Pancyprian Technical School Physics (4-hour) cheat sheet

Uniformly accelerated motion: v = u + at, s = ut + ½at², v² = u² + 2as; valid only for constant acceleration in a straight line.

Projectile motion separates into independent horizontal (constant velocity) and vertical (acceleration g) components; time of flight from vertical motion alone.

Newton's second law as a vector equation: ΣF = ma; resolve forces into perpendicular components before summing.

Friction: limiting static friction f ≤ μsN; kinetic friction f = μkN, acting opposite to relative sliding.

Inclined plane without friction: a = g sinθ; with friction on the way up a = g(sinθ + μcosθ).

Momentum form of Newton's second law: F = Δp/Δt, which remains valid when mass changes (rockets, chains).

Equilibrium of a rigid body requires both ΣF = 0 and Στ = 0 about any chosen axis; choose the axis to eliminate unknown forces.

For connected bodies over a pulley, apply ΣF = ma to each mass separately and equate the common tension/acceleration.

Work done by a constant force: W = Fs cosθ, where θ is the angle between force and displacement; perpendicular forces do no work.

Work–energy theorem: W_net = ΔKE = ½mv² − ½mu², including work by friction and gravity.

Conservative forces (gravity, spring, electrostatic) have potential energy; friction is non-conservative and dissipates mechanical energy.

Power: P = W/t = Fv cosθ; average versus instantaneous power differ when speed varies.

Conservation of linear momentum applies when net external force is zero, e.g. collisions and explosions; kinetic energy is conserved only in elastic collisions.

Perfectly inelastic collision: bodies stick, common velocity v = (m₁u₁ + m₂u₂)/(m₁ + m₂); lost KE = ½m₁u₁² + ½m₂u₂² − ½(m₁+m₂)v².

Impulse J = FΔt = Δp; the area under a force–time graph equals impulse.

Efficiency = useful output energy / total input energy × 100%; losses usually appear as thermal energy.

Angular speed ω = 2π/T = 2πf; linear speed v = ωr; centripetal acceleration a = v²/r = ω²r directed toward the centre.

Centripetal force is a net resultant, not an extra force: F = mv²/r supplied by tension, friction, gravity, or normal force.

Vertical circle: at the top mg + N = mv²/r; at the bottom N − mg = mv²/r; minimum speed at top for a string is v = √(gr).

Banked track without friction: tanθ = v²/(rg); with friction the safe speed range widens.

Conical pendulum: T cosθ = mg and T sinθ = mv²/r, giving tanθ = v²/(rg).

Angular displacement in radians: θ = s/r; tangential acceleration a_t = rα changes speed while centripetal acceleration changes direction.

Centripetal force does no work because it is always perpendicular to velocity, so speed and kinetic energy stay constant in uniform circular motion.

SHM defining condition: a = −ω²x, with displacement measured from equilibrium; ω = 2π/T.

SHM solutions: x = A sin(ωt + φ), v = ωA cos(ωt + φ), maximum speed v_max = ωA, maximum acceleration a_max = ω²A.

Mass–spring: T = 2π√(m/k), independent of amplitude and of g.

Simple pendulum: T = 2π√(l/g) for small angles (typically θ < 10°); T is independent of bob mass.

Energy in SHM: total E = ½kA² = ½mω²A²; KE and PE interchange, with KE maximum at equilibrium.

Newton's law of gravitation: F = Gm₁m₂/r², G = 6.67 × 10⁻¹¹ N·m²/kg²; field strength g = GM/r² outside a spherical mass.

Orbital speed v = √(GM/r); period T = 2π√(r³/GM), so T² ∝ r³ (Kepler's third law).

Escape speed from a body of mass M and radius R: v = √(2GM/R); for Earth it is about 11.2 km/s.

Pressure p = F/A; in a liquid p = p₀ + ρgh, and pressure depends only on depth, not on container shape.

Archimedes' principle: upthrust = weight of displaced fluid = ρ_fluid V_submerged g; floating body displaces its own weight.

Continuity equation for incompressible flow: A₁v₁ = A₂v₂.

Bernoulli's equation: p + ½ρv² + ρgh = constant along a streamline for steady, non-viscous, incompressible flow.

Viscous drag on a sphere at low speed (Stokes' law): F = 6πηrv; terminal velocity when 6πηrv = (ρ_sphere − ρ_fluid)Vg.

Hooke's law: stress ∝ strain within the elastic limit; Young's modulus E = (F/A)/(Δl/l).

Stress = force/cross-sectional area (Pa); strain is dimensionless; the area under a stress–strain curve up to fracture is toughness.

Density ρ = m/V; relative density compares a substance to water; specific heat and latent heat are intensive properties.

Temperature conversion: T(K) = θ(°C) + 273.15; absolute zero is 0 K = −273.15 °C.

Linear expansion: ΔL = αL₀Δθ; area and volume expansions use 2α and 3α for isotropic solids.

Heat transfer: Q = mcΔθ for temperature change; Q = mL at a phase change with no temperature change.

Ideal gas law: pV = nRT with R = 8.31 J/(mol·K); alternatively pV = NkT with k = 1.38 × 10⁻²³ J/K.

Combined gas law for a fixed mass: p₁V₁/T₁ = p₂V₂/T₂; isothermal pV = constant, isobaric V/T = constant, isochoric p/T = constant.

Kinetic theory: average translational KE per molecule = (3/2)kT, so internal energy of an ideal gas depends only on temperature.

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

Absolute humidity is vapour mass per unit volume; relative humidity = partial vapour pressure / saturated vapour pressure × 100%.

Coulomb's law: F = kQ₁Q₂/r² with k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C²; force is along the line joining the charges, attractive for unlike signs.

Uniform field between parallel plates: E = V/d; force on charge q is F = qE; work done moving q through V is qV.

Electric potential V = kQ/r; potential is a scalar, field is a vector pointing from high to low potential.

Capacitance C = Q/V; parallel plates C = ε₀ε_rA/d; energy stored = ½QV = ½CV².

Ohm's law V = IR applies to ohmic conductors at constant temperature; resistance of a wire R = ρl/A.

Series: R = R₁ + R₂ + …, same current; parallel: 1/R = 1/R₁ + 1/R₂ + …, same voltage.

Electrical power: P = VI = I²R = V²/R; energy in kWh = power(kW) × time(h).

Kirchhoff's rules: junction rule (charge conservation) ΣI_in = ΣI_out; loop rule (energy conservation) ΣEMF = ΣIR around any closed loop.

EMF and internal resistance: terminal voltage V = ε − Ir; maximum power transfer to load when R = r.

Force on a current-carrying conductor: F = BIl sinθ, direction by Fleming's left-hand rule; maximum when conductor is perpendicular to B.

Force on a moving charge: F = qvB sinθ; it is always perpendicular to velocity, so magnetic force does no work and cannot change speed.

Radius of a charged particle's circular path in a magnetic field: r = mv/(qB); period T = 2πm/(qB) is independent of speed.

Magnetic flux Φ = BA cosθ; flux is measured in webers (Wb), flux density B in tesla (T).

Faraday's law: induced EMF ε = −N dΦ/dt; the magnitude depends on rate of change of flux linkage, not on flux itself.

Lenz's law: induced current opposes the change producing it; this is a statement of energy conservation.

Straight conductor moving through a field: motional EMF ε = Blv (when B, l, v are mutually perpendicular).

Transformer (ideal): V_s/V_p = N_s/N_p = I_p/I_s; a step-up voltage transformer steps current down.

Wave equation: v = fλ; for a stretched string v = √(T/μ) with μ = mass per unit length.

Speed of sound in air ≈ 340 m/s at room temperature and rises with temperature; sound needs a material medium.

Sound intensity level β = 10 log₁₀(I/I₀) dB with I₀ = 1 × 10⁻¹² W/m²; doubling intensity adds about 3 dB.

Doppler effect for sound: f' = f(v ± v_o)/(v ∓ v_s), with signs chosen so approach raises the observed frequency.

String and pipe modes: string fixed both ends f_n = nv/(2L); open pipe f_n = nv/(2L); pipe closed at one end gives odd harmonics only, f_n = nv/(4L), n odd.

Snell's law: n₁ sinθ₁ = n₂ sinθ₂; refractive index n = c/v; total internal reflection requires travel from denser to less dense medium and θ > critical angle sinθ_c = n₂/n₁.

Thin lens/mirror equation: 1/f = 1/v + 1/u with sign convention stated explicitly; magnification m = v/u (or −v/u depending on convention).

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

Photon energy E = hf = hc/λ; h = 6.63 × 10⁻³⁴ J·s; 1 eV = 1.60 × 10⁻¹⁹ J.

Photoelectric equation: hf = W + KE_max, with threshold frequency f₀ = W/h; maximum KE depends on frequency, not intensity.

Stopping potential: eV_s = KE_max; the photocurrent saturates with intensity at fixed frequency.

Hydrogen energy levels: E_n = −13.6/n² eV; photon energy for a transition = E_high − E_low.

de Broglie wavelength λ = h/p = h/(mv); electron diffraction confirms wave behaviour of matter.

Radioactive decay N = N₀e^(−λt); half-life T½ = ln2/λ = 0.693/λ; activity A = λN measured in becquerel.

Nuclear notation: mass number A = protons + neutrons; isotopes share Z but differ in N; alpha decay reduces A by 4 and Z by 2.

Mass–energy equivalence E = mc²; nuclear binding energy per nucleon peaks near iron-56, explaining fission of heavy and fusion of light nuclei.

Absolute uncertainty of a single reading is typically half the smallest scale division; for a digital display use ±1 in the last digit.

Combining uncertainties: sums/differences add absolute uncertainties; products/quotients add percentage uncertainties; powers multiply percentage uncertainty by the exponent.

Report a result as value ± uncertainty with matching decimal places and an appropriate number of significant figures (usually 2–3).

Percentage difference from a stated value: |measured − accepted| / accepted × 100%.

Systematic errors shift all readings in one direction (zero error, calibration); random errors scatter and are reduced by averaging repeats.

Graph analysis: gradient of a straight-line graph gives the derived quantity; error bars show uncertainty and a line of best fit should pass through the bars.

Linearise a relationship before plotting, e.g. plot T² against l for a pendulum so that gradient = 4π²/g.

Precision describes repeatability of readings, accuracy describes closeness to the true value; a precise set can still be inaccurate.

Electrical transmission at high voltage reduces I²R line losses for the same transmitted power P = VI.

Transformers enable efficient voltage change but work only with alternating current, since they rely on changing flux.

Internal combustion engine efficiency is limited by the Carnot bound η ≤ 1 − T_cold/T_hot; real engines fall well below it.

Hydraulic systems transmit force via Pascal's principle: F₁/A₁ = F₂/A₂, giving mechanical advantage from area ratio.

Heat pumps and refrigerators move heat against the temperature gradient and require work input; coefficient of performance = Q_desired/W.

Sensors convert physical quantities to electrical signals: thermistors change resistance with temperature, LDRs with light, strain gauges with deformation.

Renewable generation: a wind turbine's available power scales as P ∝ A v³, so doubling wind speed multiplies available power by eight.

Structural design uses stress = F/A and compares it with the material's yield strength, applying a safety factor greater than one.

Write the formula symbolically, substitute values with units, then give the numerical answer with the correct unit and significant figures.

Convert all data to SI before substitution: cm to m, g to kg, minutes to seconds, kW·h to J (1 kW·h = 3.6 × 10⁶ J).

Check dimensional consistency: both sides of any physical equation must have identical base units.

When a question asks to 'show that', derive the stated result explicitly and do not simply quote it.

For multi-step problems, label intermediate results and carry them forward unrounded; round only the final answer.

In graphs, state what the gradient and intercept represent physically, not just their numerical values.

If an answer seems unphysical (negative time, speed above c, efficiency over 100%), recheck the sign convention or unit conversion.

g ≈ 9.81 m/s² near Earth's surface; gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg².

Elementary charge e = 1.60 × 10⁻¹⁹ C; electron mass 9.11 × 10⁻³¹ kg; proton mass 1.67 × 10⁻²⁷ kg.

Speed of light c = 3.00 × 10⁸ m/s; Planck constant h = 6.63 × 10⁻³⁴ J·s.

Avogadro constant N_A = 6.02 × 10²³ mol⁻¹; molar gas constant R = 8.31 J/(mol·K); Boltzmann constant k = 1.38 × 10⁻²³ J/K.

Specific heat capacity of water ≈ 4200 J/(kg·K); specific latent heat of fusion of ice ≈ 3.34 × 10⁵ J/kg.

Permittivity of free space ε₀ = 8.85 × 10⁻¹² F/m; Coulomb constant k ≈ 9.0 × 10⁹ N·m²/C².

Unit conversions: 1 N = 1 kg·m/s²; 1 J = 1 N·m; 1 W = 1 J/s; 1 Pa = 1 N/m²; 1 T = 1 Wb/m²; 1 V = 1 J/C.

Prefixes: nano 10⁻⁹, micro 10⁻⁶, milli 10⁻³, kilo 10³, mega 10⁶, giga 10⁹.