Physics cheat sheet

Kinematics (constant a): v = u + at; s = ut + ½at²; v² = u² + 2as. For projectile motion resolve into independent horizontal (a = 0) and vertical (a = −g) components.

Newton's second law: ΣF = ma; friction f ≤ μN, with equality at the point of sliding. On an incline, the component along the slope is mg sinθ and the normal force is mg cosθ.

Work–energy theorem: W_net = ΔKE = ½mv² − ½mu². For a conservative force, F = −dU/dx; gravitational potential energy near Earth is U = mgh.

Conservation of linear momentum applies when the net external force is zero: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. In perfectly inelastic collisions the objects stick together and kinetic energy is lost; in elastic collisions kinetic energy is also conserved.

Circular motion: a_c = v²/r = ω²r, directed toward the centre; the period is T = 2π/ω. Centripetal force is the net inward force, not a separate force to be added to the free-body diagram.

Rotational dynamics: τ = Iα, L = Iω, KE_rot = ½Iω². Parallel-axis theorem: I = I_cm + md². Rolling without slipping relates v = ωr.

Gravitation: F = Gm₁m₂/r²; g = GM/r²; orbital speed v = √(GM/r); Kepler's third law T² ∝ r³. Escape speed from a body of mass M and radius R is √(2GM/R).

Coulomb's law: F = kq₁q₂/r², k ≈ 8.99×10⁹ N·m²/C². Electric field E = F/q; for a point charge E = kq/r², and the field inside a conductor in electrostatic equilibrium is zero.

Electric potential V = kq/r is a scalar; potential difference W = qΔV. Field and potential are related by E = −dV/dx. Equipotential surfaces are perpendicular to field lines.

Capacitance: C = Q/V; for parallel plates C = ε₀A/d; energy stored U = ½CV² = Q²/2C. Capacitors in parallel add; in series the reciprocals add.

DC circuits: V = IR; resistors in series add, in parallel the reciprocals add. Power P = IV = I²R = V²/R. Kirchhoff's rules: junction rule (charge conservation) and loop rule (energy conservation).

Magnetic force: F = qvB sinθ on a moving charge and F = BIL sinθ on a current-carrying wire; the direction is given by the right-hand rule. A charge moving parallel to B experiences no magnetic force.

Fields of current configurations: a long straight wire gives B = μ₀I/(2πr); a solenoid gives B = μ₀nI. A charged particle moving perpendicular to a uniform B travels in a circle of radius r = mv/(qB).

Faraday's law: emf = −N dΦ/dt, where Φ = BA cosθ. Lenz's law fixes the direction: the induced current opposes the change in flux. A transformer relates voltages by V_s/V_p = N_s/N_p.

Wave relation: v = fλ; period T = 1/f. For a wave on a string v = √(T/μ), where T is tension and μ is linear mass density.

Sound in air: v ≈ 343 m/s at 20 °C and rises with temperature. Intensity level β = 10 log₁₀(I/I₀) dB, with I₀ = 10⁻¹² W/m²; a 10 dB increase means ten times the intensity.

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

Reflection and refraction: θ_i = θ_r; Snell's law n₁sinθ₁ = n₂sinθ₂. Total internal reflection occurs only when light travels from a denser to a less dense medium and θ exceeds the critical angle, where sinθ_c = n₂/n₁.

Thin lens and mirror equation: 1/f = 1/d_o + 1/d_i, with magnification m = −d_i/d_o. For a converging lens f > 0; a real image forms when d_i > 0.

Young's double-slit interference: bright fringes at d sinθ = mλ, dark fringes at d sinθ = (m + ½)λ; fringe spacing on a distant screen is Δy = λL/d.

Single-slit diffraction: minima at a sinθ = mλ (m = 1, 2, …). Diffraction grating maxima follow d sinθ = mλ, with larger m spacing for longer wavelengths.

Temperature conversion: T(K) = T(°C) + 273.15. Linear expansion ΔL = αL₀ΔT; heat transfer Q = mcΔT, and during a phase change Q = mL.

Ideal gas law: pV = nRT, with R = 8.31 J/(mol·K). Kinetic theory gives average translational KE = (3/2)k_BT per molecule; rms speed v_rms = √(3RT/M).

First law: ΔU = Q − W, where W is work done by the gas. For an isothermal process ΔU = 0; for an adiabatic process Q = 0; for an isochoric process W = 0.

Work done by a gas at constant pressure is W = pΔV; on a p–V diagram the work done in a process equals the area under the curve.

Efficiency of a heat engine: η = W/Q_H = 1 − Q_C/Q_H. Carnot efficiency η_max = 1 − T_C/T_H, using absolute temperatures.

Second law: entropy of an isolated system never decreases; no engine operating between two reservoirs can exceed the Carnot efficiency. Heat flows spontaneously from hot to cold.

For a monatomic ideal gas C_V = (3/2)R and C_P = (5/2)R; C_P − C_V = R. The ratio γ = C_P/C_V governs adiabatic processes, where pV^γ is constant.

Photon energy E = hf = hc/λ; h = 6.63×10⁻³⁴ J·s and hc ≈ 1240 eV·nm. Photoelectric effect: maximum kinetic energy K_max = hf − φ, with threshold frequency f₀ = φ/h.

de Broglie wavelength λ = h/p. For an electron accelerated through potential difference V, λ ≈ 1.23 nm/√(V in volts).

Bohr model: E_n = −13.6 eV/n² for hydrogen; photon energy on a transition equals the energy-level difference. The ground state is n = 1.

Nuclear: activity A = λN; half-life T₁/₂ = ln2/λ, and N = N₀(1/2)^(t/T₁/₂). Mass defect converts to binding energy by E = Δmc².

Radioactive decay types: alpha emission reduces Z by 2 and A by 4; beta-minus emission raises Z by 1 with A unchanged; gamma emission changes neither Z nor A.

Special relativity: γ = 1/√(1 − v²/c²); time dilation Δt = γΔt₀; length contraction L = L₀/γ; relativistic energy E = γmc², with rest energy E₀ = mc².

Mass–energy conversion: 1 u = 931.5 MeV/c². Total energy and momentum are conserved in nuclear reactions; charge and nucleon number are also conserved.

SI base units: metre (length), kilogram (mass), second (time), ampere (current), kelvin (temperature), mole (amount), candela (luminous intensity). Derived examples: 1 N = 1 kg·m/s²; 1 J = 1 N·m; 1 W = 1 J/s; 1 V = 1 J/C.

Dimensional analysis: both sides of a valid equation must have identical base-unit dimensions; use this to check formulas and to convert units.

Significant figures: for multiplication and division, the result keeps the least number of significant figures among the inputs; for addition and subtraction, the least number of decimal places.

Uncertainty propagation: for sums and differences add absolute uncertainties; for products and quotients add fractional (percentage) uncertainties; for a power xⁿ multiply the fractional uncertainty by |n|.

For repeated readings, the random uncertainty is often estimated as the standard deviation of the mean, s/√N, while the resolution uncertainty of a single reading is typically ±½ of the smallest scale division.

Graphical analysis: the slope of a best-fit line equals the rate of change of the y-variable with respect to the x-variable; the intercept gives the value at x = 0. Label axes with quantity and unit.

A valid experiment controls variables, repeats readings, and compares results with the accepted value using percentage error = |measured − accepted|/accepted × 100%.

Kinematics (constant a): v = u + at; s = ut + ½at²; v² = u² + 2as. For projectile motion resolve into independent horizontal (a = 0) and vertical (a = −g) components.

Newton's second law: ΣF = ma; friction f ≤ μN, with equality at the point of sliding. On an incline, the component along the slope is mg sinθ and the normal force is mg cosθ.

Work–energy theorem: W_net = ΔKE = ½mv² − ½mu². For a conservative force, F = −dU/dx; gravitational potential energy near Earth is U = mgh.

Conservation of linear momentum applies when the net external force is zero: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. In perfectly inelastic collisions the objects stick together and kinetic energy is lost; in elastic collisions kinetic energy is also conserved.

Circular motion: a_c = v²/r = ω²r, directed toward the centre; the period is T = 2π/ω. Centripetal force is the net inward force, not a separate force to be added to the free-body diagram.

Rotational dynamics: τ = Iα, L = Iω, KE_rot = ½Iω². Parallel-axis theorem: I = I_cm + md². Rolling without slipping relates v = ωr.

Gravitation: F = Gm₁m₂/r²; g = GM/r²; orbital speed v = √(GM/r); Kepler's third law T² ∝ r³. Escape speed from a body of mass M and radius R is √(2GM/R).