Work, energy and power An understanding of the forms of energy and energy transfers from Cambridge IGCSE/O Level Physics or equivalent is assumed.

Cambridge IGCSE / O‑Level Physics – Core Syllabus Notes

How to Use These Notes

• Each major syllabus block is presented in a definition → formula → example → typical exam question format.
• Key equations are collected in summary tables at the end of each block for quick revision.
• Worked examples mirror the style of past Cambridge papers (AO1–AO3) and include step‑by‑step reasoning.
• Practical‑skills tips (Paper 3 & 5) are provided in a dedicated appendix.

1. Quantities, Scalars & Vectors

1.1 Fundamental Quantities

• Base units (SI): metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd).
• Derived units are built from base units (e.g., newton = kg·m s⁻², joule = kg·m² s⁻²).

1.2 Scalars vs Vectors

1.3 Vector Operations

• Resultant of two vectors (graphical tip‑to‑tail method) or analytically using components.
• Dot product \(\mathbf{A}\!\cdot\!\mathbf{B}=AB\cos\theta\) (used for work).
• Cross product \(\mathbf{A}\!\times\!\mathbf{B}=AB\sin\theta\) (direction given by right‑hand rule, used for torque).

2. Kinematics – Motion in One and Two Dimensions

2.1 Motion Graphs

• Displacement‑time, velocity‑time, and acceleration‑time graphs convey the same information; the slope of a \(v\)–\(t\) graph gives acceleration, the area under a \(v\)–\(t\) graph gives displacement, etc.
• Key shapes: straight‑line (uniform acceleration), horizontal (constant velocity), curved (non‑uniform).

2.2 Equations of Motion (constant acceleration)

\[

\begin{aligned}

v &= u + at\\

s &= ut + \tfrac12 at^{2}\\

v^{2} &= u^{2} + 2as

\end{aligned}

\]

where \(u\) = initial speed, \(v\) = final speed, \(a\) = acceleration, \(s\) = displacement, \(t\) = time.

2.3 Projectile Motion (2‑D)

• Horizontal motion: \(x = u_{x}t\) (constant velocity).
• Vertical motion: \(y = u_{y}t - \tfrac12gt^{2}\) (downward acceleration \(g=9.8\;\text{m s}^{-2}\)).
• Range \(R = \dfrac{u^{2}\sin2\theta}{g}\) for launch speed \(u\) and angle \(\theta\).

Worked Example – Throwing a Ball Horizontally

A ball is thrown horizontally from a 1.5 m high table with a speed of 4.0 m s⁻¹. Find the time to reach the ground and the horizontal distance travelled.

1. Vertical motion: \(y = \tfrac12gt^{2}\) → \(1.5 = 0.5(9.8)t^{2}\) → \(t = 0.55\;\text{s}\).
2. Horizontal distance: \(x = u_{x}t = 4.0 \times 0.55 = 2.2\;\text{m}\).

3. Dynamics – Forces, Momentum & Friction

3.1 Newton’s Laws

1. First law (inertia): A body remains at rest or in uniform motion unless acted on by a net external force.
2. Second law: \(\displaystyle \sum\mathbf{F}=m\mathbf{a}\).
3. Third law: For every action there is an equal and opposite reaction.

3.2 Momentum and Impulse

\[

\mathbf{p}=m\mathbf{v},\qquad

\Delta\mathbf{p}= \mathbf{F}_{\text{net}}\Delta t

\]

Impulse (area under a force‑time graph) changes momentum.

3.3 Friction

• Static friction: \(F{s}^{\max}= \mu{s}N\) (prevents motion).
• Kinetic friction: \(F{k}= \mu{k}N\) (opposes sliding).
• \(N\) is the normal reaction; \(\mu{s},\mu{k}\) are coefficients (dimensionless).

Worked Example – Block on a Rough Incline

A 3.0 kg block rests on a 25° incline. \(\mu{s}=0.35\), \(\mu{k}=0.25\). Find the minimum force parallel to the plane needed to start the block moving up.

1. Component of weight down the plane: \(W_{\parallel}=mg\sin25^{\circ}=3.0\times9.8\times0.423=12.4\;\text{N}\).
2. Normal reaction: \(N=mg\cos25^{\circ}=3.0\times9.8\times0.906=26.6\;\text{N}\).
3. Maximum static friction: \(F{s}^{\max}= \mu{s}N=0.35\times26.6=9.3\;\text{N}\).
4. Required upward force: \(F{\text{min}} = W{\parallel}+F_{s}^{\max}=12.4+9.3=21.7\;\text{N}\).

4. Forces, Density & Pressure – Turning Effects of Forces

4.1 Translational Equilibrium

\[

\sum F{x}=0,\qquad \sum F{y}=0

\]

4.2 Rotational (Turning) Equilibrium

\[

\sum \tau =0

\]

• Torque: \(\displaystyle \boldsymbol{\tau}= \mathbf{r}\times\mathbf{F},\quad |\tau|=rF\sin\theta\).
• Units: newton‑metre (N·m).

Couples

Two equal, opposite, non‑collinear forces. Net torque \(\tau_{\text{couple}} = Fd\) (independent of reference point).

Principle of Moments (Law of the Lever)

\[

\sum rF{\text{clockwise}} = \sum rF{\text{anticlockwise}}

\]

Centre of Gravity (CG)

Point at which the total weight of a body may be considered to act. For uniform bodies CG = geometric centre; for irregular bodies locate experimentally (e.g., plumb‑line method).

Worked Example – Ladder Against a Smooth Wall

1. Diagram: ladder length \(L\), angle \(\theta\) with ground, weight \(W=mg\) acting at midpoint.
2. Translational equilibrium:

   • \(\sum F{x}: R = F{\text{wall}}\) (horizontal reaction at wall).
   • \(\sum F_{y}: N = W\) (normal reaction at ground).

3. Rotational equilibrium about the base:

   \[

   R\,L\cos\theta = W\,\frac{L}{2}\sin\theta

   \;\Rightarrow\;

   R = \frac{W}{2}\tan\theta

   \]

4. Result: the wall experiences a horizontal force \(R\) and the ground a vertical reaction \(N=W\).

5. Work, Energy and Power

5.1 Work

• Definition: Transfer of energy when a force causes a displacement.
• Constant force: \(\displaystyle W = \mathbf{F}\!\cdot\!\mathbf{s}=Fs\cos\theta\).
• Variable force: \(\displaystyle W = \int{si}^{s_f}\mathbf{F}\!\cdot\!d\mathbf{s}\).
• Units: joule (J) = N·m.

5.2 Kinetic Energy

\[

E_{k}= \frac12 mv^{2}

\]

Work‑Energy Theorem: Net work done on a particle equals the change in its kinetic energy, \(\displaystyle W{\text{net}}=\Delta E{k}\).

5.3 Potential Energy

• Gravitational (near Earth): \(E_{g}=mgh\).
• Elastic (ideal spring): \(E_{s}= \tfrac12 kx^{2}\) (Hooke’s law).

5.4 Conservation of Mechanical Energy

If only conservative forces do work:

\[

E{\text{total}} = E{k}+E_{\text{potential}} = \text{constant}

\]

When non‑conservative forces (e.g., kinetic friction) act:

\[

E{k}+E{\text{potential}}+W_{\text{nc}} = \text{constant}

\]

5.5 Power

\[

P = \frac{dW}{dt}= \frac{\Delta E}{\Delta t}

\]

• Constant force at constant speed: \(P = Fv\cos\theta\).
• Electrical form: \(P = IV = I^{2}R = \dfrac{V^{2}}{R}\).
• Units: watt (W) = J s⁻¹.

5.6 Energy in Circular Motion (A‑Level preview)

• Centrepetal force does no work (always perpendicular to displacement); kinetic energy remains constant if speed is constant.
• For a mass \(m\) moving in a circle of radius \(r\) at speed \(v\): \(\displaystyle F_{c}= \frac{mv^{2}}{r}\).

Worked Example – Block on an Incline Pulled by a Horizontal Force

Block mass \(m=5.0\;\text{kg}\), smooth incline \(\alpha=30^{\circ}\), horizontal pull \(F=40\;\text{N}\). The block moves \(s=2.0\;\text{m}\) up the plane in \(t=4.0\;\text{s}\).

1. Resolve the horizontal force:

   \[

   F_{\parallel}=F\cos\alpha=34.6\;\text{N},\quad

   F_{\perp}=F\sin\alpha=20.0\;\text{N}

   \]

2. Since the plane is smooth, the only force parallel to the plane is the component of the pull and the component of weight \(mg\sin\alpha=24.5\;\text{N}\) down the plane.

   \[

   \sum F{\parallel}=0\;\Rightarrow\;F{\parallel}=mg\sin\alpha\; \text{(no tension needed).}

   \]

3. Work done by the horizontal pull:

   \[

   W = Fs\cos\alpha = 40\times2.0\times\cos30^{\circ}=69.3\;\text{J}

   \]

4. Average power:

   \[

   P_{\text{avg}} = \frac{W}{t}= \frac{69.3}{4.0}=17.3\;\text{W}

   \]

6. Deformation – Stress, Strain & Young’s Modulus

6.1 Definitions

• Stress \(\displaystyle \sigma = \frac{F}{A}\) (N m⁻² or Pa).
• Strain \(\displaystyle \varepsilon = \frac{\Delta L}{L}\) (dimensionless).
• Young’s Modulus \(\displaystyle Y = \frac{\sigma}{\varepsilon}\) (Pa).

6.2 Hooke’s Law (Elastic region)

\[

F = kx,\qquad k = \frac{Y A}{L}

\]

Worked Example – Stretching a Wire

A steel wire (length \(L=1.2\;\text{m}\), area \(A=2.0\times10^{-6}\;\text{m}^{2}\)) is loaded with a force of 500 N. Find the extension \(\Delta L\). (Young’s modulus for steel \(Y=2.0\times10^{11}\;\text{Pa}\)).

\[

\Delta L = \frac{FL}{YA}= \frac{500\times1.2}{2.0\times10^{11}\times2.0\times10^{-6}}=1.5\times10^{-3}\;\text{m}=1.5\;\text{mm}

\]

7. Waves – General Properties & Superposition

7.1 Wave Basics

• Wave speed: \(\displaystyle v = f\lambda\) (where \(f\) is frequency, \(\lambda\) wavelength).
• Transverse vs longitudinal waves.
• Medium properties: tension \(T\) and linear mass density \(\mu\) for a string give \(v=\sqrt{T/\mu}\).

7.2 Superposition & Interference

• When two waves occupy the same region, the resultant displacement is the algebraic sum of the individual displacements.
• Constructive interference: amplitudes add (maxima coincide).
  

  Destructive interference: amplitudes subtract (minima coincide).

7.3 Standing Waves (String & Air Column)

• Fundamental frequency for a string fixed at both ends: \(\displaystyle f_{1}= \frac{1}{2L}\sqrt{\frac{T}{\mu}}\).
• Open–open pipe: \(f_{n}= n\frac{v}{2L}\) (n = 1,2,3,…).
  

  Closed–open pipe: \(f_{n}= n\frac{v}{4L}\) (n = 1,3,5,…).

Worked Example – Fundamental of a Stretched String

A string of length 0.8 m, tension 40 N, and linear density \(2.5\times10^{-3}\;\text{kg m}^{-1}\) is plucked. Find the fundamental frequency.

\[

v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{40}{2.5\times10^{-3}}}=126.5\;\text{m s}^{-1}

\]

\[

f_{1}= \frac{v}{2L}= \frac{126.5}{2\times0.8}=79.1\;\text{Hz}

\]

8. Electricity – Charge, Current, Potential & Resistance

8.1 Fundamental Concepts

• Charge \(Q\) (C), current \(I = \dfrac{dQ}{dt}\) (A).
• Potential difference \(V\) (V) and electric field \(E = V/d\).
• Resistance \(R = \dfrac{V}{I}\) (Ω).

8.2 Ohm’s Law & Power

\[

V = IR,\qquad P = IV = I^{2}R = \frac{V^{2}}{R}

\]

8.3 Energy in Electrical Circuits

\[

W = Pt = VIt

\]

Worked Example – Resistor Heating

A 10 Ω resistor carries a current of 3 A for 5 s. Find the energy dissipated as heat.

\[

P = I^{2}R = 3^{2}\times10 = 90\;\text{W}

\]

\[

W = Pt = 90\times5 = 450\;\text{J}

\]

9. DC Circuits – Series, Parallel & Kirchhoff’s Laws

9.1 Series Circuits

• Current same through each element: \(I_{\text{total}} = I\).
• Total resistance: \(R{\text{total}} = R{1}+R_{2}+ \dots\).
• Voltage divides: \(V{k}=IR{k}\).

9.2 Parallel Circuits

• Voltage same across each branch: \(V_{\text{total}} = V\).
• Reciprocal total resistance: \(\displaystyle \frac{1}{R{\text{total}}}= \frac{1}{R{1}}+\frac{1}{R_{2}}+\dots\).
• Current divides: \(I{k}=V/R{k}\).

9.3 Kirchhoff’s Rules

1. Junction rule (conservation of charge): \(\displaystyle \sum I{\text{in}} = \sum I{\text{out}}\).
2. Loop rule (conservation of energy): \(\displaystyle \sum V = 0\) around any closed loop (taking signs according to direction).

Worked Example – Mixed Circuit

A 12 V battery supplies a series combination of 4 Ω and a parallel branch of 6 Ω and 12 Ω. Find the total current and the voltage across the 6 Ω resistor.

1. Equivalent resistance of parallel branch:

   \(\displaystyle \frac{1}{R{p}} = \frac{1}{6}+\frac{1}{12}= \frac{1}{4}\Rightarrow R{p}=4\;\Omega.\)

2. Total resistance: \(R_{T}=4+4=8\;\Omega.\)
3. Total current: \(I = V/R_{T}=12/8=1.5\;\text{A}.\)
4. Voltage across parallel branch: \(V{p}=IR{p}=1.5\times4=6\;\text{V}.\)
5. Current through 6 Ω resistor: \(I{6}=V{p}/6=6/6=1.0\;\text{A}.\)

10. Particle Physics – Structure of the Atom & Radioactivity

10.1 Atomic Structure

• Protons (+e), neutrons (0), electrons (‑e).
• Atomic number \(Z\) = number of protons; mass number \(A\) = protons + neutrons.
• Isotopes: same \(Z\), different \(A\).

10.2 Radioactive Decay

• Alpha (\(\alpha\)) decay: loss of 2p + 2n, \(Q\)‑value releases kinetic energy.
• Beta (\(\beta\)) decay: neutron → proton + electron + antineutrino (or reverse).
• Gamma (\(\gamma\)) decay: emission of high‑energy photon; no change in \(Z\) or \(A\).

10.3 Half‑Life

\[

N = N{0}\left(\frac{1}{2}\right)^{t/t{1/2}}

\]

where \(N\) = remaining nuclei after time \(t\), \(t_{1/2}\) = half‑life.

Worked Example – Carbon‑14 Dating

A sample contains 25 % of the original \(\,^{14}\)C activity. Given \(t_{1/2}=5730\) yr, find its age.

\[

0.25 = \left(\frac{1}{2}\right)^{t/5730}\;\Rightarrow\; \frac{t}{5730}=2\;\Rightarrow\; t=11\,460\;\text{yr}

\]

11. A‑Level Extensions (Preview)

These topics are not required for the IGCSE/O‑Level exam but form the bridge to the full A‑Level syllabus and are useful for deeper understanding.

12. Practical Skills (Paper 3 & 5)

12.1 Required Laboratory Techniques

• Using a metre rule, vernier caliper and micrometer for length measurements (uncertainty ±0.1 mm).
• Measuring mass with a balance (±0.01 kg).
• Timing with a stopwatch (±0.2 s) and using a photogate for higher precision.
• Voltage and current measurements with a digital multimeter (±0.5 % of reading).
• Using a force sensor or spring balance to obtain forces (±0.1 N).
• Plotting graphs (e.g., \(F\)–\(x\) for springs, \(V\)–\(I\) for resistors) and determining gradients and areas under curves.

12.2 Typical Practical Questions

1. Design a method to determine the coefficient of kinetic friction for a wooden block on a metal surface using an inclined plane.
2. Analyse data from a spring‑extension experiment to calculate the spring constant and its uncertainty.
3. Investigate energy loss in a pendulum by measuring successive amplitudes and plotting \(\ln A\) versus time.
4. Determine internal resistance of a cell using the method of volt‑ampere characteristics.

12.3 Reporting Results

• State the aim, apparatus diagram (with labelled components), procedure, and safety considerations.
• Present raw data in clear tables, include uncertainties, and show calculations step‑by‑step.
• Conclude by comparing experimental values with theoretical predictions, discussing possible sources of error.

13. Summary Tables of Key Equations

13.1 Mechanics

13.2 Energy & Power

13.3 Deformation

13.4 Waves

13.5 Electricity & Circuits

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