Physical Quantities & Measurement — Athens Academy · Grade 9 Physics · Unit 1

Athens Academy · FBISE Study Notes

Physical Quantities & Measurement

Grade 9 Physics · Unit 1

Board	Federal Board (FBISE) · NCP 2022–23
Grade · Subject	Grade 9 · Physics · Unit 1
Weight	Always comes. Definitions, SI units, instruments and significant figures are tested every year.
Why it matters	This unit is the foundation of all Physics. Without it, no numerical in any later unit can be solved correctly.

1Physical and Non-Physical Quantities

Physics is the branch of science that studies matter, energy, and their interactions. It is the most fundamental of all the sciences — biology, chemistry and engineering all rest on it.

★ The Two Kinds of Quantity

Physical quantity — a quantity that can be measured and expressed as a numerical magnitude together with a unit. Example: a length of .

Non-physical quantity — a quantity that cannot be measured with an instrument. Example: taste, or a feeling.

🧠 Memory Aid

“If you can COUNT it or MEASURE it — it is Physical.”

The length of a desk is physical. The sweetness of a mango is not.

A measurement has two parts. Every measurement is a number and a unit. “The length is 1.65” means nothing; “the length is ” means something. An answer with no unit loses marks, every time.

⚠ A note on “colour”. Colour is sometimes given as an example of a non-physical quantity. Strictly this is wrong — colour can be measured, as the wavelength of light (in nanometres). Safer examples of genuinely non-physical quantities are taste, beauty and feeling.

2The International System of Units (SI)

Imagine your teacher measured the classroom in hand-spans and your friend measured it in foot-lengths. You would get different numbers for the same room. That is exactly why scientists agreed on one international system — the Système International (SI) — so that a measurement taken anywhere in the world means the same thing to everyone.

★ Base and Derived Quantities

Base (fundamental) quantity — cannot be derived from any other quantity; it is defined independently. There are seven.

Derived quantity — obtained by multiplying or dividing two or more base quantities.

Think of it as LEGO. The seven base quantities are the bricks. Derived quantities are the structures you build from them. Speed = distance ÷ time = . Force = mass × acceleration = . Every derived unit can be broken back down into base units — and examiners love to ask exactly that.

2.1 The seven base quantities

Base quantity	Symbol	SI unit
Length	l	metre
Mass	m	kilogram
Time	t	second
Electric current	I	ampere
Temperature	T	kelvin
Amount of substance	n	mole
Luminous intensity		candela

🧠 Mnemonic for the Seven

“Lazy Monkeys Time Every Kilogram Amicably, Lovingly”

Length · Mass · Time · Electric current · K…temperature (unit: kelvin) · Amount of substance · Luminous intensity

⚠ Careful with the mnemonic. The “K” stands for kelvin, which is a unit — but the quantity it stands for is Temperature. If an examiner asks you to “name the seven base quantities”, the correct answer is Temperature, not kelvin. Mixing up a quantity with its unit is an easy mark to lose.

2.2 Common derived quantities

Derived quantity	SI unit	In base units
Area (A)	square metre
Volume (V)	cubic metre
Speed / velocity (v)	metre per second
Acceleration (a)	metre per second squared
Density ()	kilogram per cubic metre
Force (F)	newton ()
Pressure (P)	pascal ()
Energy / work (E)	joule ()

⚠ Exam trap. The SI unit of mass is the kilogram — not the newton. The newton is the unit of force. And the unit of pressure is the pascal, not the newton: pressure = force ÷ area.

3Scientific Notation and Prefixes

The mass of the Earth is . Nobody wants to write that out. Scientific notation lets us write it as instead.

★ Scientific Notation (Standard Form)

M is the mantissa · n is the exponent (any integer)

🧠 The Direction Rule

“Left = Large exponent. Right = tiny (negative exponent).”

Count how many places you moved the decimal point — that count is your exponent. Moved left → n is positive. Moved right → n is negative.

⚠ The mantissa must be under 10. Writing is wrong, because . The correct form is . This single slip is the most common scientific-notation error in FBISE papers.

3.1 Prefixes — the shorthand for powers of ten

A prefix is a word or symbol attached to a unit to indicate a power-of-ten multiplier, making very large or very small values easier to express — instead of .

Prefix	Symbol	Prefix	Symbol
exa	E	deci	d
peta	P	centi	c
tera	T	milli	m
giga	G	micro
mega	M	nano	n
kilo	k	pico	p
hecto	h	femto	f
deca		atto	a

The eight on the left make things bigger; the eight on the right make things smaller. The ones you will actually use in Grade 9 are kilo (), centi (), milli (), micro () and nano (). Learn those five cold; recognise the rest.

4Scalars and Vectors

★ The Distinction

Scalar — has magnitude only. No direction is needed to describe it completely.

Vector — has magnitude and direction. Both must be stated for a complete description.

Scalar quantities	Vector quantities
Distance, speed, mass, time	Displacement, velocity, acceleration
Energy, temperature, pressure	Force, weight, momentum
Volume, density, power	Electric field strength, gravitational field strength

🧠 Memory Aid

“Walk 5 km” is distance. “Walk 5 km North” is displacement.

Add the direction and a scalar becomes a vector. That word — North, East, upward — is the whole difference.

⚠ The mark you lose every single year. If a question asks for a vector — displacement, velocity, acceleration, force or weight — you must state the direction. Writing “velocity ” with no direction is an incomplete answer. Write “ towards East.”

4.1 Adding vectors — the head-to-tail rule

You cannot simply add vectors like ordinary numbers, unless they point in the same direction. To add them: draw vector , then draw starting from the tip (head) of . The resultant runs from the tail of to the head of .

Figure 1 — Head-to-tail addition of two perpendicular vectors. The resultant is the closing side of the triangle.

★ Resultant of Two Perpendicular Vectors

R is the magnitude · is the direction, measured from

⚠ Do not just add the numbers. For the vectors above, is wrong. Only vectors pointing in the same direction may be added directly. For perpendicular vectors you must use Pythagoras — and the answer is , not .

5Measuring Instruments

★ Least Count

The least count (LC) is the smallest value an instrument can measure accurately. It sets the limit of that instrument’s precision — you can never report a reading finer than the least count.

Instrument	Measures	Least count
Metre rule	Length
Measuring tape	Larger lengths
Vernier caliper	Small lengths, diameters
Screw gauge (micrometer)	Very small lengths, wire diameters
Physical balance	Mass	milligram precision
Measuring cylinder	Volume of a liquid
Analogue stop watch	Time intervals
Digital stop watch	Time intervals

5.1 The vernier caliper

A vernier caliper measures a fraction of the smallest division on a main scale, by sliding a second scale (the vernier scale) along it.

★ Vernier Caliper

MSR = main scale reading · CVD = coinciding vernier division

5.2 The screw gauge (micrometer)

A screw gauge does the same job but by rotating a circular (thimble) scale. It is ten times more precise than a vernier caliper, which is why it is used for wire and ball diameters — things far too thin for a ruler.

★ Screw Gauge

CSR = circular scale reading · pitch = distance the spindle advances in one full turn

🧠 Memory Aid

“Vernier divides. A Screw has a PITCH.”

Vernier: → . Screw gauge: → .

⚠ Two traps. (1) The vernier LC is not — that is only the main-scale division. You must divide by the number of vernier divisions to get . (2) Always use the pitch given in the question. Assuming a pitch of when the question says will wreck every line that follows.

Zero error. Before trusting any reading, check whether the instrument reads zero when fully closed. If a vernier caliper has a zero error of , then a measured is really . Correct for zero error first, then use the reading.

6Errors, Precision and Accuracy

★ Error

An error is the uncertainty that arises in any measurement — the difference between the measured value and the true value.

Feature	Systematic error	Random error
Direction	Always in one direction — always too high, or always too low	Unpredictable — sometimes high, sometimes low
Cause	Faulty instrument, zero error, consistently poor technique	Fluctuations in conditions or in the observer’s technique
How to reduce	Find the fault and correct for it (e.g. subtract the zero error)	Take many readings and use the mean

★ Mean of Repeated Readings

Averaging cancels random error — but it will never remove systematic error.

6.1 Precision versus accuracy

★ Never Confuse These Two

Precision — how closely repeated measurements agree with each other. Are the results consistent?

Accuracy — how close a measurement is to the true value. Is the result right?

Figure 2 — The darts analogy. A measurement can be precise without being accurate — the second board is the one that catches students out.

⚠ The classic example. A weighing scale that reads too heavy every single time is precise (perfectly repeatable) but not accurate (always wrong by the same amount). That is a systematic error — and no amount of repeating the measurement will fix it.

7Significant Figures

★ Significant Figures

The accurately known digits in a measurement, plus the first doubtful digit. They tell you how much of a number you are actually entitled to trust.

Rule	Statement	Example
1	All non-zero digits (1–9) are always significant.	3.752 → 4 s.f.
2	Zeros between non-zero digits are significant.	100.8 → 4 s.f.
3	Trailing zeros after a decimal point are significant.	0.00290 → 3 s.f. (2, 9, 0)
4	Leading zeros (before the first non-zero digit) are never significant.	0.0029 → 2 s.f. (2, 9)
5	Trailing zeros in a whole number with no decimal point are ambiguous. Use scientific notation to make your intent clear.	1500 could be 2, 3 or 4 s.f. → clearly 2 s.f.

🧠 Memory Aid

“A non-zero digit is ALWAYS significant. A zero needs a REASON to count.”

0.00580 → 3 significant figures: the 5, the 8, and the trailing 0 after the 8. The two leading zeros are just placeholders — they do not count.

8Common Mistakes That Cost Marks

What students write	What the examiner expects
The SI unit of mass is the newton	The SI unit of mass is the kilogram. The newton is the unit of force.
Velocity (no direction)	Velocity towards East. Always state a direction for a vector.
0.00580 has 5 significant figures	0.00580 has 3 significant figures (5, 8, and the trailing 0).
	. Divide by 3.6 — never multiply by 1000 alone.
Vernier	. You must divide by the vernier divisions.
Precision and accuracy mean the same thing	Precision = reproducibility. Accuracy = closeness to the true value. Different things.
	. The mantissa must be less than 10.
Force	Force . Acceleration is , not .

8.1 Unit conversions you must know

Convert	Method
	Multiply by 1000
	Divide by 100
	Divide by 1000
	Divide by 3.6 (because )
	Multiply by 3.6
	Add 273 ()

⚠ The most forgotten conversion in the whole paper. means divide by 3.6. Not by 3. Not multiply by 1000. And always convert to SI units before substituting into a formula — never after.

9Worked Examples

FBISE awards method marks. Write every stage: Given → Required → Formula → Substitution → Calculation → ∴ Answer.

Example 1 — Scientific notation MEDIUM

Problem	Express the average Earth–Sun distance, , in scientific notation.
Given
Formula	, where
Substitution	Place the decimal after the first digit: 1.4953. Count the places moved: 11 places to the left, so .
Calculation	and

∴

Marks are awarded for: a mantissa between 1 and 10, the correct positive exponent, and the unit m. Writing loses the first mark; omitting the unit loses the last.

Example 2 — Vernier caliper ★ HARD

Problem	Find the diameter of a small sphere. The main scale reads and the 6th vernier division coincides. Standard vernier, .
Given	, ,
Required	Diameter
Formula
Substitution
Calculation

∴

Zero error. If this caliper had a zero error of , the corrected diameter would be . Check for zero error before you report the answer — forgetting it is the commonest way to lose this question.

Example 3 — Screw gauge MEDIUM

Problem	A screw gauge has a pitch of and 50 circular-scale divisions. The main scale reads and the circular scale reads 22. Find the least count and the total reading.
Given	pitch , divisions , ,
Formula	then
Calculation

∴ and

Always state the LC first. It carries its own mark, separate from the final reading. And use the pitch given in the question — assuming when it says destroys everything downstream.

Example 4 — Resultant of two perpendicular vectors ★ HARD

Problem	Vector towards East and vector towards North act on a body. Find the resultant and its direction.
Given	(East), (North) — the two are perpendicular
Required	and
Formula	and
Substitution	and
Calculation

∴ at North of East

Never stop at “50 N”. The resultant is a vector, so the direction is part of the answer and carries its own mark. And note is not the answer — perpendicular vectors add by Pythagoras, not arithmetic.

Example 5 — Find the error CLINIC

Problem	A car travels at . Express this in , then find the distance covered in .
Student’s working	✗ wrong ✗ wrong
The error	The student multiplied by 1000 only. But requires dividing by 3.6, because . Everything after that first line is then wrong — one root error, two lost answers.
Correct working

∴ and

Sanity check. is about 200 times the speed of sound. If your answer is physically absurd, you have made an arithmetic error — always pause and ask whether the number could be real.

10Revision Checklist

Tick a box only when you can write the answer from memory. Clear all boxes the night before the exam.

I can define physical and non-physical quantities, with two safe examples of each.

I know every measurement needs a number AND a unit — an answer without a unit loses marks.

I can name all seven base quantities and their SI units.

I know the “K” in the mnemonic means the quantity Temperature, whose unit is the kelvin.

I can define base and derived quantities and give two examples of each.

I can express force, pressure and energy in base units.

I know the SI unit of mass is the kilogram, not the newton.

I can write a number in scientific notation with .

I know: decimal moved left → positive exponent; right → negative exponent.

I know the five prefixes that matter: .

I can define scalar and vector and classify any quantity as one or the other.

I always state a direction when the answer is a vector.

I can add two perpendicular vectors: , .

I know that perpendicular gives — not .

I can define least count and state it for the metre rule, vernier caliper and screw gauge.

I can compute for a vernier () and a screw gauge ().

I can take a reading: , and correct for zero error.

I can distinguish systematic from random error, and say how to reduce each.

I can explain precision vs accuracy using the darts analogy, and give the scale example.

I know all five significant-figure rules, including that 1500 is ambiguous.

I can convert by dividing by 3.6.

I always convert to SI units before substituting into a formula.

Final word from Athens Academy

This unit looks like a list of definitions, and students treat it that way — which is exactly why they lose marks on it. Every later unit in Physics stands on what you learn here. If you cannot state a unit, correct a zero error, or read a vernier scale, then no numerical you meet for the rest of the year can be fully right. Learn the seven base quantities, learn the two least-count formulae, and learn to write the direction after every vector. The foundation is where the marks are cheapest — take all of them.

— Athens Academy

Athens Academy · Physics · Grade 9 · Unit 1 · Physical Quantities & Measurement

🔬 Chapter 1 · Physical Quantities and Measurement