Cambridge AS Level Physics 9702 Complete Theory and Practical Guide

Physical Quantities and Units

• Every physical quantity is composed of a numerical magnitude and a unit.

• The International System of Units (SI) defines six fundamental base quantities and their corresponding units:

  • Mass: kilogram ($kg$)

  • Length: metre ($m$)

  • Time: second ($s$)

  • Electric Current: ampere ($A$)

  • Thermodynamic Temperature: kelvin ($K$)

  • Amount of Substance: mole ($mol$)

• Derived units are obtained by using the defining equations for a quantity and substituting the base units into that equation:

  • Force ($F = ma$): $1\,N = 1\,kg\,m\,s^{-2}$

  • Work or Energy ($W = Fs$): $1\,J = 1\,kg\,m^2\,s^{-2}$

  • Pressure ($P = \frac{F}{A}$): $1\,Pa = 1\,kg\,m^{-1}\,s^{-2}$

  • Voltage ($V = \frac{W}{Q} = \frac{W}{I \times t}$): $1\,V = 1\,kg\,m^2\,s^{-3}\,A^{-1}$

• Physical quantities are categorized as either scalars or vectors:

  • Scalars possess magnitude only. Examples include mass, distance, speed, energy, density, power, and temperature.

  • Vectors possess both magnitude and direction. Examples include displacement, velocity, acceleration, force, and momentum.

• Vector resolution and addition for a vector $V$ at an angle $\theta$ to the horizontal involve the following equations:

  • Horizontal component ($V_x$): $V_x = V \cos(\theta)$

  • Vertical component ($V_y$): $V_y = V \sin(\theta)$

  • Resultant magnitude ($V$): $V = \sqrt{V_x^2 + V_y^2}$

  • Angle ($\theta$): $\theta = \tan^{-1}(\frac{V_y}{V_x})$

Kinematics and Dynamics

• Equations of motion (SUVAT) apply only when acceleration ($a$) is constant along a straight line. The variables are defined as follows: $s$ (displacement), $u$ (initial velocity), $v$ (final velocity), $a$ (acceleration), and $t$ (time):

  1. $v = u + at$

  2. $s = ut + \frac{1}{2}at^2$

  3. $s = \frac{1}{2}(u + v)t$

  4. $v^2 = u^2 + 2as$

• Projectile motion treats horizontal and vertical components as independent:

  • Horizontal motion: Operates at constant velocity because horizontal acceleration is zero ($a_x = 0$). Distance is calculated as $(u \cos(\theta)) \times t$.

  • Vertical motion: Operates at constant acceleration due to gravity ($a_y = -g = -9.81\,m\,s^{-2}$). SUVAT equations are used with the vertical initial velocity component $u_y = u \sin(\theta)$.

• Newton's Laws of Motion:

  • Newton's First Law: A body remains in its state of rest or uniform motion in a straight line unless acted upon by a resultant external force.

  • Newton's Second Law: The resultant force is directly proportional to the rate of change of momentum: $F = \frac{\Delta p}{\Delta t} = ma$.

  • Newton's Third Law: If body A exerts a force on body B, body B exerts an equal and opposite force on body A.

• Linear momentum ($p$) is the product of mass and velocity ($p = mv$), measured in units of $kg\,m\,s^{-1}$ or $N\,s$.

• The Principle of Conservation of Momentum states that if no external forces act, the total momentum before a collision equals the total momentum after: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$.

• Collisions are classified by energy conservation:

  • Elastic Collisions: Momentum, total energy, and kinetic energy are all conserved. The relative speed of approach equals the relative speed of separation: $u_1 - u_2 = v_2 - v_1$.

  • Inelastic Collisions: Momentum and total energy are conserved, but kinetic energy is not conserved (it is converted to heat or sound). The speed of approach is greater than the speed of separation.

Forces, Density, Pressure, and Equilibrium

• Center of Gravity is the specific point through which the entire weight of a body is considered to act.

• The Moment of a Force is defined as the force multiplied by the perpendicular distance from the pivot ($M = F \times d$).

• A Couple consists of a pair of equal and opposite parallel forces that produce rotation only. The torque of a couple is calculated as the force multiplied by the perpendicular distance between the forces.

• The Principle of Moments states that for a body in equilibrium, the sum of clockwise moments about any point must equal the sum of anticlockwise moments about that same point.

• Density ($\rho$) is the ratio of mass to volume: $\rho = \frac{m}{V}$.

• Pressure ($P$) is the force per unit area: $P = \frac{F}{A}$.

• Hydrostatic Pressure in a fluid column is calculated by: $\Delta P = \rho g h$.

Work, Energy, Power, and Deformation

• Work Done ($W$) is calculated as: $W = F s \cos(\theta)$.

• Gravitational Potential Energy changes ($\Delta E_p$) are calculated as: $\Delta E_p = m g \Delta h$.

• Kinetic Energy ($E_k$) is given by: $E_k = \frac{1}{2} m v^2$.

• Power ($P$) is the rate of work done or energy transfer: $P = \frac{\Delta W}{\Delta t} = F v$.

• Efficiency ($\eta$) is expressed as a percentage: $\eta = (\frac{\text{Useful Power Output}}{\text{Total Power Input}}) \times 100\%$.

• Deformation of Solids:

  • Hooke's Law: Force is directly proportional to extension ($F = k x$), provided the limit of proportionality is not exceeded.

  • Stress ($\sigma$): The force per unit cross-sectional area, measured in $Pa$ or $N\,m^{-2}$: $\sigma = \frac{F}{A}$.

  • Strain ($\epsilon$): The extension per unit original length, which is dimensionless: $\epsilon = \frac{x}{L}$.

  • Young Modulus ($E$): The ratio of stress to strain within the elastic limit: $E = \frac{\sigma}{\epsilon} = \frac{F L}{A x}$.

  • Elastic Strain Energy: The energy stored in a deformed material, equivalent to the area under the force-extension graph: $E = \frac{1}{2} F x = \frac{1}{2} k x^2$.

Waves and Superposition

• Fundamental Wave Equations:

  • Velocity: $v = f \lambda$

  • Period ($T$): $T = \frac{1}{f}$

  • Intensity ($I$): The power per unit area: $I = \frac{P}{4 \pi r^2}$

  • Key Relationship: Intensity is proportional to the square of the amplitude: $I \propto A^2$

• Wave Types:

  • Transverse Waves: Oscillations occur perpendicular to the direction of energy propagation (e.g., light, electromagnetic waves, water waves).

  • Longitudinal Waves: Oscillations occur parallel to the direction of energy propagation (e.g., sound waves). These waves consist of compressions and rarefactions.

• The Doppler Effect in sound describes the observed frequency shift when a source moves relative to an observer:

  • $f_o = \frac{f_s v}{v \pm v_s}$

  • A minus sign is used in the denominator when the source moves toward the observer; a plus sign is used when moving away.

• Electromagnetic Spectrum Wavelengths:

  • Radio Waves: > 10^{-1}\,m

  • Microwaves: $10^{-2}\,m$

  • Infrared: $10^{-5}\,m$

  • Visible Light: $400\text{--}700\,nm$

  • Ultraviolet: $10^{-8}\,m$

  • X-rays: $10^{-10}\,m$

  • Gamma Rays: < 10^{-12}\,m

• Superposition and Interference:

  • Principle of Superposition: When multiple waves overlap, the resultant displacement is the vector sum of the individual displacements.

  • Young's Double Slit (Two-Slit Interference): $\lambda = \frac{a x}{D}$, where $a$ is slit separation, $x$ is fringe separation, and $D$ is the distance to the screen.

  • Diffraction Grating: $d \sin(\theta) = n \lambda$, where $d$ is grating spacing ($\frac{1}{N}$ lines per meter) and $n$ is the order number.

  • Stationary (Standing) Waves: Formed by the superposition of identical waves traveling in opposite directions. They feature Nodes (zero amplitude) and Antinodes (maximum amplitude). The distance between adjacent nodes is $\frac{\lambda}{2}$.

Electricity and DC Circuits

• Fundamental Electrical Quantities:

  • Charge: $\Delta Q = I \Delta t$

  • Current: Defined as $I = n A v q$, where $n$ is charge carrier density, $A$ is cross-sectional area, $v$ is drift velocity, and $q$ is the charge of the carrier.

  • Potential Difference / Electromotive Force (EMF): $V = \frac{W}{Q}$

  • Ohm's Law: $V = I R$

  • Resistivity ($\rho$): $R = \frac{\rho L}{A}$

  • Electrical Power: $P = V I = I^2 R = \frac{V^2}{R}$

• Circuit Rules:

  • Kirchhoff's First Law: Conservation of Charge states the sum of currents entering a junction equals the sum of currents leaving ($\sum I_{in} = \sum I_{out}$).

  • Kirchhoff's Second Law: Conservation of Energy states the sum of e.m.f.s in a closed loop equals the sum of potential differences ($\sum E = \sum V$).

  • Series Resistors: $R_{total} = R_1 + R_2 + R_3$

  • Parallel Resistors: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

  • Internal Resistance ($r$): $E = I(R + r) = V_{terminal} + I r$

• Potential Divider Equation for output voltage across $R_2$:

  • $V_{out} = V_{in} \times [\frac{R_2}{R_1 + R_2}]$

  • Commonly utilized with LDRs for light sensors and thermistors for temperature sensors.

Particle Physics and Radioactivity

• Subatomic particle composition and charge:

  • Proton: Composition $uud$; Charge $+1\,e$

  • Neutron: Composition $udd$; Charge $0$

  • Antiproton: Composition $\bar{u}\bar{u}\bar{d}$; Charge $-1\,e$

  • Antineutron: Composition $\bar{u}\bar{d}\bar{d}$; Charge $0$

• Quark Properties:

  • Up ($u$): Charge $+\frac{2}{3}\,e$

  • Down ($d$): Charge $-\frac{1}{3}\,e$

  • Anti-up ($\bar{u}$): Charge $-\frac{2}{3}\,e$

  • Anti-down ($\bar{d}$): Charge $+\frac{1}{3}\,e$

• Radioactive Decay Equations:

  • Alpha Decay ($\alpha$): Emission of a Helium nucleus ($_{2}^{4}He$). Equation: $_{Z}^{A}X \rightarrow _{Z-2}^{A-4}Y + _{2}^{4}He$

  • Beta-Minus Decay ($\beta^-$): A neutron transforms into a proton via the weak interaction ($d \rightarrow u + e^- + \bar{\nu}_e$). Equation: $_{Z}^{A}X \rightarrow _{Z+1}^{A}Y + _{-1}^{0}e + \bar{\nu}_e$

  • Beta-Plus Decay ($\beta^+$): A proton transforms into a neutron ($u \rightarrow d + e^+ + \nu_e$). Equation: $_{Z}^{A}X \rightarrow _{Z-1}^{A}Y + _{+1}^{0}e + \nu_e$

  • Gamma Decay ($\gamma$): Nucleus releases excess energy as high-frequency photons; mass number and atomic number remain unchanged.

Practical Lab Skills and Measurement Rules

• Precision and raw data recording standards:

  • Metre rule or tape measure: Record to $1\,mm$ (e.g., $12.3\,cm$ or $0.123\,m$).

  • Vernier Calliper: Record to $0.01\,cm$ or $0.1\,mm$.

  • Micrometer Screw Gauge: Record to $0.01\,mm$.

  • Digital Stopwatch: Record to $0.01\,s$ or $0.1\,s$. When timing oscillations, time multiple cycles (e.g., $20\,T$) before calculating the period $T$.

• Calculated Values: The number of significant figures (s.f.) must be equal to or one more than the minimum s.f. of the raw data used.

• Table Headers: Must include quantity name/symbol and units separated by a solidus or brackets (e.g., $L/cm$, $T^2/s^2$, $1/R\,(\Omega^{-1})$).

• Uncertainty Calculations:

  • Absolute Uncertainty ($\Delta x$): Usually the smallest scale division or half the range for repeated values.

  • Percentage Uncertainty: $\frac{\Delta x}{x_{mean}} \times 100\%$

  • Combining Uncertainties if $y = a \pm b$: $\Delta y = \Delta a + \Delta b$

  • Combining Uncertainties if $y = a \times b$ or $y = \frac{a}{b}$: $\Delta \% y = \Delta \% a + \Delta \% b$

  • Combining Uncertainties if $y = a^n$: $\Delta \% y = n \times (\Delta \% a)$

• Graph Execution Requirements:

  • Scale Selection: Data points must occupy more than half the grid. Scales must be simple/linear (e.g., $1, 2, 5, 10$ units per square). Ratios like $3, 7, 13$ are prohibited.

  • Plotting: Use small 'x' marks. Centers must be accurate within half a small square.

  • Line of Best Fit (LBF): Points should be distributed evenly above and below the line along its entire length. Forced origins are avoided unless theoretical justification exists.

  • Gradient ($m$): Calculated using a large right-angled triangle with a hypotenuse at least half the length of the LBF. Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

  • Y-Intercept ($c$): Read directly from the $x = 0$ axis or calculate via $c = y_1 - m x_1$.

Practical Limitations and Improvements

• Limitation: Two readings are insufficient to draw a valid conclusion.

  • Improvement: Take more readings for various values and plot a graph of $y$ against $x$.

• Limitation: Difficulty measuring the diameter of thin wire due to non-uniform thickness.

  • Improvement: Measure diameter at multiple positions along the wire using a micrometer and calculate the average.

• Limitation: Parallax error when measuring height or length with a ruler.

  • Improvement: Position a set-square flush against the scale and the object, or ensure reading is taken at eye level.

• Limitation: Friction at the pivot or pulley affects acceleration and force readings.

  • Improvement: Apply lubricant, use an air track, or use light gates with a digital timer.

• Limitation: Terminal velocity is not reached or oscillation stops too quickly due to damping.

  • Improvement: Use a taller column of liquid or use video recording with a grid scale to analyze motion frame by frame.