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Class 11 Physics Chapter 1 Units and Measurements — Formulas & Key Points

Units and Measurements is the bedrock of Physics, teaching you to quantify observations and ensure dimensional consistency. This chapter introduces the SI system, dimensional analysis, significant figures, and error calculation — skills you will use in every Physics problem through Class 11 and 12. This formula sheet organizes all key formulas, definitions, and rules into tables for quick revision, along with common pitfalls and solved examples.

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Key takeaways

✓Seven SI base units form the foundation: metre, kilogram, second, ampere, kelvin, mole, and candela.
✓Dimensional analysis checks formula correctness and derives relationships using [M], [L], [T] notation.
✓Significant figures reflect measurement precision; all non-zero digits and trapped zeros count.
✓Relative error equals absolute error divided by the true value, often expressed as percentage.
✓Dimensional formulas are unique for each physical quantity and independent of the system of units.
✓Least count determines the smallest measurement possible with an instrument.
✓Combination of errors follows specific rules for addition, multiplication, division, and powers.

SI Base Units and Derived Units — Complete Table

The International System of Units (SI) rests on seven base units. All other physical quantities are expressed as combinations of these base units, called derived units. Memorizing this table is non-negotiable for dimensional analysis and unit conversions. The 2019 redefinition of SI units now ties each base unit to a fundamental constant (like the speed of light or Planck constant), ensuring universal reproducibility. For CBSE exams, you must know the base unit symbol and the quantity it measures. Derived units like newton, joule, watt, and pascal appear frequently in mechanics and thermodynamics chapters.

Length: metre (m) — base unit
Mass: kilogram (kg) — base unit
Time: second (s) — base unit
Electric current: ampere (A) — base unit
Temperature: kelvin (K) — base unit
Amount of substance: mole (mol) — base unit
Luminous intensity: candela (cd) — base unit
Force: newton (N) = kg·m·s⁻² — derived
Energy: joule (J) = kg·m²·s⁻² — derived
Power: watt (W) = kg·m²·s⁻³ — derived
Pressure: pascal (Pa) = kg·m⁻¹·s⁻² — derived

Dimensional Formulas of Key Physical Quantities

Dimensional formulas express any physical quantity in terms of the base dimensions: mass [M], length [L], time [T], electric current [I], temperature [K], amount of substance [N], and luminous intensity [J]. For Class 11 mechanics, you primarily use [M], [L], and [T]. Dimensional analysis helps verify equations, derive relations between quantities, and convert units. Remember: angles, strain, and refractive index are dimensionless. Pure numbers have no dimensions. The table below lists the dimensional formulas you must memorize for NCERT Class 11 Physics Chapter 1 and beyond.

Dimensional Formula Table

Below is a quick-reference table of dimensional formulas for quantities covered in CBSE Class 11 Physics. Each formula shows the powers of M, L, and T. Use this table to solve dimensional analysis problems, verify equations, and derive new relationships. Notice that velocity and speed share the same dimension [LT⁻¹], while acceleration is [LT⁻²]. Energy, work, and torque all carry [ML²T⁻²], which is why they share the same SI unit, the joule. Memorize the dimensions of force, energy, power, and momentum — these appear in nearly every mechanics problem. Dimensionless quantities like strain, angle (radian), and coefficient of friction have [M⁰L⁰T⁰].

Velocity, Speed: [M⁰L¹T⁻¹]
Acceleration: [M⁰L¹T⁻²]
Force: [M¹L¹T⁻²]
Work, Energy, Torque: [M¹L²T⁻²]
Power: [M¹L²T⁻³]
Momentum: [M¹L¹T⁻¹]
Impulse: [M¹L¹T⁻¹]
Pressure, Stress: [M¹L⁻¹T⁻²]
Density: [M¹L⁻³T⁰]
Frequency: [M⁰L⁰T⁻¹]
Angular velocity: [M⁰L⁰T⁻¹]
Gravitational constant G: [M⁻¹L³T⁻²]
Planck constant h: [M¹L²T⁻¹]
Surface tension: [M¹L⁰T⁻²]
Coefficient of viscosity: [M¹L⁻¹T⁻¹]

Significant Figures — Rules and Operations

Significant figures indicate the precision of a measurement. All non-zero digits are significant. Zeros between non-zero digits are significant. Leading zeros (before the first non-zero digit) are not significant; they only indicate decimal position. Trailing zeros in a number with a decimal point are significant. For example, 0.00340 has three significant figures (3, 4, 0). When multiplying or dividing, the result should have as many significant figures as the measurement with the fewest. When adding or subtracting, the result should have as many decimal places as the measurement with the fewest decimal places. Rounding follows the rule: if the digit to be dropped is greater than 5, round up; if less than 5, round down; if exactly 5, round to the nearest even number.

123.45 has 5 significant figures.
0.00456 has 3 significant figures (4, 5, 6).
100.0 has 4 significant figures; 100 (no decimal) may have 1, 2, or 3 depending on context.
Multiplication/Division: result takes the least number of significant figures among operands.
Addition/Subtraction: result takes the least number of decimal places among operands.
Scientific notation clarifies significant figures: 1.20 × 10³ has 3 significant figures.

Error Analysis — Absolute, Relative, and Percentage Error

Measurements always carry some uncertainty. Absolute error is the magnitude of the difference between the measured value and the true value. Mean absolute error is the average of absolute errors from repeated measurements. Relative error is the ratio of mean absolute error to the mean value, giving a dimensionless measure of precision. Percentage error is relative error expressed as a percentage. For a set of measurements a₁, a₂,..., aₙ, the mean value is a_mean = (a₁ + a₂ +... + aₙ)/n. The absolute error in each measurement is Δaᵢ = |aᵢ − a_mean|. The mean absolute error is Δa_mean = (Δa₁ + Δa₂ +... + Δaₙ)/n. Relative error = Δa_mean / a_mean. Percentage error = (Δa_mean / a_mean) × 100%. In CBSE exams, error propagation in addition, multiplication, and powers is frequently tested.

Absolute error: Δa = |a_measured − a_true|
Mean absolute error: Δa_mean = Σ|aᵢ − a_mean| / n
Relative error: Δa_mean / a_mean
Percentage error: (Δa_mean / a_mean) × 100%
For sum Z = A + B: ΔZ = ΔA + ΔB
For product Z = A × B: ΔZ/Z = ΔA/A + ΔB/B
For quotient Z = A / B: ΔZ/Z = ΔA/A + ΔB/B
For power Z = Aⁿ: ΔZ/Z = n × (ΔA/A)

Conversion Factors and Common Prefixes

Unit conversion is a daily task in Physics problem-solving. You must be fluent in converting between units like kilometre and metre, gram and kilogram, hour and second. The NCERT Class 11 Physics textbook uses SI units, but many problems present data in CGS or other systems. Know the standard SI prefixes: kilo (10³), centi (10⁻²), milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹), and pico (10⁻¹²). For astronomy, remember the astronomical unit (1 AU ≈ 1.496 × 10¹¹ m), light year (≈ 9.46 × 10¹⁵ m), and parsec (≈ 3.08 × 10¹⁶ m). Common conversions: 1 litre = 10⁻³ m³, 1 hour = 3600 s, 1 atm = 1.013 × 10⁵ Pa. Practice converting compound units like km/h to m/s by multiplying by 5/18.

1 km = 10³ m; 1 cm = 10⁻² m; 1 mm = 10⁻³ m
1 kg = 10³ g; 1 g = 10⁻³ kg
1 hour = 3600 s; 1 minute = 60 s
1 litre = 10⁻³ m³ = 1000 cm³
1 km/h = (5/18) m/s
1 light year ≈ 9.46 × 10¹⁵ m
1 AU ≈ 1.496 × 10¹¹ m
1 eV = 1.6 × 10⁻¹⁹ J
1 calorie = 4.186 J

Principle of Homogeneity of Dimensions

The principle of homogeneity states that in any correct physical equation, the dimensions of all terms on both sides must be identical. This means you cannot add or equate quantities with different dimensions. For example, you cannot add length to time or equate force to energy. This principle is the foundation of dimensional analysis and allows you to check whether a given formula is potentially correct. However, dimensional correctness does not guarantee the formula is physically correct — dimensionless constants (like factors of 2 or π) cannot be determined by dimensional analysis alone. Use this principle to eliminate wrong options in multiple-choice questions and verify your derived formulas in numerical problems.

Every term in a physical equation must have the same dimensions.
You cannot add [L] to [T]; dimensions must match for addition or subtraction.
Exponents and arguments of logarithms, trigonometric functions must be dimensionless.
Dimensional analysis cannot determine dimensionless constants or proportionality factors.
Useful to derive relationships between physical quantities.
Helps convert units from one system to another.

Least Count and Vernier Callipers, Screw Gauge

The least count (LC) is the smallest measurement an instrument can reliably read. For a metre scale, LC = 1 mm. For a vernier callipers, LC = 1 MSD − 1 VSD, where MSD is the value of one main scale division and VSD is the value of one vernier scale division. Typically, LC of vernier callipers is 0.1 mm (or 0.01 cm). For a screw gauge (micrometer screw), LC = pitch / number of divisions on circular scale. A standard screw gauge has pitch 1 mm and 100 divisions, so LC = 0.01 mm. Zero error occurs when the instrument does not read zero when it should. Positive zero error means the reading is higher than true; subtract it. Negative zero error means the reading is lower; add the correction. Always report measurements as: measured value ± least count, to reflect precision and ensure dimensional consistency with significant figures rules.

Least count of metre scale: 1 mm
Least count of vernier callipers: typically 0.1 mm
Formula: LC (vernier) = 1 MSD − 1 VSD
Least count of screw gauge: pitch / circular scale divisions (typically 0.01 mm)
Zero error must be corrected: true reading = observed reading − zero error
Always record measurements with units and within the precision limit (least count).

Memory Tricks, Mnemonics, and Common Mistakes

Remembering SI base units: 'My Lazy Teacher Assigned Costly Kitchen Chores' stands for Metre, Kilogram (note: not gram), Second, Ampere, Kelvin, Mole, Candela. Dimensional formula for force: 'MLT minus two' — [M¹L¹T⁻²]. For energy: 'MLT minus two, but L squared' — [M¹L²T⁻²]. Common mistakes: confusing mass (kg) with weight (N); writing dimensions of angle as [L/L] = [L⁰] instead of recognizing it as dimensionless; forgetting that coefficients and pure numbers have no dimensions; mixing up absolute error (has units) with relative error (dimensionless). When rounding significant figures, students often forget the 'round to even' rule for exact 5. In error propagation, forgetting to add relative errors for both multiplication and division. Always write units in final answers — marks are often deducted for missing units in CBSE board exams.

Mnemonic for base units: My Lazy Teacher Assigned Costly Kitchen Chores (m, kg, s, A, K, mol, cd).
Angle is dimensionless; radian is a derived unit without dimension.
Relative error and percentage error are dimensionless.
In significant figures, 100 (no decimal) is ambiguous; write 1.00 × 10² for 3 sf.
Do not confuse least count with zero error.
In dimensional analysis, coefficients (like 2, π) cannot be found.
Always include units in final answers to avoid mark loss.

Three Solved Mini-Examples Applying Formulas

These worked examples demonstrate dimensional analysis, error calculation, and significant figures — the three pillars of Units and Measurements. Practice these methods until they become automatic. In board exams, dimensional analysis can save time by eliminating impossible answer choices. Error propagation is tested in practicals and theory both. Significant figures ensure your numerical answers reflect measurement precision. Each example below uses formulas from the tables above and follows NCERT problem-solving style. Study the step-by-step logic, then try similar problems from your NCERT exercises and previous year CBSE question papers. For deeper practice and instant doubt-solving, parents often turn to CBSETUTOR.ai, where students upload problem photos and receive step-by-step AI-tutor solutions 24×7 at just ₹999/month for any class from 6 to 12, with a 3-day free trial to start.

One-Glance Last-Minute Revision Box

Use this rapid-fire checklist 10 minutes before your exam. Seven SI base units: m, kg, s, A, K, mol, cd. Dimensional formula shortcuts: Force [MLT⁻²], Energy [ML²T⁻²], Power [ML²T⁻³], Pressure [ML⁻¹T⁻²]. Significant figures: non-zero always count; zeros between non-zeros count; leading zeros never count; trailing zeros in decimals count. Error rules: for sum, add absolute errors; for product/quotient, add relative errors; for power Aⁿ, multiply relative error by n. Least count of vernier callipers typically 0.1 mm; screw gauge typically 0.01 mm. Conversion: 1 km/h = 5/18 m/s. Dimensional analysis checks correctness but cannot find numerical constants. Homogeneity principle: all terms in an equation must have identical dimensions. Always write units in final answers. Practice these points until recall is instant, and you will breeze through Chapter 1 questions in both theory and numericals.

7 SI base units: m, kg, s, A, K, mol, cd
[Force] = MLT⁻², [Energy] = ML²T⁻², [Power] = ML²T⁻³, [Pressure] = ML⁻¹T⁻²
Significant figures: all non-zero, trapped zeros, trailing zeros in decimals
Multiplication/Division → least sf; Addition/Subtraction → least decimal places
Error: ΔZ/Z = ΔA/A + ΔB/B for Z = A×B or A/B; ΔZ/Z = n(ΔA/A) for Z = Aⁿ
Vernier LC ≈ 0.1 mm; Screw gauge LC ≈ 0.01 mm
1 km/h = 5/18 m/s; 1 litre = 10⁻³ m³
Homogeneity: every term in a valid equation has the same dimension
Dimensional analysis cannot determine pure numerical factors
Always include units in answers to avoid mark deduction

Frequently asked questions

What are the seven SI base units I must memorize for CBSE Class 11 Physics Chapter 1?+

The seven SI base units are metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity. Remember the mnemonic 'My Lazy Teacher Assigned Costly Kitchen Chores' to recall them in order.

How do I find the least count of a vernier callipers?+

Least count of vernier callipers = 1 main scale division (MSD) − 1 vernier scale division (VSD). Typically, 1 MSD = 1 mm and 10 VSD = 9 MSD, so 1 VSD = 0.9 mm. Hence LC = 1 mm − 0.9 mm = 0.1 mm or 0.01 cm. Always check the instrument specification for your practical exam.

What is the difference between absolute error and relative error?+

Absolute error is the magnitude of deviation from the true or mean value, measured in the same units as the quantity (e.g., ±0.5 cm). Relative error is the ratio of absolute error to the mean value, making it dimensionless (e.g., 0.5/50 = 0.01). Percentage error is relative error multiplied by 100, expressed as 1%.

Can dimensional analysis tell me the exact formula with numerical constants?+

No. Dimensional analysis can verify whether a formula is dimensionally consistent and help derive the relationship between quantities, but it cannot determine dimensionless constants like 2, π, or e. For example, dimensional analysis shows T ∝ √(L/g), but it cannot tell you T = 2π√(L/g).

How many significant figures are in the number 0.00340?+

The number 0.00340 has three significant figures: 3, 4, and the trailing 0. Leading zeros (0.00) are not significant; they only indicate the position of the decimal. The trailing zero after 4 is significant because it comes after the decimal point and a non-zero digit.

What is the rule for significant figures when adding or subtracting measurements?+

When adding or subtracting, the result should be rounded to the same number of decimal places as the measurement with the fewest decimal places. For example, 12.3 (1 decimal) + 0.456 (3 decimals) = 12.756, which rounds to 12.8 (1 decimal place).

How do I propagate errors in multiplication and division?+

For Z = A × B or Z = A / B, the relative error in Z is the sum of the relative errors in A and B: ΔZ/Z = ΔA/A + ΔB/B. Convert this to absolute error by multiplying by Z. This rule applies to products and quotients alike.

Why is angle considered dimensionless even though it is measured in radians?+

Angle (in radians) is defined as arc length divided by radius: θ = s/r. Since both s and r have dimension [L], the ratio is [L]/[L] = [L⁰] = dimensionless. Radian is a derived unit but carries no dimension. Similarly, strain and refractive index are dimensionless.

How do I convert km/h to m/s quickly?+

Multiply the speed in km/h by 5/18 to get m/s. For example, 90 km/h = 90 × (5/18) = 25 m/s. This works because 1 km = 1000 m and 1 hour = 3600 s, so 1 km/h = (1000 m)/(3600 s) = 5/18 m/s.

What is the principle of homogeneity of dimensions?+

The principle states that in any valid physical equation, every term must have the same dimensions. You cannot add or equate quantities with different dimensions. This allows you to check formulas for correctness. However, it does not prove correctness, as dimensionless constants may still be wrong.

Where can I get instant help solving Units and Measurements problems with step-by-step explanations?+

CBSETUTOR.ai offers 24×7 AI tutor support. Upload a photo of any problem from NCERT Class 11 Physics Chapter 1, and receive detailed, step-by-step solutions instantly. One flat fee of ₹999/month covers all classes (6–12), and you can start with a 3-day free trial — perfect for clearing doubts at midnight before exams.

Related resources

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