PAF Physics Master Guide: Exhaustive Study Notes for CAE and GDP Candidates

Physics Test Requirements and Syllabus Scope

The physics section for the CAE and GDP tests focuses on short numerical problems that can be solved with a single formula, alongside direct definitions and unit identification. Calculators are strictly prohibited, so numerical values are designed to be resolved through manual calculation. Candidates are expected to have a comprehensive grasp of the F.Sc Part I and Part II syllabus. While both streams cover general physics, the CAE stream places additional high-yield emphasis on vectors, equilibrium, motion, force, waves, physical optics, electrostatics, and magnetism. Familiarity with the definitions of differentiation and integration is also required for the engineering-specific streams.

Measurements and Dimensional Analysis

Physical quantities are divided into base quantities and derived quantities. Base quantities are mutually independent and include length, mass, time, temperature, electric current, luminous intensity, and amount of substance. Derived quantities are formed through the multiplication or division of base quantities, such as area, force, or speed. Measurement quality is defined by precision, which concerns consistency and least count, and accuracy, which describes proximity to the actual value.

Dimensions represent the powers to which base quantities are raised for a given physical quantity, such as force having the dimension $[MLT^{-2}]$. The principle of homogeneity states that any valid physical equation must have identical dimensions on both sides. Measurements also involve significant figures, which include all reliably known digits plus the first uncertain one. Common instruments include Vernier callipers with a least count of $0.01\,\text{cm}$ and screw gauges with a least count of $0.001\,\text{cm}$.

Vectors and Equilibrium

Scalars are quantities characterized solely by magnitude, while vectors require both magnitude and direction. A unit vector possesses a magnitude of $1$ and indicates direction. Rectangular components represent a vector's projections along the $x$, $y$, and $z$ axes. The addition of vectors is typically handled via the head-to-tail rule, while concurrent forces in equilibrium can be analyzed using Lami's theorem.

Vector products include the dot (scalar) product and the cross (vector) product. The dot product is defined as:

$A \cdot B = AB \cos(\theta)$

This is used in calculating work ($W = F \cdot d$). The cross product is defined as:

$A \times B = AB \sin(\theta) \mathbf{\hat{n}}$

This determines quantities like torque ($\tau = r \times F$). Equilibrium occurs when the net force (linear) and net torque (angular) on a body are both zero. The first condition for equilibrium is $\sum F = 0$, and the second condition is $\sum \tau = 0$.

Motion, Force, and Projectile Dynamics

Motion is described by displacement (vector change in position), velocity (rate of change of displacement), and acceleration (rate of change of velocity). Momentum is the product of mass and velocity ($p = mv$), while impulse is the change in momentum ($J = F \Delta t$). Projectile motion is the two-dimensional movement of an object under the influence of gravity alone.

Newton's laws of motion are fundamental: the first law establishes inertia; the second law relates force to the rate of change of momentum ($F = ma$); and the third law states that action and reaction are equal and opposite. The law of conservation of momentum specifies that the total momentum of an isolated system remains constant. Newton's law of gravitation determines the force between two masses:

$F = G \frac{m_1 m_2}{r^2}$

Work, Energy, and Power

Work is performed when a force causes displacement in its direction ($W = F \cdot d \cos(\theta)$). Energy, the capacity to do work, exists as kinetic energy ($KE = \frac{1}{2} m v^2$) and potential energy ($PE = mgh$). Power measures the rate of doing work ($P = \frac{W}{t}$). The work-energy theorem states that work done equals the change in kinetic energy. The law of conservation of energy dictates that energy cannot be created or destroyed, only transformed. Escape velocity is the speed required to leave a planet's gravity permanently:

$v_e = \sqrt{2 g R}$

Circular Motion and Gravitation

Circular motion involves angular displacement, angular velocity ($\omega = \frac{d\theta}{dt}$), and centripetal acceleration ($a_c = \frac{v^2}{r}$). Centripetal force is necessary to maintain a circular path. Moment of inertia ($I = \sum m r^2$) measures resistance to angular acceleration, and angular momentum ($L = I \omega$) remains constant in the absence of external torque. Kepler's laws describe planetary orbits as elliptical, with the square of the orbital period proportional to the cube of the semi-major axis ($T^2 \propto r^3$).

Fluid Dynamics and Oscillations

Fluid dynamics covers ideal fluids (incompressible and non-viscous) and different flow patterns like steady (streamline) flow. Viscosity provides internal resistance to flow, and terminal velocity ($v_t$) is reached when drag force equals gravitational pull. The equation of continuity ($A_1 v_1 = A_2 v_2$) and Bernoulli's equation describe fluid behavior under conservation of mass and energy. Stokes' law calculates viscous drag on spheres:

$F = 6 \pi \eta r v$

Simple Harmonic Motion (SHM) is characterized by acceleration proportional to displacement and directed toward a mean position. Resonance occurs when a system is driven at its natural frequency, leading to maximum amplitude. Hooke's law defines the restoring force of a spring as $F = -kx$.

Waves and Physical Optics

Waves transfer energy without matter. Transverse waves vibrate perpendicular to travel, while longitudinal waves vibrate parallel. The Doppler effect explains the shift in frequency due to relative motion. Stationary waves exhibit fixed nodes and antinodes. Light phenomena include interference (coherent superposition), diffraction (bending around obstacles), and polarization, which proves light is a transverse wave. Malus's law calculates the intensity of polarized light through an analyzer:

$I = I_0 \cos^2(\theta)$

Optical Instruments and Thermodynamics

Optical instruments include simple and compound microscopes and astronomical telescopes. Magnifying power is calculated as the ratio of angle subtended by the image to that of the object. Myopia is corrected by diverging lenses, while hypermetropia is corrected by converging lenses. Thermodynamics deals with internal energy, which is the sum of kinetic and potential energies of molecules. Isothermal processes occur at constant temperature, while adiabatic processes involve no heat exchange. The first law is $Q = \Delta U + W$, and the second law explains that entropy in an isolated system never decreases.

Electrostatics and Current Electricity

Electrostatics involves electric charge ($q = ne$), field strength ($E = \frac{F}{q}$), and potential ($V = \frac{W}{q}$). Coulomb's law defines the force between point charges:

$F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2}$

Gauss's law relates flux to enclosed charge. Current electricity covers the flow of charge ($I = \frac{Q}{t}$), resistance ($R = \frac{V}{I}$), and resistivity. Ohm's law states $V = IR$ at constant temperature. Kirchhoff's rules govern charge preservation at junctions and energy conservation in loops. The Wheatstone bridge is used for measuring unknown resistance.

Electromagnetism and Induction

Magnetic flux density ($B$) is measured in Tesla. The Lorentz force describes a charge in combined fields:

$F = q\mathbf{E} + q(\mathbf{v} \times \mathbf{B})$

Electromagnetic induction involves producing EMF through changing magnetic flux. Faraday's law states induced EMF is proportional to the rate of flux change, while Lenz's law indicates induced current opposes the change that created it. Transformers and inductors utilize these principles to manage alternating current.

Modern and Nuclear Physics

Modern physics includes the photoelectric effect ($KE_{\text{max}} = hf - \Phi$), time dilation, and the de Broglie wavelength ($\lambda = \frac{h}{p}$). Einstein's energy equation is $E = mc^2$. Atomic spectra reveal quantized energy levels and are described by the Bohr postulates. Nuclear physics studies the nucleus, isotopes, and binding energy. Radioactivity involves spontaneous decay governed by:

$N = N_0 e^{-\lambda t}$

Nuclear fission splits heavy nuclei, while nuclear fusion combines light nuclei, both releasing immense energy.

Physical Constants Reference

$g \approx 9.8\,\text{m/s}^2$

$G \approx 6.673 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2$

$c = 3 \times 10^{8}\,\text{m/s}$

$h = 6.63 \times 10^{-34}\,\text{J}\cdot\text{s}$

$e = 1.6 \times 10^{-19}\,\text{C}$

$\epsilon_0 = 8.85 \times 10^{-12}\,\text{C}^2/(\text{N}\cdot\text{m}^2)$

$\mu_0 = 4 \pi \times 10^{-7}\,\text{T}\cdot\text{m/A}$

$k \approx 9 \times 10^{9}\,\text{N}\cdot\text{m}^2/\text{C}^2$

$m_e \approx 9.1 \times 10^{-31}\,\text{kg}$

$m_p \approx 1.67 \times 10^{-27}\,\text{kg}$

$N_A = 6.022 \times 10^{23}\,\text{mol}^{-1}$