UEE Physics cheat sheet

Newton's second law: ΣF = ma in an inertial frame; for variable mass use F = dp/dt with p = mv.

Kinematics with constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as; only valid when a is constant.

Projectile motion: horizontal and vertical motions are independent; time of flight from vertical component, range R = u²sin2θ/g over level ground.

Work-energy theorem: W_net = ΔKE; conservative forces admit a potential with F = −dU/dx, and mechanical energy is conserved only when non-conservative work is zero.

Momentum: Σp is conserved when net external force is zero; elastic collisions conserve KE, inelastic collisions conserve p only.

Circular motion: a_c = v²/r = ω²r directed toward the centre; centripetal force is a net force, not an extra force to add.

Gravitation: F = Gm₁m₂/r²; orbital speed v = √(GM/r) and period T² = 4π²r³/(GM) for circular orbits.

Rotational dynamics: τ = Iα, L = Iω conserved when net external torque is zero; rolling without slipping requires v = ωr.

Static equilibrium requires both ΣF = 0 and Στ = 0 about any chosen point; choose the pivot to eliminate unknown forces.

Ideal gas law: pV = nRT = NkT; R ≈ 8.31 J mol⁻¹ K⁻¹ and k = R/N_A ≈ 1.38×10⁻²³ J K⁻¹.

First law: ΔU = Q − W where W is work done by the gas; for a quasistatic process W = ∫p dV.

Kinetic theory: mean translational KE per molecule = (3/2)kT; rms speed v_rms = √(3RT/M) = √(3kT/m).

Internal energy of an ideal gas depends only on temperature: U = (f/2)nRT, with f = 3 monatomic and f = 5 diatomic at ordinary temperatures.

Isothermal work by an ideal gas: W = nRT ln(V₂/V₁); adiabatic processes obey pV^γ = constant with γ = C_p/C_v.

Efficiency of a heat engine: η = W/Q_H = 1 − Q_C/Q_H; Carnot limit η_max = 1 − T_C/T_H with absolute temperatures.

Second law: entropy of an isolated system never decreases; ΔS = Q_rev/T for reversible paths, and ΔS = 0 for a reversible cycle.

Mean free path λ = 1/(√2 π d² n) for hard-sphere molecules of diameter d and number density n.

SHM: a = −ω²x with ω = 2π/T; solutions x = A cos(ωt + φ), v_max = ωA, a_max = ω²A.

Mass-spring ω = √(k/m); simple pendulum ω = √(g/L) valid for small angular amplitude.

Energy in SHM: E = ½kA²; KE and PE each average to half the total over a full cycle.

Damped oscillation amplitude decays as e^(−bt/2m); resonance occurs when driving frequency approaches the natural frequency.

Wave equation: v = fλ; for a stretched string v = √(T/μ) with tension T and linear density μ.

Superposition gives standing waves with nodes spaced λ/2; string fixed at both ends has f_n = nv/(2L).

Sound intensity level β = 10 log₁₀(I/I₀) dB with I₀ = 10⁻¹² W m⁻²; intensity falls as 1/r² from a point source.

Doppler effect for sound: f' = f(v ± v_o)/(v ∓ v_s), with signs set by whether observer and source approach.

Two-source interference: constructive when path difference = nλ, destructive when (n + ½)λ.

Coulomb's law: F = kq₁q₂/r² with k = 1/(4πε₀) ≈ 8.99×10⁹ N m² C⁻²; field of a point charge E = kq/r².

Gauss's law: ∮E·dA = Q_enclosed/ε₀; useful for spherical, cylindrical and planar symmetry.

Potential is scalar: V = kq/r for a point charge, and E = −dV/dx in one dimension.

Capacitance: C = Q/V; parallel-plate C = ε₀A/d; energy stored U = ½CV² = Q²/(2C).

DC circuits: series resistors add, parallel conductances add; Kirchhoff's junction and loop rules follow from charge and energy conservation.

Magnetic force: F = qv × B and F = BIL sinθ on a current-carrying wire; a moving charge in a uniform B follows a circular path of radius r = mv/(qB).

Biot–Savart/Ampere results: long straight wire B = μ₀I/(2πr); solenoid B = μ₀nI; Faraday's law EMF = −dΦ/dt with Φ = BA cosθ.

Lenz's law: induced current opposes the change in flux; motional EMF for a rod of length L moving at speed v is BLv.

RC circuits: charging Q = Q₀(1 − e^(−t/RC)); time constant τ = RC.

Snell's law: n₁ sinθ₁ = n₂ sinθ₂; total internal reflection occurs when the angle exceeds θ_c = arcsin(n₂/n₁) for n₁ > n₂.

Thin lens: 1/f = 1/u + 1/v with sign conventions; magnification m = −v/u, and real images form on the opposite side of a converging lens.

Mirror equation uses the same form 1/f = 1/u + 1/v with f = R/2 for a spherical mirror.

Young's double slit: fringe spacing Δy = λD/d; bright fringes at path difference nλ.

Single-slit diffraction: first minimum at sinθ = λ/a; diffraction grating maxima at d sinθ = nλ.

Rayleigh criterion: two sources are just resolved when the angular separation is 1.22λ/D for a circular aperture.

Thin-film interference: reflected rays shift by λ/2 when reflecting off a higher-index medium; condition depends on whether both or one reflection is hard.

Malus's law: transmitted intensity through a polariser is I = I₀ cos²θ; Brewster angle satisfies tanθ_B = n₂/n₁.

Dispersion: refractive index varies with wavelength, so shorter wavelengths bend more in glass, producing spectra from prisms.

Photoelectric effect: maximum KE = hf − φ, with threshold frequency f₀ = φ/h; intensity controls current, frequency controls stopping voltage.

Photon energy E = hf = hc/λ; hc ≈ 1240 eV·nm, so λ(nm) ≈ 1240/E(eV).

de Broglie wavelength λ = h/p; for an electron accelerated through V volts, λ ≈ 1.23/√V nm.

Bohr model: E_n = −13.6 eV/n² for hydrogen; photon energy equals the energy-level difference.

Heisenberg uncertainty: Δx Δp ≥ ħ/2, with ħ = h/(2π).

Radioactive decay: N = N₀e^(−λt), half-life t½ = ln2/λ, activity A = λN.

Mass-energy: E = mc²; nuclear binding energy from the mass defect Δm via E = Δmc² (1 u ≈ 931.5 MeV).

Work function, stopping potential and Planck constant are linked by eV_s = hf − φ, the basis of the Millikan experiment.

Report a measurement as value ± uncertainty with matching decimal places; the uncertainty usually carries one or two significant figures.

For independent random uncertainties, combine in quadrature: δz = √((∂z/∂x)²δx² + (∂z/∂y)²δy²).

Simple rules: absolute uncertainties add for sums and differences; fractional uncertainties add for products and quotients.

A repeated-reading uncertainty is often estimated as half the smallest scale division for analogue instruments, or the resolution for digital ones.

Graphical analysis: plot to obtain a straight line, then extract gradient and intercept with uncertainty from the best-fit and extreme lines.

Systematic errors shift all readings in one direction and are not reduced by averaging; random errors scatter and are reduced by averaging.

Percentage difference compares an experimental value with an accepted value: |exp − acc|/acc × 100%.

Significant figures: keep the least precise input in multiplication and division; the final answer should not exceed the precision of the data.

Vectors: resolve into components, use scalar product A·B = AB cosθ and vector product |A × B| = AB sinθ.

Differentiation rules: product, quotient and chain rules; d/dx(sin x) = cos x and d/dx(e^x) = e^x.

Kinematics and rates: v = dx/dt and a = dv/dt = v dv/dx; the last form links acceleration to position without time.

Integration gives area under a curve: displacement from a v–t graph, work from an F–x graph, charge from an I–t graph.

Small-angle approximations: sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 − θ²/2, valid for θ in radians.

Exponential decay y = y₀e^(−kt) linearises as ln y = ln y₀ − kt, giving a straight-line graph of gradient −k.

Logarithms: log(ab) = log a + log b and log(a^n) = n log a; used for log–log and semi-log plots.

Simple harmonic and wave solutions come from sinusoidal functions; phase difference in radians relates to path difference by Δφ = 2πΔx/λ.