STPM Physics 960 cheat sheet

SI base units: kg, m, s, A, K, mol, cd. Derived units are products of base units, e.g. N = kg m s⁻², J = kg m² s⁻², Pa = kg m⁻¹ s⁻².

Dimensional analysis: each physical quantity has dimensions expressed as [M^a L^b T^c I^d Θ^e N^f J^g]. Equations must be dimensionally homogeneous; dimensionless quantities include angle, strain, refractive index, relative density.

Uncertainty rules: for y = a ± b, absolute uncertainties add: Δy = Δa + Δb. For y = ab or a/b, fractional uncertainties add: Δy/y = Δa/a + Δb/b. For y = aⁿ, Δy/y = |n| Δa/a.

Random uncertainty from repeated readings: mean = Σx/N; standard uncertainty of the mean ≈ s/√N, where s is the sample standard deviation. Systematic uncertainty shifts all readings in one direction and is not reduced by averaging.

Precision vs accuracy: precision describes closeness of repeated readings to each other (small random spread); accuracy describes closeness of the mean to the true value (small systematic error).

Reading instruments: for analogue scales, uncertainty is typically half the smallest division (or the smallest division for a ruler end-to-end); for digital displays, uncertainty is at least ±1 in the last digit.

Vectors: resolve into perpendicular components, add components separately, then recombine with Pythagoras and tan θ = (Σy)/(Σx). For two vectors P and Q at angle θ, resultant magnitude R² = P² + Q² + 2PQ cos θ.

Scalar product A·B = AB cos θ (work, power); vector product |A×B| = AB sin θ (torque, magnetic force). Direction of A×B is given by the right-hand rule.

Graphical analysis: straight-line form y = mx + c. Common linearisations: T² = (4π²/g)L for pendulum; V = (nR T)/P for ideal gas; ln k = ln A − Ea/(RT) for Arrhenius; pV = constant for Boyle's law.

Percentage difference and percentage error: % error = |measured − accepted| / accepted × 100%; % uncertainty = (Δx/x) × 100%.

Equations for uniform acceleration: v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t. Valid only when acceleration is constant.

Projectile motion: horizontal component u cos θ constant; vertical component u sin θ − gt. Time of flight T = 2u sin θ/g on level ground; range R = u² sin 2θ/g; maximum height H = u² sin²θ/(2g).

Newton's laws: (1) a body remains at rest or in uniform motion unless acted on by a net force; (2) F_net = dp/dt = ma for constant mass; (3) action and reaction are equal, opposite and act on different bodies.

Momentum p = mv; impulse J = FΔt = Δp. Conservation of linear momentum applies when net external force is zero; for collisions, total momentum before = total momentum after in any direction.

Collisions: elastic collisions conserve kinetic energy as well as momentum; perfectly inelastic collisions stick together and lose maximum kinetic energy. Coefficient of restitution e = relative speed of separation / relative speed of approach.

Work W = Fs cos θ; kinetic energy KE = ½mv²; gravitational potential energy near Earth ΔPE = mgh; elastic PE = ½kx². Work-energy theorem: net work = ΔKE.

Power P = dW/dt = Fv cos θ. Efficiency = useful output power / total input power × 100%.

Friction: limiting static friction f_s ≤ μ_s N; kinetic friction f_k = μ_k N, acting opposite to relative motion. On an incline, component of weight along slope = mg sin θ, normal reaction = mg cos θ.

Equilibrium of a rigid body: ΣF = 0 and Στ = 0 about any point. Principle of moments: sum of clockwise moments = sum of anticlockwise moments.

Centre of mass: x_cm = Σm_i x_i / Σm_i. For a uniform body, centre of mass is at its geometric centre.

Circular motion: angular velocity ω = 2π/T = 2πf; linear speed v = rω; centripetal acceleration a = v²/r = rω²; centripetal force F = mv²/r = mrω².

Banked track: for no friction, tan θ = v²/(rg). Vertical circular motion: minimum speed at top for string tension ≥ 0 is v = √(gr).

Newton's law of gravitation: F = GMm/r². Gravitational field strength g = GM/r². Gravitational potential V = −GM/r; potential energy U = −GMm/r.

Orbital motion: for a satellite in circular orbit, GMm/r² = mv²/r, so v = √(GM/r) and T² = (4π²/GM) r³ (Kepler's third law).

Escape velocity from Earth's surface: v_esc = √(2GM/R) = √(2gR) ≈ 11.2 km s⁻¹.

Simple harmonic motion: a = −ω²x; x = A sin(ωt) or A cos(ωt); v = ±ω√(A² − x²); v_max = ωA; a_max = ω²A. Period T = 2π/ω.

SHM systems: mass-spring T = 2π√(m/k); simple pendulum T = 2π√(L/g) for small angles. Energy in SHM: total E = ½mω²A²; KE = ½mω²(A² − x²); PE = ½mω²x².

Damping: light damping reduces amplitude gradually; critical damping returns to equilibrium fastest without overshoot; overdamping returns slowly without oscillation. Resonance occurs when driving frequency ≈ natural frequency; amplitude maximum and phase difference = 90° at resonance.

For a forced oscillator, amplitude depends on damping: greater damping lowers and broadens the resonance peak.

Thermal equilibrium: two bodies in contact with no net heat flow have the same temperature. Zeroth law: if A is in equilibrium with B and B with C, then A is in equilibrium with C.

Heat capacity C = Q/Δθ; specific heat capacity c = Q/(mΔθ). Latent heat L = Q/m for phase change at constant temperature.

Ideal gas equation: pV = nRT = NkT, where R = 8.31 J mol⁻¹ K⁻¹, k = 1.38 × 10⁻²³ J K⁻¹, N = nN_A.

Kinetic theory: pV = ⅓ N m <c²>; mean translational KE per molecule = (3/2)kT; root-mean-square speed c_rms = √(3RT/M) = √(3kT/m).

Internal energy of an ideal gas depends only on temperature (no intermolecular potential energy). For monatomic gas, U = (3/2)nRT; for diatomic near room temperature, U = (5/2)nRT.

First law of thermodynamics: ΔU = Q − W, where W is work done by the gas. Work done by gas W = ∫p dV; at constant pressure W = pΔV.

Thermodynamic processes: isothermal (T constant, ΔU = 0, Q = W); adiabatic (Q = 0, ΔU = −W); isobaric (p constant, W = pΔV); isochoric (V constant, W = 0, ΔU = Q).

Molar heat capacities: C_p − C_v = R for an ideal gas; ratio γ = C_p/C_v. For monatomic gas C_v = (3/2)R, γ = 5/3; diatomic C_v = (5/2)R, γ = 7/5.

Adiabatic change: pV^γ = constant; TV^(γ−1) = constant; T^γ p^(1−γ) = constant.

Thermodynamic efficiency of a heat engine: η = W/Q_H = 1 − Q_C/Q_H. Carnot efficiency η_max = 1 − T_C/T_H (temperatures in kelvin).

Wave equation: v = fλ. For a stretched string, v = √(T/μ), where T is tension and μ is mass per unit length. For sound in air, v ≈ 331 + 0.6θ m s⁻¹ (θ in °C).

Superposition: resultant displacement is the vector sum of individual displacements. Constructive interference when path difference = nλ (phase difference = 2nπ); destructive when path difference = (n + ½)λ (phase difference = (2n + 1)π).

Two-source interference: fringe spacing for Young's double slits y = λD/d, where D is slit-to-screen distance and d is slit separation. For sound, path difference condition same as above.

Diffraction grating: d sin θ = nλ, where d is grating spacing (1/N for N lines per metre). Maxima occur at angles satisfying this equation; n = 0 is the central maximum.

Single-slit diffraction: first minimum at a sin θ = λ (a = slit width). Width of central maximum ∝ λ/a.

Stationary waves on a string: nodes and antinodes; adjacent nodes separated by λ/2. Fundamental frequency f₁ = v/(2L); harmonics f_n = nf₁ for string fixed at both ends. For pipe open at both ends, f_n = nv/(2L); for pipe closed at one end, f_n = (2n−1)v/(4L) (odd harmonics only).

Sound intensity level: β = 10 log₁₀(I/I₀) dB, with I₀ = 1 × 10⁻¹² W m⁻². Intensity I = P/(4πr²) for a point source; doubling distance reduces intensity by factor 4 (≈ −6 dB).

Doppler effect for sound: observed frequency f' = f (v ± v_o)/(v ∓ v_s), where upper signs apply for approach and lower for receding. v is speed of sound relative to medium.

Polarisation: transverse waves can be polarised; longitudinal waves cannot. Malus's law: I = I₀ cos² θ for polarised light passing through an analyser at angle θ to the polariser.

Coulomb's law: F = (1/4πε₀) q₁q₂/r², with 1/4πε₀ = 8.99 × 10⁹ N m² C⁻². Electric field E = F/q; for a point charge E = (1/4πε₀) q/r².

Electric potential V = (1/4πε₀) q/r; potential energy U = qV. Equipotential surfaces are perpendicular to field lines. E = −dV/dx.

Capacitance C = Q/V. Parallel-plate capacitor C = ε₀ε_r A/d. Energy stored W = ½QV = ½CV² = Q²/(2C).

Capacitors in parallel: C_total = C₁ + C₂ + …; in series: 1/C_total = 1/C₁ + 1/C₂ + … . For resistors, the rules are reversed: series R_total = R₁ + R₂ + …; parallel 1/R_total = 1/R₁ + 1/R₂ + … .

DC circuits: Ohm's law V = IR; power P = VI = I²R = V²/R. Kirchhoff's laws: sum of currents at a junction = 0; sum of emfs = sum of p.d.s around a loop. Terminal p.d. V = E − Ir.

Magnetic force on a moving charge F = qvB sin θ; on a current-carrying wire F = BIL sin θ. Direction given by Fleming's left-hand rule.

Magnetic field of a long straight wire B = μ₀I/(2πr); inside a solenoid B = μ₀nI, where n is turns per metre.

Electromagnetic induction: magnetic flux Φ = BA cos θ; induced emf ε = −dΦ/dt = −N dΦ/dt (Faraday's law); direction opposes change (Lenz's law). Motional emf ε = Bℓv for a rod moving perpendicular to field.

Transformer: V_s/V_p = N_s/N_p = I_p/I_s (ideal). Step-up increases voltage, decreases current; power is conserved in an ideal transformer.

Semiconductors: intrinsic conductivity increases with temperature (negative temperature coefficient). Doping with pentavalent atoms gives n-type (majority electrons); trivalent atoms give p-type (majority holes).

p-n junction diode: forward bias reduces depletion layer, current flows above threshold voltage (≈0.7 V for silicon, ≈0.3 V for germanium). Reverse bias widens depletion layer; only small leakage current flows until breakdown.

Diode applications: half-wave rectifier uses one diode; full-wave bridge rectifier uses four diodes. A capacitor in parallel with the load smooths the output.

Zener diode: operates in reverse breakdown at a stable voltage; used for voltage regulation. Connect in reverse bias with a series resistor.

Transistor (BJT): I_E = I_B + I_C; current gain β = I_C/I_B. For an npn transistor in active mode, base-emitter forward biased, collector-base reverse biased. Used as switch (cut-off/saturation) or amplifier (active region).

Operational amplifier: ideal properties include infinite open-loop gain, infinite input impedance, zero output impedance. Inverting amplifier gain = −R_f/R_in; non-inverting gain = 1 + R_f/R₁. Virtual short: V₊ ≈ V₋ when negative feedback is applied.

Op-amp applications: comparator (no feedback, output saturates to ±V_s), summing amplifier, integrator, differentiator. Slew rate limits high-frequency response.

Logic gates: AND (Y = A·B), OR (Y = A + B), NOT (Y = Ā), NAND, NOR, XOR (Y = A⊕B). NAND and NOR are universal gates; any logic function can be built from them.

Boolean algebra: De Morgan's theorems: (A·B)' = A' + B'; (A + B)' = A'·B'. Truth tables list all input combinations and corresponding outputs.

Photoelectric effect: Einstein's equation hf = Φ + K_max, where Φ is the work function and K_max = eV_s (stopping potential). Threshold frequency f₀ = Φ/h; maximum kinetic energy independent of intensity, depends on frequency.

Photon energy E = hf = hc/λ; h = 6.63 × 10⁻³⁴ J s; c = 3.00 × 10⁸ m s⁻¹. Electronvolt: 1 eV = 1.60 × 10⁻¹⁹ J.

Wave-particle duality: de Broglie wavelength λ = h/p = h/(mv). For an electron accelerated through potential difference V, λ = h/√(2meV) ≈ 1.23/√V nm.

Atomic energy levels: energy absorbed or emitted as photon when electron transitions between levels: hf = E_high − E_low. Ionisation energy is energy needed to remove an electron from ground state.

Line spectra: emission spectra are bright lines on dark background; absorption spectra are dark lines on continuous spectrum. Hydrogen series: Lyman (UV), Balmer (visible), Paschen (IR).

Nuclear structure: nucleus contains protons (Z) and neutrons (A−Z). Isotopes have same Z, different A. Nuclear radius r ≈ r₀ A^(1/3), r₀ ≈ 1.2 fm.

Radioactive decay: dN/dt = −λN; N = N₀ e^(−λt); half-life T₁/₂ = ln 2/λ ≈ 0.693/λ. Activity A = λN, unit becquerel (Bq).

Nuclear reactions: conservation of nucleon number (A) and charge (Z). Mass defect Δm converts to energy via E = Δmc²; 1 u = 931.5 MeV.

Binding energy per nucleon peaks near iron-56; fusion releases energy for light nuclei, fission for heavy nuclei.

Planning an experiment: identify independent, dependent and control variables; choose apparatus with appropriate resolution and range; describe how to vary and measure each variable; include safety precautions.

Recording data: tabulate raw readings with units and consistent decimal places (same as instrument resolution). Repeated readings for the dependent variable reduce random error.

Graph plotting: label axes with quantity and unit; use sensible scales (e.g. 1 cm = 1 unit); plot points with small crosses; draw line of best fit, not necessarily through origin; identify anomalous points.

Uncertainty analysis: combine uncertainties using rules for sums, differences, products, quotients and powers. For a gradient, Δm/m = (Δy/y) + (Δx/x) approximately, or use max/min gradients.

Common systematic errors: zero error, parallax, calibration error, reaction time (for stopwatch), heat loss (thermal experiments), contact resistance (electrical experiments).

Improving accuracy: use a fiducial mark for timing oscillations (time many oscillations and divide), use a set square to align apparatus, use a micrometer for small diameters, avoid parallax by eye level.

Significant figures: final answer should have same number of significant figures as the least precise measurement used, or one more; uncertainty usually quoted to 1 significant figure (2 if leading digit is 1).

Experimental techniques: use a rheostat as potential divider for varying voltage; use a signal generator and CRO to measure frequency and amplitude; use a travelling microscope for small displacements; use a data logger for fast or repetitive measurements.

Conclusion and evaluation: state whether results support the hypothesis; compare gradient with theoretical value; discuss sources of error and suggest improvements; calculate percentage error and comment on whether it is within experimental uncertainty.