Physics Grade 11 · Unit 1 · Physical Quantities and Measurements

Physics Grade 11 · Unit 1 · Physical Quantities and Measurements · Athens Academy Islamabad

Physics · Grade 11 · FBISE · Exam March 2027

Learning Objectives

By the end of this unit you should be able to:

Make reasonable estimates of physical quantities using prior experience and sound physical reasoning.
Express derived units — force, work, power, pressure and electric charge — as products or quotients of the SI base units.
Define the dimension of a physical quantity and write the dimensions of base and derived quantities.
Classify quantities as dimensional or dimensionless variables and dimensional or dimensionless constants.
Use dimensional analysis to check the homogeneity of an equation and to derive a possible formula.
State the limitations of dimensional analysis.
Distinguish between the precision and the accuracy of measured data.
Define uncertainty and find absolute, fractional and percentage uncertainty.
Apply the rules for propagating uncertainty in sums, products, powers, averages and timing experiments.

1. Estimation of Physical Quantities

Think of a shopkeeper in Raja Bazaar who looks at a sack of rice and says, "that is about forty kilos." He has not put it on a scale. He has held many sacks before, and he knows roughly what forty kilos feels like. He is not guessing wildly — he is reasoning from experience. Physics does exactly the same thing, and gives it a name.

Definition

Estimation is a rough educated guess of the value of a physical quantity, made by using prior experience and sound physical reasoning.

Physics is built on measurement, but we cannot always measure everything directly. In daily life we constantly ask questions like "how tall is that building?" or "how long will this download take?" — and we answer them by estimating. Scientists and engineers do the same when they have little or no actual data. An estimate is not a random number: it means picking the correct physical principle and then making a sensible guess about the quantities involved.

The most useful technique is to break a big thing into smaller things, estimate one small piece, and then multiply. To estimate the height of a building, count its floors, estimate the height of a single floor — imagine how many people standing on each other's shoulders would reach the ceiling — and then multiply. The reverse works just as well: to find the thickness of one sheet of paper, measure a thick stack with a ruler and divide by the number of sheets. The same idea lets us estimate areas and volumes by modelling an object as a simple sphere or box, and then estimate mass from volume using density, since $m = \rho V$ .

Memory hook

Estimation is "break big into small, or build small into big."

Where students lose marks

An estimate uses round figures — $m \approx 70~\text{kg}$ , $g \approx 10~\text{m s}^{-2}$ . Do not waste exam time hunting for "exact" data in an estimation question. The examiner wants sensible reasoning, not decimal places.

2. Derived Units in Terms of Base Units

A base unit is one of the seven fundamental SI units: the kilogram (kg), metre (m), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd). Everything else in physics is built out of these, the way every Urdu word is built out of the same alphabet. A derived unit is formed by multiplying or dividing base units together.

To find the base-unit form of any derived unit, replace every quantity in its defining formula with its own base unit. Take force:

$\text{Force} = \text{mass} \times \text{acceleration} = \text{mass} \times \frac{\text{velocity}}{\text{time}} = \text{mass} \times \frac{\text{displacement}}{\text{time}^{2}}$

Putting the SI units in place gives the base-unit form of the newton:

$1~\text{N} = 1~\text{kg m s}^{-2}$

where $\text{kg}$ is the unit of mass, $\text{m}$ the unit of displacement and $\text{s}$ the unit of time. Doing the same for work, since $W = F \times d$ , builds the base-unit form of the joule:

$1~\text{J} = 1~\text{N} \times 1~\text{m} = 1~\text{kg m}^{2}~\text{s}^{-2}$

Table 2.1 — Common derived units expressed in SI base units
Derived quantity	SI unit	Symbol	In terms of base units
Force	newton	N	$\text{kg m s}^{-2}$
Work / Energy	joule	J	$\text{N m} = \text{kg m}^{2}~\text{s}^{-2}$
Power	watt	W	$\text{J s}^{-1} = \text{kg m}^{2}~\text{s}^{-3}$
Pressure	pascal	Pa	$\text{N m}^{-2} = \text{kg m}^{-1}~\text{s}^{-2}$
Electric charge	coulomb	C	$\text{A s}$

Closer to home

The WAPDA meter on the wall of your house reads energy in kilowatt-hours (kWh). A kilowatt is $1000~\text{W}$ and an hour is $3600~\text{s}$ , so

$1~\text{kWh} = 1000 \times 3600 = 3.6 \times 10^{6}~\text{J}$

Every single "unit" on your electricity bill is really 3.6 million joules.

3. Dimensions of Physical Quantities

Length, width, height, distance, displacement and radius are all measured in metres. That is not a coincidence — they share the same underlying nature. Physics captures that shared nature with the idea of a dimension.

Definition

The dimension of a physical quantity is its qualitative nature, written as a capital letter inside square brackets $[\;]$ , showing how the quantity is related to the base quantities mass, length and time.

So every one of those length-type quantities has the dimension $[\text{L}]$ . The dimensions of the seven base quantities are given in Table 3.1.

Table 3.1 — Dimensions of the seven base quantities
Sr. No.	Base physical quantity	Dimension
1	Mass	$[\text{M}]$
2	Length	$[\text{L}]$
3	Time	$[\text{T}]$
4	Electric current	$[\text{I}]$
5	Temperature	$[\theta]$
6	Intensity of light	$[\text{J}]$
7	Amount of substance	$[\text{N}]$

Dimensions of derived quantities come from multiplying or dividing the dimensions of the base quantities they are built from. For example area is $[\text{L}^{2}]$ , volume is $[\text{L}^{3}]$ , velocity is $[\text{LT}^{-1}]$ and acceleration is $[\text{LT}^{-2}]$ .

3.1 Four kinds of quantity

Every physical quantity can be sorted by asking just two yes/no questions. Figure 1 shows the result.

Fig. 1 — Classifying quantities by dimension and by magnitude.

Dimensional variables have dimensions and a changing magnitude — length, velocity, acceleration, force, energy. Dimensional constants have dimensions but a fixed magnitude — Planck's constant $h$ , the gravitational constant $G$ , the speed of light $c$ and the ideal gas constant $R$ . Dimensionless variables have no dimensions but a changing magnitude — plane angle, solid angle, strain and the coefficient of friction. Dimensionless constants have neither dimensions nor a changing magnitude — the pure numbers $1, 2, 3, \ldots$ , the exponential constant $e = 2.718$ , and $\pi$ .

Memory hook

Two yes/no questions sort every quantity: "Does it have dimensions?" and "Does its value change?"

3.2 Advantages of dimensional analysis

Dimensional analysis treats dimensions as ordinary algebraic quantities. Two rules follow immediately: terms can be added or subtracted only when they have the same dimensions, and both sides of any valid equation must have identical dimensions. This gives us three useful powers — we can (i) check the homogeneity of an equation, (ii) derive a possible formula, and (iii) find the units of a physical quantity.

(i) Checking the homogeneity of an equation

Principle of homogeneity: both sides of a correct physical equation must have the same dimensions; otherwise the equation cannot be physically correct. Let us test the first equation of motion, $v_f = v_i + at$ .

$\text{L.H.S.} = [v_f] = [\text{LT}^{-1}]$

$\text{R.H.S.} = [v_i] + [a][t] = [\text{LT}^{-1}] + [\text{LT}^{-2}][\text{T}] = [\text{LT}^{-1}] + [\text{LT}^{-1}] = [\text{LT}^{-1}]$

Both sides reduce to $[\text{LT}^{-1}]$ , so the equation is dimensionally correct. Notice that adding two terms of the same dimension does not change the dimension — the "2" that would appear if you added them numerically is a dimensionless constant and carries no dimension at all.

(ii) Deriving a possible formula

If we can guess which quantities a result depends on, dimensional analysis can recover the shape of the formula for us. Let us derive the wavelength $\lambda$ of matter waves, assuming it depends on Planck's constant $h$ , the particle's mass $m$ and its velocity $v$ :

$\lambda = k\, h^{a} m^{b} v^{c} \qquad \ldots (1)$

where $k$ is a dimensionless constant. Inserting the dimensions $[h] = [\text{ML}^{2}\text{T}^{-1}]$ , $[m] = [\text{M}]$ , $[v] = [\text{LT}^{-1}]$ and $[\lambda] = [\text{L}]$ :

$[\text{M}^{0}\text{L}^{1}\text{T}^{0}] = [\text{ML}^{2}\text{T}^{-1}]^{a}\,[\text{M}]^{b}\,[\text{LT}^{-1}]^{c}$

$[\text{M}^{0}\text{L}^{1}\text{T}^{0}] = [\text{M}]^{a+b}\,[\text{L}]^{2a+c}\,[\text{T}]^{-a-c} \qquad \ldots (2)$

Equating the powers of M, L and T on both sides gives three simultaneous equations:

$\text{M}: \; a + b = 0 \qquad \text{L}: \; 2a + c = 1 \qquad \text{T}: \; -a - c = 0$

Solving these together gives $a = 1$ , $b = -1$ , $c = -1$ . Substituting back into equation (1):

$\lambda = k\,h^{1} m^{-1} v^{-1} = \frac{k\,h}{mv}$

This is the form of the de Broglie wavelength. Dimensional analysis has recovered everything except the value of the dimensionless constant $k$ — which is exactly the method's honest limit.

Worked Example 3.1 — Time period of a simple pendulum HARD

Given	$T$ may depend on mass $m$ (kg), length $l$ (m), angle $\theta$ (rad, dimensionless) and $g$ (m s $^{-2}$ ), with $[g] = [\text{LT}^{-2}]$
Required	The form of the formula for the time period $T$ (s), by dimensions
Formula	$T = k\, m^{a} l^{b} \theta^{c} g^{d}$
Substitution	$[\text{M}^{0}\text{L}^{0}\text{T}^{1}] = [\text{M}]^{a}\,[\text{L}]^{b}\,[\text{LT}^{-2}]^{d} = [\text{M}]^{a}\,[\text{L}]^{b+d}\,[\text{T}]^{-2d}$ ( $\theta$ is dimensionless, so it drops out)
Calculation	$\text{M}: a = 0$ · $\text{T}: -2d = 1 \Rightarrow d = -\tfrac{1}{2}$ · $\text{L}: b + d = 0 \Rightarrow b = \tfrac{1}{2}$

∴

T = k\,l^{1/2} g^{-1/2} = k\sqrt{\dfrac{l}{g}}

— experiment gives

k = 2\pi

, so

T = 2\pi\sqrt{l/g}

Examiner tip:

\theta

is an angle — a dimensionless variable — so its power cannot be fixed by dimensions. Say so explicitly in your answer to secure the method mark.

Common error: forgetting that mass cancels out (

a = 0

). A pendulum's period does not depend on the mass of the bob at all.

3.3 Limitations of dimensional analysis

It cannot distinguish between quantities that share the same dimensions — $[\text{ML}^{2}\text{T}^{-2}]$ could be work, energy or torque.
It cannot derive formulae containing trigonometric, exponential or logarithmic functions.
It cannot find the dimensionless constant in a derived formula — the $2\pi$ or the $\tfrac{1}{2}$ must come from experiment or from a fuller theory.
A dimensionally correct equation is not guaranteed to be physically correct. But the reverse is watertight: a dimensionally wrong equation is always wrong.

4. Precision and Accuracy

In ordinary Urdu and English these two words mean almost the same thing. In physics they mean two completely different things, and the examiner will test whether you know the difference.

Definition

Precision refers to how closely repeated measurements of the same quantity agree with each other. Accuracy refers to how closely a measurement agrees with the standard or true value.

The clearest picture is an archery target. Accuracy is how close the arrows land to the bull's-eye. Precision is how close the arrows land to one another — whether or not they are anywhere near the centre.

Fig. 2 — Precision versus accuracy on a bull's-eye target.

Now suppose a person who truly weighs $160.0~\text{lb}$ steps on three different weighing scales:

Table 4.1 — Three scales weighing the same 160.0 lb person
Scale	Three readings	Verdict
A	170.1, 169.9, 170.0 lb	Precise but not accurate
B	161, 162, 158 lb	Accurate but not precise
C	159.9, 160.0, 160.1 lb	Both precise and accurate

Scale A is beautifully consistent — and consistently wrong by ten pounds. That is precision without accuracy, and it is the trap in this topic. Precision is linked to the least count of the instrument: the smaller the least count, the more precise the reading, and precision is shown by the absolute uncertainty. Accuracy is shown by the fractional or percentage uncertainty — the smaller this is, the more accurate the measurement.

Table 4.2 — The five examinable differences
Precision	Accuracy
Closeness of repeated readings to each other	Closeness of a reading to the true value
Shown by absolute uncertainty	Shown by fractional or percentage uncertainty
Depends on the least count of the instrument	Depends on correct calibration of the instrument
Can be precise yet wrong (Scale A)	Can be accurate yet scattered (Scale B)
Improved by using a finer-scale instrument	Improved by removing systematic error

5. Uncertainties

Definition

Uncertainty is the range of possible values within which the true value of a measurement lies. Every measurement carries some uncertainty, caused by the limitations of the instrument and the skill of the experimenter.

The difference between the true value and the observed value is called the error. Errors arise from negligence or inexperience, from faulty apparatus, or from an unsuitable method, and are classed as personal, systematic and random errors. When we write a reading as $3.06 \pm 0.02~\text{mm}$ , we are saying we are confident the true value lies somewhere between $3.04~\text{mm}$ and $3.08~\text{mm}$ .

5.1 Absolute, fractional and percentage uncertainty

The absolute uncertainty equals the least count of the instrument. A metre rule has a least count of $1~\text{mm} = 0.1~\text{cm}$ , which gives $\pm 0.05~\text{cm}$ at each end of the object being measured — so $\pm 0.1~\text{cm}$ overall, because you must read the scale twice. Suppose a slab runs from the $20.5~\text{cm}$ mark to the $58.0~\text{cm}$ mark:

$(58.0 \pm 0.05)~\text{cm} - (20.5 \pm 0.05)~\text{cm} = (37.5 \pm 0.1)~\text{cm}$

Fig. 3 — Reading a length with a metre rule. Two readings are taken, so the two

\pm 0.05~\text{cm}

uncertainties add to

\pm 0.1~\text{cm}

From the absolute uncertainty we get the other two forms:

$\text{Fractional uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}}$

$\text{Percentage uncertainty} = \text{fractional uncertainty} \times 100\%$

For the slab, the fractional uncertainty is $0.1 / 37.5 \approx 0.003$ and the percentage uncertainty is about $0.3\%$ . The smaller the fractional uncertainty, the greater the accuracy of the measurement.

5.2 Rules for combining uncertainties

Let $x$ and $y$ be measured quantities with uncertainties $\Delta x$ and $\Delta y$ , combined to give a result $z$ . The whole topic rests on one decision, made first: use absolute uncertainty for sums and differences, and percentage uncertainty for products, quotients and powers.

Table 5.1 — The five uncertainty rules
Operation	Rule	Example
(a) Addition / subtraction	Add the absolute uncertainties: $\Delta z = \pm(\Delta x + \Delta y)$	$(24.0 \pm 0.1) + (30.0 \pm 0.1) \Rightarrow \Delta z = \pm 0.2~\text{cm}$
(b) Multiplication / division	Add the percentage uncertainties	$\%z = \%x + \%y$
(c) Power of a quantity	Multiply the percentage uncertainty by the power	$z = x^{3} \Rightarrow \%z = 3 \times \%x$
(d) Average of many readings	The mean deviation from the average is the uncertainty	see worked steps below
(e) Timing experiment	Divide the least count by the number of vibrations	$\Delta T = \dfrac{\text{L.C.}}{N}$

(d) Average of many measurements. Suppose three readings of a radius give $1.50,\ 1.51,\ 1.52~\text{cm}$ . The average is $1.51~\text{cm}$ . The deviations from that average are $0.01,\ 0,\ 0.01~\text{cm}$ , so the mean deviation is $(0.01 + 0 + 0.01)/3 \approx 0.007~\text{cm}$ . Therefore $r = (1.51 \pm 0.007)~\text{cm}$ .

(e) Timing experiment. Suppose 20 vibrations of a pendulum are timed as $35.2~\text{s}$ using a stopwatch of least count $0.1~\text{s}$ . Then

$T = \frac{35.2}{20} = 1.76~\text{s}, \qquad \Delta T = \frac{0.1}{20} = 0.005~\text{s}$

so $T = (1.76 \pm 0.005)~\text{s}$ . Dividing by the number of vibrations is exactly why we time many swings instead of one — it shrinks the uncertainty by a factor of $N$ . That single line is worth remembering as the reason behind the whole experiment.

6. Worked Examples

Worked Example 6.1 — Energy needed to climb one floor MEDIUM

Given	Mass of an adult $m \approx 70~\text{kg}$ · height of one floor $h \approx 3~\text{m}$ · $g \approx 10~\text{m s}^{-2}$
Required	Estimated energy $E$ required, in joules (J)
Formula	$E = mgh$ where $m$ is in kg, $g$ in m s $^{-2}$ and $h$ in m
Substitution	$E = 70 \times 10 \times 3$
Calculation	$E = 2100~\text{kg m}^{2}~\text{s}^{-2}$

∴

E \approx 2100~\text{J}

Examiner tip: write the unit chain

\text{kg m}^{2}~\text{s}^{-2} = \text{J}

so the marker can see you understand why the joule is the SI unit of energy.

Worked Example 6.2 — Resistance from voltage and current MEDIUM

Given	$V = (7.3 \pm 0.1)~\text{V}$ · $I = (2.73 \pm 0.05)~\text{A}$
Required	Resistance $R$ (Ω) and its uncertainty $\Delta R$ (Ω)
Formula	$R = \dfrac{V}{I}$ — a division, so add the percentage uncertainties
Substitution	$R = \dfrac{7.3}{2.73} = 2.7~\Omega$ ; $\%V = \dfrac{0.1}{7.3} \times 100 \approx 1\%$ ; $\%I = \dfrac{0.05}{2.73} \times 100 \approx 2\%$
Calculation	$\%R = 1\% + 2\% = 3\%$ ; $\Delta R = 3\% \times 2.7 \approx 0.08~\Omega$

∴

R = (2.7 \pm 0.08)~\Omega

Examiner tip: for a division you add percentage uncertainties, never absolute ones.

Common error: rounding

R

before finding

\Delta R

. Keep the working value, find the percentage, then convert back to an absolute uncertainty at the end.

Worked Example 6.3 — Area of a disc HARD

Given	Radius $r = 2.25~\text{cm}$ · $\Delta r = \pm 0.01~\text{cm}$
Required	Area $A$ (cm²) and its uncertainty $\Delta A$ (cm²)
Formula	$A = \pi r^{2}$ — a power of 2, so the percentage is multiplied by 2
Substitution	$A = 3.14 \times (2.25)^{2} = 15.90~\text{cm}^{2}$ ; $\%r = \dfrac{0.01}{2.25} \times 100 \approx 0.4\%$
Calculation	$\%A = 2 \times 0.4\% = 0.8\%$ ; $\Delta A = 0.8\% \times 15.90 \approx 0.13~\text{cm}^{2}$

∴

A = (15.90 \pm 0.13)~\text{cm}^{2}

Examiner tip: because

A

depends on

r^{2}

, the percentage uncertainty doubles. A small error in

r

matters twice as much in the area.

Common error: doubling the absolute uncertainty in

r

instead of its percentage. The power rule acts on the percentage only.

Worked Example 6.4 — Percentage uncertainty in electrical power HARD

Given	$I = (3.0 \pm 0.1)~\text{A}$ · $R = (13 \pm 0.5)~\Omega$
Required	Percentage uncertainty in the power $P = I^{2}R$ (W)
Formula	$\%P = 2(\%I) + \%R$ — apply the power rule to $I$ first, then add
Substitution	$\%I = \dfrac{0.1}{3.0} \times 100 \approx 3\% \Rightarrow 2 \times 3\% = 6\%$ ; $\%R = \dfrac{0.5}{13} \times 100 \approx 3.8\%$
Calculation	$\%P = 6\% + 3.8\% = 9.8\%$

∴ Percentage uncertainty in

P \approx 10\%

Examiner tip: identify the power on each quantity before you add anything. Writing

\%P = \%I + \%R = 6.8\%

forgets the square on the current and loses the marks.

7. Key Terminology and Quick Reference

Table 7.1 — Formulae, units and the mistakes that cost marks
Quantity / rule	Key relation	Common exam mistake
Newton (force)	$\text{N} = \text{kg m s}^{-2}$	Writing $\text{kg m s}^{-1}$
Joule (energy)	$\text{J} = \text{kg m}^{2}~\text{s}^{-2}$	Confusing it with the watt
Watt (power)	$\text{W} = \text{kg m}^{2}~\text{s}^{-3} = \text{J s}^{-1}$	Forgetting the extra division by time
Pascal (pressure)	$\text{Pa} = \text{kg m}^{-1}~\text{s}^{-2}$	Writing $\text{L}^{+1}$ instead of $\text{L}^{-1}$
Homogeneity	Both sides must share the same dimensions	Treating "dimensionally correct" as "proven correct"
Sum / difference	$\Delta z = \Delta x + \Delta y$ (absolute)	Adding percentage uncertainties for a sum
Product / quotient	$\%z = \%x + \%y$	Adding absolute uncertainties instead
Power $z = x^{n}$	$\%z = n \times \%x$	Multiplying the absolute uncertainty by $n$
Average of readings	Uncertainty = mean deviation	Using the largest deviation instead of the mean
Timing experiment	$\Delta T = \dfrac{\text{least count}}{N}$	Using the least count without dividing by $N$

Unit alert

The single most common slip in this whole unit is mixing absolute and percentage uncertainty. The rule is short enough to say out loud: absolute for $+$ and $-$ ; percentage for $\times$ , $\div$ and powers. And always carry the SI unit — Ω, cm², J — through to the final line.

8. Sixty-Minute Revision Checklist

Define estimation and give the "break big into small" strategy.
Estimate an energy, mass or length using sensible round figures.
Express force, work, power, pressure and charge in SI base units.
Derive $\text{N} = \text{kg m s}^{-2}$ and $\text{J} = \text{kg m}^{2}~\text{s}^{-2}$ step by step.
Define the dimension of a physical quantity.
Write the dimensions of all seven base quantities from memory.
Find the dimensions of area, volume, velocity and acceleration.
Define dimensional variable and dimensional constant, with one example each.
Define dimensionless variable and dimensionless constant, with one example each.
State the principle of homogeneity of dimensions.
Check $v_f = v_i + at$ for dimensional correctness.
Derive $\lambda = kh/(mv)$ for matter waves by dimensions.
Derive $T = k\sqrt{l/g}$ for a simple pendulum.
State the four limitations of dimensional analysis.
Give the five differences between precision and accuracy.
Interpret the four bull's-eye targets (a)–(d).
Define uncertainty and error; list the three types of error.
Find absolute, fractional and percentage uncertainty for a metre-rule reading.
Rule: add absolute uncertainties for $+$ and $-$ .
Rule: add percentage uncertainties for $\times$ and $\div$ .
Rule: for a power, multiply the percentage uncertainty by the power.
Uncertainty in an average = mean deviation from the average.
Timing: $\Delta T =$ least count $\div$ number of vibrations.
Re-work Example 6.2 ( $R$ from $V$ and $I$ ) without looking.
Re-work Example 6.3 (area of a disc — power rule on $r$ ).
Re-work Example 6.4 ( $\%P$ for $P = I^{2}R$ ).

Final Word from Athens Academy

You now hold every examinable idea in Unit 1. Learn the four definitions — estimation, dimension, precision and accuracy, uncertainty — until you can write them with your eyes closed. Then practise the uncertainty rules until the question "absolute or percentage?" answers itself before you have finished reading the problem.

Measurement is the foundation every later unit stands on. Master it here and kinematics, dynamics and electricity all become easier, because the habits of writing units, checking dimensions and carrying uncertainty will already be automatic.

Trust your preparation and write clearly. Best of luck — Exam March 2027.

Athens Academy Islamabad · Physics · Grade 11 · Unit 1 — Physical Quantities and Measurements · FBISE · Exam March 2027

🔬 Chapter 1 · Physical Quantities & Measurements