Units and Measurements

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Units and Measurements — Key Facts for BUET Fundamental quantities: length (m), mass (kg), time (s), electric current (A), temperature (K), amount of substance (mol), luminous intensity (cd) SI prefixes: kilo k (10³), mega M (10⁶), milli m (10⁻³), micro μ (10⁻⁶), nano n (10⁻⁹), pico p (10⁻¹²) Dimensional formula: express quantity in [M], [L], [T]; e.g. [velocity] = [M⁰L¹T⁻¹] Dimensional analysis: check equation validity, derive relations (cannot find dimensionless constants) ⚡ Exam tip: BUET physics always has 1–2 questions on dimensional analysis and error calculation!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Units and Measurements — BUET Study Guide

SI fundamental units:

• Length: metre (m) — distance light travels in 1/299,792,458 seconds
• Mass: kilogram (kg) — mass of platinum-iridium cylinder kept at Sèvres
• Time: second (s) — 9,192,631,770 periods of radiation from caesium-133
• Electric current: ampere (A) — current which produces 2×10⁻⁷ N/m between parallel wires
• Temperature: kelvin (K) — 1/273.16 of thermodynamic temperature of triple point of water
• Amount of substance: mole (mol) — contains Avogadro number of particles
• Luminous intensity: candela (cd) — based on frequency 540×10¹² Hz

Dimensional formula: Express physical quantity in terms of [M], [L], [T] Examples:

• Velocity: [v] = [M⁰L¹T⁻¹]
• Acceleration: [a] = [M⁰L¹T⁻²]
• Force: [F] = [MLT⁻²]
• Energy: [E] = [ML²T⁻²]

Uses of dimensional analysis:

• Check consistency of equations
• Derive relation between physical quantities
• Convert units from one system to another
• Find dimensions of constants in equations

Limitations of dimensional analysis:

• Cannot determine dimensionless constants (e.g., factor of ½ in KE = ½mv²)
• Cannot handle trigonometric, exponential, logarithmic functions
• Cannot distinguish quantities with same dimensions

Error analysis:

• Absolute error: Δx = |x_mean − x_true|
• Relative error: Δx/x
• Percentage error: (Δx/x) × 100%
• Error propagation: for z = x + y, Δz = Δx + Δy; for z = xy, Δz/z = Δx/x + Δy/y

Significant figures:

• Addition: round to fewest decimal places
• Multiplication: round to fewest significant figures
• Rounding: if digit dropped > 5, round up; if exactly 5 followed by zeros, round to nearest even

Measurement instruments:

• Vernier callipers: Least Count = 1 MSD − 1 VSD = 0.1 mm = 0.01 cm
• Screw gauge: Least Count = pitch/100 = 0.01 mm = 10 μm

Accuracy vs Precision: Accuracy = how close to true value Precision = how reproducible measurements are

• Key formula: [velocity] = [M⁰L¹T⁻¹]; Error propagation for product: Δz/z = Δx/x + Δy/y
• Common trap: Dimensional analysis cannot determine the constant ½ in kinetic energy formula
• Exam weight: 1–2 questions per exam (4–8 marks); always appears

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Units and Measurements — Comprehensive BUET Notes

Systems of units:

• CGS: centimetre-gram-second
• MKS: metre-kilogram-second
• FPS: foot-pound-second
• SI: modern international standard (adopted 1960)

Supplementary quantities:

• Plane angle: radian (rad)
• Solid angle: steradian (sr)

Dimensionless quantities:

• Pure numbers, ratios, angles
• sin θ, cos θ, tan θ where θ is dimensionless
• log, ln, exponential functions require dimensionless argument

Checking dimensional consistency: For equation to be dimensionally correct, both sides must have same dimensions Example: v² = u² + 2as [LHS] = [L²T⁻²]; [RHS] = [L²T⁻²] + [LT⁻²][L] = [L²T⁻²] + [L²T⁻²] = dimensionally consistent ✓

Unit conversion: Use conversion factors that equal 1 Example: 60 km/h = 60 × (1000 m)/(3600 s) = 16.67 m/s

Random vs systematic errors:

• Random error: reduced by averaging multiple readings
• Systematic error: instrumental zero error, calibration error

Zero error:

• Positive zero error: instrument reads positive when should read zero — subtract correction
• Negative zero error: instrument reads negative when should read zero — add correction

Least count error: Inherent uncertainty in measurement = LC of instrument Absolute error = ± LC/2 is common convention

Propagation of errors: For z = x + y or z = x − y: Δz = Δx + Δy (absolute errors add) For z = xy or z = x/y: Δz/z = Δx/x + Δy/y (relative errors add)

Error in sum: If A = x + y: ΔA = √(Δx² + Δy²) for random errors

Vernier callipers calculation: Main scale reading (MSR) = n × 1 mm Vernier scale division (VSD) coinciding = k Reading = MSR + k × LC = n + k × 0.1 mm

Screw gauge calculation: Main scale reading (MSR) = n × 0.5 mm Circular scale reading = k × LC = k × 0.01 mm Total reading = MSR + circular scale reading

Significant figure rules:

• Non-zero digits are always significant
• Leading zeros are not significant
• Captive zeros (between non-zeros) are significant
• Trailing zeros are significant only if decimal point is shown

Scientific notation: Express numbers as a × 10^n where 1 ≤ a < 10 Example: 0.0000543 = 5.43 × 10⁻⁵

Order of magnitude: Approximate value to nearest power of 10

Dimensional formula for derived quantities:

• Area: [L²]
• Volume: [L³]
• Density: [ML⁻³]
• Velocity: [LT⁻¹]
• Acceleration: [LT⁻²]
• Force: [MLT⁻²]
• Work/Energy: [ML²T⁻²]
• Power: [ML²T⁻³]
• Pressure: [ML⁻¹T⁻²]

Physical constants:

• Speed of light: c = 3 × 10⁸ m/s

• Gravitational constant: G = 6.67 × 10⁻¹¹ Nm²/kg²

• Planck’s constant: h = 6.626 × 10⁻³⁴ Js

• Boltzmann constant: k = 1.38 × 10⁻²³ J/K

• Remember: Fundamental dimensions are [M], [L], [T]; can’t find dimensionless constants from dimensional analysis; always check dimensions before solving

• Previous years: “Find dimension of gravitational constant G” [2023 BUET]; “Convert 72 km/h to m/s” [2024 BUET]; “Find percentage error in area if side error is 2%” [2024 BUET]

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📊 BUET Admission Exam Essentials

Detail	Value
Questions	Varies by year (~40-50 MCQ)
Time	Usually 2–3 hours
Marks	Varies by section
Subjects	Mathematics (highest weight), Physics, Chemistry
Negative	Usually no negative marking in BUET
Mode	Written + MCQ depending on year

🎯 High-Yield Topics for BUET Physics

• Mechanics (Laws of Motion, Work-Energy, Rotational) — very high weight
• Electrostatics and Current Electricity — very high weight
• Modern Physics (Photoelectric effect, Atoms) — high weight
• Heat and Thermodynamics — medium-high weight
• Optics (Reflection, Refraction) — medium weight

📝 Previous Year Question Patterns

• Units and Measurements: 1–2 questions per exam, 4–8 marks
• Common patterns: dimensional analysis, error calculation, unit conversion
• Weight: medium — appears in every exam

💡 Pro Tips

• Always check dimensional consistency before solving problems
• For error propagation, add absolute errors for sums/differences; add relative errors for products/quotients
• Remember: dimensionless quantities cannot be determined by dimensional analysis
• For unit conversion, use conversion factors (e.g., 1 km = 1000 m) systematically
• Vernier callipers LC = 1 MSD − 1 VSD = 0.1 mm; Screw gauge LC = pitch/100 = 0.01 mm

🔗 Official Resources

• BUET Official
• BUET Admission Portal

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