Introduction: measurement, units and systems of units
At a vegetable stall in Pune the seller drops tomatoes onto the pan of a balance until it settles at 1 kg.
Down the road a tailor runs a tape round your shoulders and writes 17 inches in a notebook.
At a petrol pump the meter clicks up in litres, and your grandmother may still talk about a gold chain of "two tolas".
Each of these people is doing the same thing: comparing an amount of something with an agreed amount of the same kind of thing, and counting how many of those agreed amounts it contains.
That act of comparison is measurement, and the agreed amount is a unit.
Physics rests on it.
A statement such as "the stone fell fast" cannot be tested, argued about or used to design a bridge.
"The stone fell 20 m in 2.0 s" can.
Why physics needs measurement
Physics tries to find rules that hold everywhere: in a school lab in Kochi, in a satellite factory in Bengaluru and on the Moon.
A rule is only useful if two people in different places can test it and get the same answer.
That is possible only when they measure the same quantities in the same agreed units.
Measurement also lets you compare.
Without numbers you can say a cricket ball is heavier than a tennis ball, but not by how much.
With measurement you can say the cricket ball is about three times heavier, and a calculation can then predict how far each will travel when hit with the same bat speed.
Finally, measurement keeps physics honest.
A theory predicts a number; an experiment measures it.
If the two disagree by more than the uncertainty of the measurement, the theory is wrong.
Much of this unit is about stating that uncertainty properly.
A measured value is a number with a unit
Every measured value has two parts written together: a number that says how many units the quantity contains, and the unit itself.
"The length of the bench is 2.5" means nothing.
2.5 m and 2.5 cm are very different benches.
If Q is the quantity, n the number and u the unit, then
The quantity Q is what it is; it does not change because you choose another unit.
So if you use a smaller unit, you need a larger number, and the other way round:
A bench 2.5 m long is also 250 cm long: the centimetre is a hundred times smaller than the metre, so the number is a hundred times larger.
This simple rule is how every unit conversion in this course works, from km h⁻¹ to m s⁻¹ in kinematics to joules and ergs in dimensional analysis.
Base quantities and derived quantities
There are hundreds of physical quantities: length, area, speed, force, pressure, energy, power, charge, and many more.
If every one of them needed its own independently kept standard, measurement would be chaos.
Luckily, quantities are linked by equations.
Speed is distance divided by time.
Area is length times length.
Force is mass times acceleration.
So once you have agreed units for a few quantities, the units of all the others follow from those equations.
- The few quantities chosen as the starting point are called base (or fundamental) quantities. Their units, chosen by international agreement, are the base units.
- Every other quantity is a derived quantity, and its unit, built by multiplying and dividing base units, is a derived unit.
For example, speed is length divided by time, so its unit is metre per second, written m s⁻¹.
Density is mass divided by volume, so its unit is kilogram per cubic metre, kg m⁻³.
Force is mass times acceleration, so its unit is kg m s⁻².
That combination is used so often that it has its own name, the newton (N), but the name is only a short label for kg m s⁻²; the newton is still a derived unit.
Which quantities are chosen as base quantities is partly a matter of convenience.
The International System of Units (next section) uses seven: length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity.
Speed could in principle have been chosen as a base quantity instead of length, but then length would have become a derived quantity.
What matters is that the base quantities are independent of one another (none can be written in terms of the others) and that together they are enough to build every other quantity.
What makes a good standard
A unit is defined by a standard: a reference that fixes exactly how big one unit is.
A good standard is
- well defined, so there is no argument about what it is;
- accessible and reproducible, so laboratories anywhere can make or check copies;
- unchanging with time, temperature and place;
- internationally accepted, so a kilogram in Delhi is a kilogram in Tokyo.
Older standards often failed one of these tests.
The length of a "foot" or a "hand" depended on whose foot or hand.
For more than a century the kilogram was the mass of one particular metal cylinder kept near Paris.
India's own copy of it, National Prototype of the Kilogram No. 57, arrived in 1958 and is kept at the CSIR-National Physical Laboratory in New Delhi, which looks after the country's national standards of measurement.
The picture below shows what such a standard looks like: a small metal cylinder kept under glass, handled as little as possible.
Comparisons over the years showed that the cylinder and its copies were drifting apart by tiny amounts.
A standard that might change is not good enough for modern science, so the base units are now defined through constants of nature such as the speed of light and the Planck constant, which are the same everywhere and for ever.
You will see how in the next section.
Systems of units
A complete set of units, the base units together with all the derived units built from them, is called a system of units.
Different countries once used different systems: the centimetre-gram-second (CGS) system, the British foot-pound-second (FPS) system, and the metre-kilogram-second (MKS) system.
Today science, industry and trade use one system, the SI, which is the subject of the next section.
Using one system matters in practice.
A value written in one system and read as if it were in another can wreck an engineering project, because the numbers look sensible but describe quantities many times too big or too small.
Writing the unit with every value, every time, is the simplest protection.
Connecting it
- Class X: you already used m s⁻¹, N, J, W and Ω without asking where they came from. This unit explains how they are built from a few base units.
- Next sections: the SI and its base units; significant figures and uncertainty (how sure a measured number is); then dimensions, which turn the idea of "derived from base quantities" into a powerful checking tool.
- Later units: every numerical answer in kinematics, laws of motion, gravitation, fluids, heat and waves ends with a unit. A missing or wrong unit is the most common way to throw away an otherwise correct final value point.
- JEE goes further into the history of the 2019 redefinitions and into measurement techniques; for your school examination you need the ideas in this unit, not the experimental details.
How this unit is examined
In Class XI there is no board examination.
Your school sets and marks the annual examination, using CBSE's curriculum and its question paper design, which is the same design as the Class XII board paper.
Chapters 1 to 4 (units and measurements, the two kinematics chapters and the laws of motion) share 23 of the 70 theory marks, so this chapter usually brings a few 1-mark items (MCQ and Assertion-Reason), a 2- or 3-mark item on significant figures, errors or dimensions, and sometimes part of a long answer.
Instrument readings (vernier callipers, screw gauge, spherometer) also come up in the practical examination and its viva.
- The theory paper has five sections: A (16 one-mark items: 12 MCQs and 4 Assertion-Reason), B (five 2-mark items), C (seven 3-mark items), D (two 4-mark case studies) and E (three 5-mark long answers). All questions are compulsory, with internal choice in some.
- Answers earn value points of ½ or 1 mark. Write each point as a separate line.
- In a numerical, the formula earns ½, the substitution ½, and the answer with its unit ½. The unit belongs to that final value point, so always write it.
- A correct method by another route earns full marks, and an error is penalised only once: a wrong value carried correctly into the next step does not lose marks again.
- No calculator and no formula sheet: every definition, rule and formula in these notes is recalled, and any constant you need is given in the question.
- In an internal choice ("(a) ... OR (b) ..."), answer one alternative only, fully.
- Correct physics in your own words earns the marks, except for definitions and laws, where the textbook wording is safest.
Where an entry gives a sentence in italics, that sentence is the textbook wording to learn for a definition question.
- Unit: the agreed reference amount with which a quantity is compared. Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit.
- Result of a measurement: The result of a measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit.
- Fundamental or base units: The units for the fundamental or base quantities are called fundamental or base units.
- Derived units: units built from the base units. The units of all other physical quantities can be expressed as combinations of the base units. Such units obtained for the derived quantities are called derived units.
- System of units: A complete set of these units, both the base units and derived units, is known as the system of units.
- Number and unit: Q = n₁u₁ = n₂u₂, where Q is the quantity, n the numerical value and u the unit; the numerical value is inversely proportional to the size of the unit.
Classify each of the following as a base quantity or a derived quantity in SI, giving the reason for the derived ones: electric current, density, luminous intensity, force. [2]
1. Credit line: ½ for each correct classification with reason.
2. Electric current: base quantity (½).
3. Density: derived, since it is mass ÷ volume, unit kg m⁻³ (½).
4. Luminous intensity: base quantity (½).
5. Force: derived, since it is mass × acceleration, unit kg m s⁻² = N (½).
Answer: current and luminous intensity are base quantities; density and force are derived.
A name such as "newton" does not make a unit a base unit.
A cricket pitch is 22 yards long between the stumps.
Taking 1 yard = 0.9144 m, express this length in metres and in centimetres. [2]
1. Use n₁u₁ = n₂u₂ (½).
2. 22 yard = 22 × 0.9144 m (½)
3. = 20.1168 m ≈ 20.12 m (½)
4. = 20.12 × 100 cm = 2012 cm (½)
Answer: about 20.12 m, or 2012 cm.
The length is the same in each line; only the number changes, growing as the unit gets smaller.
Express the SI units of (a) pressure and (b) energy in terms of the base units kg, m and s. [2]
1. (a) Pressure = force ÷ area, so its unit is N m⁻² (½). With N = kg m s⁻²: Pa = kg m s⁻² × m⁻² = kg m⁻¹ s⁻² (½).
2. (b) Work (energy) = force × distance, so its unit is N m (½) = kg m s⁻² × m = kg m² s⁻², called the joule, J (½).
Answer: 1 Pa = 1 kg m⁻¹ s⁻²; 1 J = 1 kg m² s⁻².
Start from the defining equation, then replace each quantity by its unit.
Answer the following giving reasons. (a) "The table is 1.5" is not an acceptable result of a measurement.
(b) The numerical value of the length of a rod increases when it is expressed in millimetres instead of metres, although the rod is unchanged.
(c) Units for speed, force and energy do not each need their own separately kept standard. [3]
1. Credit line: 1 + 1 + 1.
2. (a) A measured value is a number together with a unit; without the unit the number does not say how many of which reference amount the length contains (1).
3. (b) The product of number and unit stays fixed, n₁u₁ = n₂u₂; the millimetre is 1000 times smaller than the metre, so the number becomes 1000 times larger (1).
4. (c) These are derived quantities, linked to the base quantities by equations (speed = length/time and so on), so their units are combinations of the base units and need no standard of their own (1).
Answer: as above.
Each reason names the idea: "number with a unit", "n inversely proportional to u", "derived from base units".
When asked whether a unit is base or derived, write the defining equation of the quantity in one line (pressure = force/area).
That one line is the reason the marking expects, and it also gives you the unit in base units at once.
- Thinking a named unit must be a base unit. The newton, joule, watt and pascal all have names but are derived units; only the seven SI base units (metre, kilogram, second, ampere, kelvin, mole, candela) are base units.
- Converting the wrong way. A smaller unit needs a larger number. If your number got smaller when you changed from metres to centimetres, you divided where you should have multiplied.
- Leaving out the unit in a final answer, or writing a unit for the wrong quantity (m s⁻² for a speed). The unit is part of the last value point.
- Calling a base quantity "more important". Base quantities are simply chosen as the independent starting set; every derived quantity is just as real.
Check your understanding
1. Which one of the following is a base quantity in SI?
- (A) Force
- (B) Volume
- (C) Amount of substance
- (D) Speed [1]
2. Which of the following is not the unit of a derived quantity?
- (A) newton
- (B) pascal
- (C) kelvin
- (D) joule [1]
3. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Assertion (A): The SI unit of force is a derived unit.
Reason (R): The unit of force can be written as a combination of the base units kg, m and s.
- (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- (C) Assertion (A) is true, but Reason (R) is false.
- (D) Both Assertion (A) and Reason (R) are false. [1]
4. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Assertion (A): A new base unit must be added to SI every time a new physical quantity is defined.
Reason (R): Physical quantities are all independent of one another, so none can be expressed in terms of the others.
- (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- (C) Assertion (A) is true, but Reason (R) is false.
- (D) Both Assertion (A) and Reason (R) are false. [1]
5. A sheet of paper is 0.10 mm thick. Express the thickness in metres and find how many such sheets make a stack 5.0 cm high. [2]
6. (a) Define unit of a physical quantity. (1)
(b) Differentiate between fundamental units and derived units, giving one example of each. (2) [3]
7. Express the SI units of (a) power and (b) momentum in terms of the base units. [2]
8. How will the numerical value of the mass of a stone be affected if it is expressed in grams instead of kilograms? Justify your answer. [2]
Answers
1. (C) Amount of substance (1) Its unit, the mole, is one of the seven SI base units; force, volume and speed are built from length, mass and time.
2. (C) kelvin (1) The kelvin is the base unit of thermodynamic temperature; the other three are derived (N = kg m s⁻², Pa = kg m⁻¹ s⁻², J = kg m² s⁻²).
3. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (1) Force = mass × acceleration, so its unit is kg m s⁻²; being built from base units is exactly what makes it derived.
4. (D) Both Assertion (A) and Reason (R) are false. (1) Quantities are linked by equations, so a new quantity is usually derived from the existing base units; and the reason is false for the same cause.
5. Credit line: conversion 1; number 1.
- 0.10 mm = 0.10 × 10⁻³ m = 1.0 × 10⁻⁴ m (½ + ½)
- Number = height ÷ thickness = (5.0 × 10⁻² m)/(1.0 × 10⁻⁴ m) (½) = 500 sheets (½)
6. (a) Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit (1).
(b) Credit line: two matched pairs, 1 each.
Fundamental (base) units
Derived units
Units of the base quantities, fixed by international agreement and not built from other units (½)
Units of all other quantities, obtained by combining base units (½)
Example: metre, kilogram, second (any one) (½)
Example: m s⁻¹, newton, joule (any one) (½)
7. Credit line: ½ + ½ each.
(a) Power = work ÷ time, unit J s⁻¹ (½) = kg m² s⁻² × s⁻¹ = kg m² s⁻³ (the watt) (½).
(b) Momentum = mass × velocity (½), unit kg m s⁻¹ (½).
8. The numerical value becomes 1000 times larger (1). The mass itself does not change; n₁u₁ = n₂u₂, and the gram is 1000 times smaller than the kilogram, so the number must rise 1000 times (1).