FSc Notes Punjab Board: 1st Year Physics Measurements (Chapter 1)

Physics is an experimental science, and it stresses accurate measurement. Every measurement needs a number, a unit and an honest idea of how far you can trust it. Chapter 1 teaches exactly that.

A tailor with a fine, sharp tape and a tailor with a stretched, worn tape can both say "I measured this carefully". Only one claim is trustworthy. Physics gives you the tools to tell which.

This note follows our Physics 11 lessons from Introduction to Physics to Dimensions of Physical Quantities. They follow the new PECTAA Physics 11 book (2025-26), Chapter 1. Keep your own book open too.

What to know

Physics is the most fundamental branch of the physical sciences. The book calls it an experimental science that stresses accurate measurement. Its foundation is physical quantities like mass, length, time, velocity and force.

Physical quantities are of two kinds. Base quantities do not depend on other quantities. Examples: length, mass and time. Derived quantities depend on base quantities. Examples: velocity, acceleration and force. Measuring a quantity gives a number and a unit.

SI units. In 1960 an international committee set up the System International, the SI. It has two kinds of units:

• 7 base units for the 7 base quantities: length, mass, time, electric current, temperature, intensity of light and amount of substance. Examples: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), candela (cd), mole (mol).
• Derived units, built from the base units. Example: the pascal is kg m⁻¹ s⁻². The units of plane angle (radian, rad) and solid angle (steradian, sr) are also derived units. The book has listed them so since 1995.

Key points

Scientific notation. Keep exactly one non-zero digit before the decimal point. Then multiply by a power of ten.

• 134.7 = 1.347 × 10². The point moves 2 places left.
• 0.0023 = 2.3 × 10⁻³. The point moves 3 places right.
• 5.0 × 10⁴ cm = 5.0 × 10⁴ × 10⁻² m = 5.0 × 10² m. Add the powers: 4 + (−2) = 2.
• 1 km² = (10³ m)² = 10⁶ m². The power applies to the whole prefixed unit.

Writing units.

• Compound prefixes are not allowed. Write 1 pF, not 1 µµF.
• Unit names do not start with a capital letter, like newton and metre. Symbols named after scientists do: N, Pa, W.
• Write the prefix right before the unit with no space: mL, not m L.
• Use symbols, not short forms: A, not amp. Symbols have no plural: 100 mm.
• Use m s⁻¹, not m/s.

Significant figures. These are the digits you are sure of, plus one doubtful last digit.

• A zero between digits counts. 2.05 has 3.
• Leading zeros never count. 0.00467 has 3.
• Trailing zeros after a decimal point count. 7.4000 has 5. 3.570 has 4.
• 8.70 × 10⁴ has 3.
• A plain integer like 8,000 kg is unclear. If the scale reads to 10 kg, write 8.00 × 10³ kg. That shows 3 figures.

Rounding.

• Below 5: keep the last digit.
• Above 5: add one.
• Exactly 5 with nothing after it: round up only if the kept digit is odd. So 43.75 becomes 43.8. And 73.650 becomes 73.6. And 64.350 becomes 64.4.
• The book's other example: 58.8546 becomes 58.9. The first dropped digit is 8, so add one.

Uncertainty in measurement. Every instrument has a smallest division. This limits how well you can measure. So every measured value has an uncertainty.

• The absolute uncertainty is taken as one smallest division. A metre rule marked in millimetres gives 1 mm, or 0.1 cm.
• Fractional uncertainty = absolute uncertainty ÷ measured value.
• Percentage uncertainty = fractional uncertainty × 100.
• The book measures a book's edges at 10.0 cm and 33.5 cm. The length is 23.5 cm ± 0.1 cm.
• Significant figures only show uncertainty from reading the scale. The book adds that personal errors and hidden systematic errors can make the total uncertainty larger.

Combining uncertainties.

• Add or subtract values: add the absolute uncertainties. Example: 25.6 − 15.4 = 10.2, with 0.1 + 0.1 = 0.2 cm.
• Multiply or divide values: add the percentage uncertainties.
• For a power: multiply the percentage uncertainty by the power.

Precision and accuracy.

• Precision = the least count of the instrument. A metre rule is ± 0.1 cm. Vernier callipers are ± 0.01 cm.
• Accuracy depends on the percentage uncertainty = (uncertainty ÷ value) × 100.

Dimensions. Use [M], [L] and [T] in square brackets.

• Speed: [LT⁻¹]
• Force: [MLT⁻²]
• Work or energy: [ML²T⁻²]
• Pressure: [ML⁻¹T⁻²]

In a correct equation, both sides have the same dimensions. Pure numbers like 2, ½ and π have no dimensions.

Worked examples

Example 1. A length is 25.5 cm ± 0.1 cm. Find the percentage uncertainty.

1. Fractional uncertainty = 0.1 ÷ 25.5 = 0.004.
2. Percentage = 0.004 × 100 = 0.4%.

Example 2. A length is 0.45 cm ± 0.01 cm. Compare it with Example 1.

1. Precision: ± 0.01 cm is finer than ± 0.1 cm. So it is more precise.
2. Percentage: 0.01 ÷ 0.45 × 100 is about 2.2%.
3. 2.2% is bigger than 0.4%. So it is less accurate.

Example 3. Check the equation s = vᵢt + ½at².

1. [s] = [L]
2. [vᵢt] = [LT⁻¹][T] = [L]
3. [½at²] = [LT⁻²][T²] = [L]

Every term is [L]. The equation passes.

Common mistakes in the exam

1. Counting leading zeros as significant. They are only placeholders.
2. Thinking the more precise instrument is always more accurate. Accuracy depends on the size of the value too.
3. Subtracting uncertainties. Absolute uncertainties are always added, even when you subtract the values.
4. Giving 2π or ½ dimensions. Pure numbers have none.
5. Forgetting what the brackets mean. The book writes dimensions as [L], [M] and [T]. It writes speed as [LT⁻¹].
6. Saying "dimensionally correct" means "correct". The equation s = vᵢt + at² passes the check but is missing the ½.
7. Applying the power to the metre only. 1 km² is 10⁶ m², not 10³ m².

Exam technique

• Show every step of a percentage calculation. Write the fraction first, then multiply by 100.
• For dimension checks, write each term in its own line in brackets.
• When you round, say which digit you looked at.
• Write the answer with its uncertainty, like 23.5 cm ± 0.1 cm.
• Use scientific notation to show the exact number of significant figures.

Practice questions

1. How many significant figures does 0.00467 m have? Answer: 3.
2. Write 0.0023 s in scientific notation. Answer: 2.3 × 10⁻³ s.
3. Two readings are x₁ = 15.4 ± 0.1 cm and x₂ = 25.6 ± 0.1 cm. Find x₂ − x₁ with its uncertainty. Answer: 10.2 ± 0.2 cm. The absolute uncertainties add.
4. A length is 25.5 cm ± 0.1 cm. Find the percentage uncertainty. Answer: 0.1 ÷ 25.5 × 100 = 0.4%.
5. Find the dimensions of force using force = mass × acceleration. Answer: [M] × [LT⁻²] = [MLT⁻²].
6. Round 58.8546 to three significant figures. Answer: 58.9. The first dropped digit is 8, which is more than 5, so round up.
7. Convert 1 km² to m². Answer: (10³ m)² = 10⁶ m².

Quick revision

• SI: 7 base units. Radian and steradian are derived units.
• Percentage uncertainty = absolute uncertainty ÷ value × 100.
• Precision = least count. Accuracy = percentage uncertainty.
• Dimensions can disprove a formula, never prove it.

Study the full lessons: SI Units (/learn/fsc-1/physics/si-units), Scientific Notation and Writing Units (/learn/fsc-1/physics/scientific-notation-and-units), Significant Figures (/learn/fsc-1/physics/significant-figures), Errors and Uncertainties (/learn/fsc-1/physics/errors-and-uncertainties), Combining Uncertainties (/learn/fsc-1/physics/combining-uncertainties), Precision and Accuracy (/learn/fsc-1/physics/precision-and-accuracy) and Dimensions of Physical Quantities (/learn/fsc-1/physics/dimensions).

Ready to practise this? Continue in 1st Year Physics.