Rules for Counting Sig. Figs

All non-zero digits are significant: 1245 → 4 sig figs
Zeros between non-zero digits are significant: 1005 → 4 sig figs
Leading zeros are NOT significant: 0.0045 → 2 sig figs
Trailing zeros after decimal point ARE significant: 2.500 → 4 sig figs
Trailing zeros without decimal point — ambiguous: 1200 could be 2, 3, or 4

📌 Rounding Off Rules

If digit to be dropped < 5: round down (leave preceding digit unchanged)
If digit to be dropped > 5: round up
If digit = 5 and followed by non-zero: round up
If digit = exactly 5 with nothing after: round to even (banker's rounding)
In addition/subtraction: result has fewest decimal places
In multiplication/division: result has fewest significant figures

4. Errors in Measurement

4.1

Types of Error & Their Expression

⚡ Error Formulae

Absolute error: Δa = |a_measured − a_true| Mean absolute error: Δa_mean = (Δa₁ + Δa₂ + ... + Δaₙ) / n Relative error: Δa / a_mean (dimensionless) Percentage error: (Δa / a_mean) × 100% Error in sum/difference: ΔZ = ΔA + ΔB Error in product/quotient: ΔZ/Z = ΔA/A + ΔB/B Error in power: ΔZ/Z = n × (ΔA/A) (for Z = Aⁿ)

Systematic errors: consistent bias in one direction (faulty instrument, wrong technique). Random errors: unpredictable, vary in magnitude and direction — reduced by taking multiple readings.

5. Measuring Instruments — Vernier Caliper & Screw Gauge

5.1

Precision Measuring Tools

Vernier caliper reads to 0.1 mm; screw gauge reads to 0.01 mm

Fig. 2 — Vernier caliper reading. The main scale gives the whole mm reading (2 mm). The Vernier scale division that coincides with a main scale line (4th here) gives the fractional part (0.4 mm). Total = 2.4 mm.

⚡ Vernier Caliper & Screw Gauge — Key Formulae

Vernier Caliper: Least Count (LC) = 1 MSD − 1 VSD Standard Vernier: 1 MSD = 1 mm; 10 VSD = 9 MSD → LC = 0.1 mm Reading = Main Scale Reading + (Vernier Division coinciding × LC) Zero error: if Vernier zero is ahead of main zero → +ve error (subtract from reading) Screw Gauge (Micrometer): Pitch = distance moved per complete rotation of thimble (usually 0.5 mm or 1 mm) Least Count = Pitch / Number of circular scale divisions Standard screw gauge: Pitch = 0.5 mm; 50 circular divisions → LC = 0.01 mm Reading = PSR + (CSR × LC) (PSR = Pitch Scale Reading; CSR = Circular Scale Reading)

Screw gauge has smaller least count (0.01 mm) than Vernier caliper (0.1 mm) — it is more precise. Used to measure thickness of a wire, diameter of a sphere, etc.

📝 CDS PYQ

Units, Dimensions & Measurements

Q1. The dimensional formula of pressure is:

(a) MLT⁻²
(b) ML⁻¹T⁻²
(c) ML²T⁻²
(d) ML⁻²T⁻²

Answer: (b) ML⁻¹T⁻²
Pressure = Force / Area = [MLT⁻²] / [L²] = [ML⁻¹T⁻²]. SI unit = Pascal (Pa). This is the most directly repeated dimensional formula question in CDS. Pressure has the same dimension as energy density (energy per unit volume) — also ML⁻¹T⁻².

Q2. Which of the following pairs has the same dimensional formula?

(a) Force and Impulse
(b) Work and Torque
(c) Power and Pressure
(d) Momentum and Force

Answer: (b) Work and Torque — both [ML²T⁻²]
Work = Force × distance; Torque = Force × arm — both are force times length, giving [ML²T⁻²]. They have identical dimensions but completely different physical meanings. This distinction is a classic CDS and general science trap. Other option analysis: Impulse = [MLT⁻¹] ≠ Force [MLT⁻²]; Power = [ML²T⁻³] ≠ Pressure [ML⁻¹T⁻²].

Q3. The least count of a screw gauge with pitch 0.5 mm and 50 circular scale divisions is:

(a) 0.5 mm
(b) 0.1 mm
(c) 0.01 mm
(d) 0.001 mm

Answer: (c) 0.01 mm
Least Count = Pitch / Number of circular divisions = 0.5 mm / 50 = 0.01 mm. This is why a screw gauge is more precise than a Vernier caliper (LC = 0.1 mm). This formula is directly tested — know it cold.

Q4. The unit of Planck's constant h has the same dimension as:

(a) Momentum
(b) Energy
(c) Angular Momentum
(d) Force

Answer: (c) Angular Momentum — both [ML²T⁻¹]
From E = hf: h = E/f = [ML²T⁻²] / [T⁻¹] = [ML²T⁻¹]. Angular momentum L = Iω = [ML²][T⁻¹] = [ML²T⁻¹]. Both have the same dimension. This is a high-value CDS question — the connection between Planck's constant and angular momentum is conceptually deep and frequently tested.

📚 Formula Sheet — PC01

📍 Key Dimensional Formulae

Force: [MLT⁻²]
Energy/Work/Torque: [ML²T⁻²]
Power: [ML²T⁻³]
Momentum/Impulse: [MLT⁻¹]
Pressure: [ML⁻¹T⁻²]

📍 More Key Formulae

Angular momentum = h (Planck's): [ML²T⁻¹]
Surface tension: [MT⁻²]
Viscosity: [ML⁻¹T⁻¹]
G (gravitational): [M⁻¹L³T⁻²]
Strain, angle, RIndex: dimensionless

⚥ Measuring Instruments

Vernier LC = 1 MSD − 1 VSD = 0.1 mm
Screw gauge LC = Pitch / n = 0.01 mm
Reading = MS + VS × LC
+ve zero error → subtract from reading
−ve zero error → add to reading

📚 SI Derived Units

1 N = kg m s⁻²
1 J = N·m = kg m² s⁻²
1 W = J/s
1 Pa = N/m²
1 Hz = s⁻¹

⚡ Quick Revision — PC01

📍 7 Base SI Units

m (metre), kg (kilogram), s (second)
A (ampere), K (kelvin)
mol (mole), cd (candela)
Mnemonic: My Kind Sister Taught A Little Child

📍 Same-Dim Pairs

Work = Torque: [ML²T⁻²]
Ang. momentum = Planck's h: [ML²T⁻¹]
Impulse = Momentum: [MLT⁻¹]
Pressure = Energy density: [ML⁻¹T⁻²]

🚨 CDS Traps

Leading zeros NOT significant (0.005 → 1 sig fig)
Dimensional analysis cannot find constants (½, π)
Angle and strain are dimensionless
Screw gauge more precise than Vernier

📝 Practice Exercise

E-01

The dimension of gravitational constant G is:

(a) M⁻¹L³T⁻²
(b) MLT⁻²
(c) ML²T⁻²
(d) ML⁻¹T⁻²

E-02

A Vernier caliper has 10 Vernier divisions equal to 9 main scale divisions of 1 mm each. Its least count is:

(a) 0.01 mm
(b) 0.1 mm
(c) 1 mm
(d) 0.001 mm

E-03

Which of the following is dimensionless?

(a) Strain
(b) Stress
(c) Young's modulus
(d) Pressure

E-04

The number of significant figures in 0.00500 is:

(a) 6
(b) 5
(c) 3
(d) 1

Answers: E-01: (a) M⁻¹L³T⁻² [from F = GMm/r²; G = Fr²/Mm = MLT⁻²·L²/(M·M) = M⁻¹L³T⁻²] | E-02: (b) 0.1 mm [LC = 1 MSD − 1 VSD = 1 − 9/10 = 0.1 mm] | E-03: (a) Strain [= change in length / original length = L/L = dimensionless; Stress, Young's modulus, and Pressure all have dimensions] | E-04: (c) 3 [Leading zeros not significant; 5, 0, 0 after decimal → 3 sig figs]

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