understand that all physical quantities consist of a numerical magnitude and a unit

Physical Quantities, Units & Measurement

1.1 What is a physical quantity?

A physical quantity is a measurable property of the physical world. It is always written as a product of a numerical value (pure number) and a unit that gives the number meaning:

Q = N × U

• Q – the quantity (e.g. length, force).
• N – the numerical value (no units).
• U – the unit.

Omitting the unit removes the physical meaning (e.g. “3 kg m s⁻²” is a force, “3” alone is meaningless).

1.2 SI Base Units

1.3 SI Prefixes (Common Powers of Ten)

Example: \(0.003\;\text{km}=0.003\times10^{3}\;\text{m}=3\;\text{m}\).

1.4 Derived Quantities & Dimensional Checks

Dimensional check example: In \(F=ma\) the left‑hand side has units N = kg m s⁻². The right‑hand side gives \(m\)(kg) × \(a\)(m s⁻²) = kg m s⁻² – the same, confirming the equation is homogeneous.

1.5 Scalars & Vectors

• Scalar – described by magnitude only (e.g. mass, temperature).
• Vector – described by magnitude + direction (e.g. displacement, velocity, force). Written in bold (\(\mathbf{F}\)) or with an arrow (\(\vec{F}\)).

Vector addition follows the “head‑to‑tail” rule; subtraction is performed by adding the opposite vector.

1.6 Uncertainties, Precision & Accuracy

• Systematic error – reproducible bias; affects accuracy.
• Random error – unpredictable fluctuations; affects precision.
• Precision – closeness of repeated measurements to each other.
• Accuracy – closeness of a measurement to the true value.

Propagation of uncertainties (syllabus rules)

• Addition / Subtraction: \(\displaystyle \Delta Q=\sqrt{(\Delta A)^2+(\Delta B)^2}\)
• Multiplication / Division: \(\displaystyle \frac{\Delta Q}{|Q|}= \sqrt{\left(\frac{\Delta A}{|A|}\right)^2+\left(\frac{\Delta B}{|B|}\right)^2}\)

1.7 Significant Figures & Rounding

• Report measured numbers with the correct number of significant figures (SF) – all certain digits plus one estimated digit.
• Multiplication / division – result has the same number of SF as the factor with the fewest SF.
• Addition / subtraction – result is rounded to the same number of decimal places as the term with the fewest decimal places.

1.8 Unit Symbols, Case Sensitivity & Naming

• Symbols are case‑sensitive: N = newton, n = nano‑ (10⁻⁹).
• Do not confuse I (electric current) with I_v (luminous intensity).
• Unit names are written in lower case (metre, second) except when derived from a proper name (newton, joule).

2 Kinematics

2.1 Distance & Displacement

• Distance – scalar, total length travelled (units: m).
• Displacement – vector, straight‑line change in position (units: m, direction shown).

2.2 Speed & Velocity

• Average speed \(= \dfrac{\text{total distance}}{\text{time}}\) (scalar, m s⁻¹).
• Average velocity \(= \dfrac{\text{displacement}}{\text{time}}\) (vector, m s⁻¹).
• Instantaneous speed/velocity – slope of a distance‑time or displacement‑time graph at a point.

2.3 Acceleration

Acceleration is the rate of change of velocity:

\(a = \dfrac{\Delta v}{\Delta t}\) (units: m s⁻²)

2.4 Equations of Uniformly Accelerated Motion (UAM)

For constant acceleration \(a\):

All symbols are vectors unless otherwise stated; use magnitudes when solving scalar problems.

2.5 Graphical Interpretation

• Slope of a distance‑time graph = speed.
• Slope of a velocity‑time graph = acceleration.
• Area under a velocity‑time graph = displacement.
• Area under an acceleration‑time graph = change in velocity.

2.6 Practical Example

Drop a steel ball from a known height, measure the time with a stopwatch, and use \(s = \tfrac{1}{2}gt^{2}\) (with \(g≈9.8\;\text{m s}^{-2}\)) to determine the experimental value of \(g\). Discuss random and systematic errors.

3 Dynamics

3.1 Newton’s Laws of Motion

1. First law (Inertia) – a body remains at rest or in uniform motion unless acted on by a net external force.
2. Second law – \(\mathbf{F}_{\text{net}} = m\mathbf{a}\). The direction of \(\mathbf{a}\) is the same as the net force.
3. Third law – For every action there is an equal and opposite reaction: \(\mathbf{F}{AB} = -\mathbf{F}{BA}\).

3.2 Momentum & Impulse

• Linear momentum: \(\mathbf{p}=m\mathbf{v}\) (kg m s⁻¹).
• Impulse: \(\mathbf{J}= \Delta\mathbf{p}= \mathbf{F}_{\text{avg}}\Delta t\).
• Conservation of momentum (isolated system): \(\mathbf{p}{\text{initial}} = \mathbf{p}{\text{final}}\).

3.3 Collisions

• Elastic – kinetic energy conserved; both momentum and kinetic energy are conserved.
• Inelastic – kinetic energy not conserved; only momentum conserved. If the bodies stick together, it is a perfectly inelastic collision.

Worked example (2‑D collision): A 2.0 kg ball moving at \(3.0\;\text{m s}^{-1}\) east collides elastically with a 1.0 kg ball at rest. Use conservation of momentum in \(x\) and \(y\) and kinetic‑energy conservation to find the final velocities.

3.4 Friction

• Static friction: \(F{\text{s}} \le \mu{\text{s}}N\).
• Kinetic friction: \(F{\text{k}} = \mu{\text{k}}N\).
• Typical values of \(\mu\) depend on the surfaces involved; they are determined experimentally.

3.5 Terminal Velocity

When the upward drag force equals the weight, the net force is zero and the object falls at constant speed:

\(mg = kv\) ⟹ \(v_{\text{t}} = \dfrac{mg}{k}\)

where \(k\) is a drag constant (experimentally measured).

4 Forces, Density & Pressure

4.1 Turning Effects of Forces

• Moment (torque) \(\tau = rF\sin\theta\) (units: N m). \(r\) is the perpendicular distance from the pivot to the line of action of the force.
• Couple – pair of equal, opposite forces whose lines of action are parallel but not collinear; produces a pure rotation.
• Equilibrium of moments: \(\sum\tau = 0\).

4.2 Density

Density \(\rho = \dfrac{m}{V}\) (units: kg m⁻³). Useful for converting between mass and volume, especially in buoyancy problems.

4.3 Pressure

• Definition: \(p = \dfrac{F}{A}\) (Pa = N m⁻²).
• Hydrostatic pressure in a fluid of density \(\rho\) at depth \(h\): \(p = \rho gh\).
• Archimedes’ principle – buoyant force equals the weight of the displaced fluid: \(F{\text{b}} = \rho{\text{fluid}} V_{\text{sub}} g\).

4.4 Practical Illustration

Use a lever (rigid bar on a fulcrum) to demonstrate moments: \(F{1}r{1}=F{2}r{2}\). Measure masses and distances, calculate expected equilibrium, and discuss experimental uncertainties.

5 Work, Energy & Power

5.1 Work

Work done by a constant force \(F\) acting through a displacement \(s\) at an angle \(\theta\):

\(W = Fs\cos\theta\) (units: J = N m).

If the force varies, \(W = \int \mathbf{F}\cdot d\mathbf{s}\).

5.2 Kinetic & Gravitational Potential Energy

• Kinetic energy: \(K = \tfrac{1}{2}mv^{2}\).
• Gravitational potential energy (near Earth): \(U = mgh\) (reference level chosen arbitrarily).

5.3 Work‑Energy Theorem

The net work done on an object equals the change in its kinetic energy:

\(W_{\text{net}} = \Delta K\).

5.4 Conservation of Mechanical Energy

If only conservative forces do work, total mechanical energy is constant:

\(K{i}+U{i}=K{f}+U{f}\).

5.5 Power

Power is the rate of doing work:

\(P = \dfrac{W}{t} = Fv\) (units: W = J s⁻¹).

5.6 Efficiency

\(\displaystyle \eta = \frac{\text{useful energy output}}{\text{energy input}}\times100\%.\)

5.7 Practical Activity

Measure the elastic potential energy stored in a spring (Hooke’s law) by hanging masses, recording the extension, and comparing \(\tfrac{1}{2}kx^{2}\) with the gravitational potential energy lost.

6 Deformation of Solids

6.1 Stress & Strain

• Stress \(\sigma = \dfrac{F}{A}\) (Pa).
• Strain \(\varepsilon = \dfrac{\Delta L}{L_{0}}\) (dimensionless).

6.2 Hooke’s Law

For elastic deformation: \(\sigma = E\varepsilon\)  or \(F = kx\) (where \(k = EA/L_{0}\)). \(E\) is Young’s modulus (Pa).

6.3 Elastic vs Plastic Behaviour

• Elastic region – deformation fully reversible.
• Plastic region – permanent deformation after the yield point.

6.4 Energy Stored in a Stretched Wire

Elastic potential energy: \(U = \tfrac{1}{2}kx^{2}\) (same form as a spring).

6.5 Experiment

Hang known masses from a wire, measure the extension, plot \(F\) vs \(\Delta L\), determine the gradient (the spring constant) and calculate \(E\).

7 Waves

7.1 Basic Terminology

• Wave – disturbance that transfers energy without permanent displacement of the medium.
• Transverse – particle motion ⟂ to direction of propagation (e.g. light, string).
• Longitudinal – particle motion ∥ to direction of propagation (e.g. sound).
• Frequency \(f\) (Hz), period \(T = 1/f\) (s).
• Wavelength \(\lambda\) – distance between successive points in phase.
• Wave speed \(v = f\lambda\).

7.2 Wave Equation

For a wave on a string under tension \(T\) with linear mass density \(\mu\):

\(v = \sqrt{\dfrac{T}{\mu}}\).

7.3 Intensity

Intensity \(I = \dfrac{P}{A}\) (W m⁻²). For sound, \(I \propto p^{2}\) where \(p\) is the pressure amplitude.

7.4 Doppler Effect

Observed frequency when source and observer move along the line of propagation:

\(f' = f\left(\dfrac{v \pm v{\text{O}}}{v \pm v{\text{S}}}\right)\) (choose signs according to motion towards/away).

7.5 Electromagnetic Spectrum

From longest to shortest wavelength: radio → microwave → infra‑red → visible → ultraviolet → X‑ray → γ‑ray. All travel at \(c = 3.00\times10^{8}\;\text{m s}^{-1}\) in vacuum.

7.6 Practical Demonstration

Use a ripple tank to visualise transverse water waves, measure \(\lambda\) and \(v\) and verify \(v = f\lambda\). For sound, use a tuning fork and a microphone to explore the Doppler shift.

8 Superposition & Stationary Waves

8.1 Principle of Superposition

When two or more waves occupy the same region, the resultant displacement is the algebraic sum of the individual displacements.

8.2 Standing (Stationary) Waves

• Formed by the superposition of two identical waves travelling in opposite directions.
• Nodes – points of zero displacement; antinodes – points of maximum displacement.
• For a string fixed at both ends, allowed wavelengths: \(\lambda_{n}= \dfrac{2L}{n}\) (n = 1,2,3,…).
• Corresponding frequencies: \(f_{n}= n\,\dfrac{v}{2L}\).

8.3 Diffraction & Interference

• Single‑slit diffraction minima at \(\sin\theta = m\lambda/a\) (m = 1,2,…, a = slit width).
• Double‑slit interference bright fringes at \(\sin\theta = n\lambda/d\) (n = 0,±1,±2,…, d = slit separation).

8.4 Illustration

Sketch a vibrating string showing the first three normal modes (n = 1,2,3) with labelled nodes and antinodes.

9 Electricity

9.1 Charge & Current

• Charge \(Q\) (C). Electrons carry \(-e\) ( \(e = 1.60\times10^{-19}\;\text{C}\) ).
• Current \(I = \dfrac{Q}{t}\) (A).

9.2 Potential Difference & Power

Potential difference (voltage) \(V = \dfrac{W}{Q}\) (V). Electrical power:

\(P = VI = I^{2}R = \dfrac{V^{2}}{R}\).

9.3 Resistance & Ohm’s Law

• Resistance \(R = \rho\frac{L}{A}\) (Ω), where \(\rho\) is resistivity.
• Ohm’s law: \(V = IR\) (valid for ohmic conductors).

9.4 Series & Parallel Circuits

• Series: \(R{\text{eq}} = R{1}+R_{2}+…\); same current through each element; voltages add.
• Parallel: \(\dfrac{1}{R{\text{eq}}}= \dfrac{1}{R{1}}+\dfrac{1}{R_{2}}+…\); same voltage across each element; currents add.

9.5 I‑V Characteristics

Plot current \(I\) versus voltage \(V\) for:

• Ohmic resistor – straight line through the origin (gradient = 1/R).
• Non‑ohmic devices (e.g. filament lamp, diode) – curved lines.

9.6 Practical Lab

Build a simple series‑parallel circuit with a battery, resistors, and a voltmeter/ammeter. Measure \(V\) and \(I\) for each configuration, calculate \(R\) and compare with the theoretical values using the series/parallel formulas.

10 DC Circuits

10.1 Kirchhoff’s Laws

• Current law (KCL) – algebraic sum of currents at a junction is zero: \(\sum I{\text{in}} = \sum I{\text{out}}\).
• Voltage law (KVL) – algebraic sum of potential differences around any closed loop is zero: \(\sum V = 0\).

10.2 Potential Divider

For two resistors \(R{1}\) and \(R{2}\) in series across a supply \(V\):

\(V{R{2}} = V\;\dfrac{R{2}}{R{1}+R_{2}}\).

10.3 Internal Resistance of a Cell

Terminal voltage: \(V = \mathcal{E} - Ir\) where \(\mathcal{E}\) is emf and \(r\) the internal resistance.

10.4 Circuit‑Symbol Practice

10.5 Sample Problem

A circuit consists of a 12 V battery (internal resistance 0.5 Ω) supplying three resistors: 4 Ω and 6 Ω in series, the combination in parallel with a 3 Ω resistor. Determine the terminal voltage and the current through each resistor using KVL and KCL.

11 Particle Physics & Nuclear Phenomena

11.1 Structure of the Atom

• Protons (p, +e), neutrons (n, neutral) in the nucleus.
• Electrons (e⁻, –e) in shells.
• Atomic number \(Z\) = number of protons; mass number \(A\) = \(Z\) + neutrons.

11.2 Radioactivity

• Alpha (α) decay: \(^{A}{Z}X \rightarrow\; ^{A-4}{Z-2}Y + ^{4}_{2}\alpha\). Emits He‑2 nuclei, low penetration, high ionisation.
• Beta (β⁻) decay: \(^{A}{Z}X \rightarrow\; ^{A}{Z+1}Y + \beta^{-} + \bar{\nu}\). Emits electrons, moderate penetration.
• Gamma (γ) decay: \(^{A}{Z}X^{*} \rightarrow\; ^{A}{Z}X + \gamma\). High‑energy photons, very penetrating.

11.3 Half‑Life & Decay Law

Number of nuclei remaining after time \(t\):

\(N = N{0}e^{-\lambda t}\) or \(N = N{0}\left(\dfrac{1}{2}\right)^{t/T_{1/2}}\)

where \(\lambda\) is the decay constant and \(T_{1/2}\) the half‑life.

11.4 Binding Energy

Mass defect \(\Delta m = (Z m{p} + (A-Z)m{n} - m{\text{nucleus}})\). Binding energy \(E{b}= \Delta m c^{2}\) (MeV). Provides a measure of nuclear stability.

11.5 Fundamental Particles (Brief)

• Quarks: up (u), down (d), strange (s), charm (c), bottom (b), top (t). Combine to form hadrons.
• Baryons = three quarks (e.g. proton = uud, neutron = udd).
• Mesons = quark–antiquark pair (e.g. pion).

11.6 Sample Calculation

The half‑life of carbon‑14 is 5730 years. How much of a 5.0 g sample remains after 10 000 years? Use the decay law and the relationship \(N\propto\) mass.

Link‑In: Why the Foundations Matter

The concepts introduced in Section 1 (units, significant figures, vectors, uncertainties) are used repeatedly throughout the syllabus. Mastery of these basics ensures that every subsequent calculation – from kinematic equations to nuclear decay – is physically meaningful and correctly reported.

Common Mistakes to Avoid

1. Leaving out units or mixing incompatible units (e.g. cm with s).
2. Confusing symbols that differ only by case (N vs n, I vs i).
3. Ignoring vector direction when adding forces or velocities.
4. Failing to check dimensional consistency before using an equation.
5. Incorrect handling of significant figures in intermediate steps.
6. Using the wrong sign in the Doppler formula or in Newton’s third‑law pairs.

Practice Questions (Mixed Topics)

1. Express the speed of a car travelling \(150\;\text{km}\) in \(2\;\text{h}\) in SI units (m s⁻¹). State the answer with the correct number of significant figures.
2. A 2.00 kg block is pulled up a 30° incline by a constant force of 25 N parallel to the incline. The block moves 1.5 m. Calculate the work done by the pulling force and the increase in gravitational potential energy. Comment on any difference.
3. Two identical springs (spring constant \(k\)) are attached in series and support a 5 kg mass. Find the total extension. Then repeat with the springs in parallel and compare the extensions.
4. A sound source of frequency 500 Hz moves towards a stationary observer at 20 m s⁻¹. The speed of sound is 340 m s⁻¹. What frequency does the observer hear?
5. In a circuit, a 9 V battery (internal resistance 1 Ω) supplies a 3 Ω resistor in series with a parallel combination of 6 Ω and 12 Ω. Determine the current supplied by the battery and the terminal voltage.
6. The half‑life of a radioactive isotope is 2.0 h. A sample initially contains \(4.0\times10^{6}\) nuclei. How many nuclei remain after 5.0 h?