MTTC Physics (019) cheat sheet

SI base units: length m, mass kg, time s, current A, temperature K, amount mol, luminous intensity cd; every derived unit (N, J, W, T, F) reduces to a combination of these.

Accuracy is closeness to the accepted value; precision is closeness of repeated readings to each other. Systematic error shifts the mean away from truth, random error widens the spread and shrinks when trials are averaged.

Percent error = |measured − accepted| ÷ accepted × 100%; percent difference uses the average of two measured values in the denominator because neither is treated as the accepted value.

Significant figures: multiplication and division keep the fewest sig figs among the inputs; addition and subtraction keep the fewest decimal places. Exact counting numbers do not limit sig figs.

Graph linearization: a plot of displacement vs t² gives slope = ½a; a plot of 1/f vs 1/d_o yields focal length information. Slope units are y-units per x-unit, and intercepts often reveal a systematic offset.

Uncertainty combination: for z = xy or z = x/y, fractional uncertainties add in quadrature, δz/z = √[(δx/x)² + (δy/y)²]; for sums and differences, absolute uncertainties add in quadrature.

Dimensional analysis: both sides of a valid equation must reduce to identical base dimensions with matching exponents. This catches algebra errors but cannot detect dimensionless numerical factors.

A controlled experiment changes one independent variable, measures one dependent variable, and holds all others constant so that alternative explanations are eliminated; a control group establishes the baseline for comparison.

Single-measurement uncertainty is typically half the smallest scale division for an analog instrument and one unit in the last recorded digit for a digital display; interpolation between marks lowers random uncertainty.

Constant-acceleration kinematics: v = v₀ + at; Δx = v₀t + ½at²; v² = v₀² + 2aΔx; average velocity = (v₀ + v)/2 only when acceleration is uniform.

Projectile motion without air drag: a_x = 0 so horizontal velocity is constant, a_y = −g so vertical motion is free fall. Level-ground range R = v₀² sin 2θ/g is maximized at 45°.

Newton's laws: the net force equals ma; static friction satisfies f_s ≤ μ_s N while kinetic friction is f_k = μ_k N. On an incline, the gravity components are mg sin θ along the slope and mg cos θ perpendicular to it.

Circular motion: a_c = v²/r directed toward the center, requiring net inward force mv²/r. A frictionless banked curve is designed so that tan θ = v²/(rg); speed is constant while velocity direction changes continuously.

Work and energy: W = Fd cos θ; KE = ½mv²; W_net = ΔKE; W_nonconservative = ΔE_mechanical; U_gravity = mgh and U_spring = ½kx². Power P = W/t = Fv.

Momentum: p = mv; impulse J = FΔt = Δp. In a perfectly inelastic collision v′ = (m₁v₁ + m₂v₂)/(m₁ + m₂); elastic collisions conserve both momentum and kinetic energy, while inelastic ones conserve momentum only.

Rotational dynamics and equilibrium: τ = rF sin θ, Στ = Iα, L = Iω is conserved when external torque is zero, KE_rot = ½Iω², and rolling without slipping requires v = ωr. Static equilibrium requires ΣF = 0 and Στ = 0 about any chosen pivot.

Simple harmonic motion: ω = √(k/m) gives T = 2π√(m/k) for a mass-spring system independent of amplitude; a small-angle pendulum has T = 2π√(L/g). Total energy ½kA² trades between potential and kinetic.

Gravitation and orbits: F = Gm₁m₂/r² with G = 6.67×10⁻¹¹ N·m²/kg²; g = GM/r²; circular orbital speed v = √(GM/r); escape speed v = √(2GM/r), independent of the escaping mass.

Wave relation v = fλ. Frequency is fixed by the source and unchanged when a wave enters a new medium; speed and wavelength change. On a stretched string v = √(T/μ), and sound in air v ≈ 331 + 0.6T(°C) m/s, near 343 m/s at 20 °C.

Standing waves: string fixed at both ends and open pipe give f_n = nv/(2L) for n = 1, 2, 3…; a pipe closed at one end gives f_n = nv/(4L) for odd n only. Nodes sit at fixed ends, antinodes at open ends.

Beats arise at f_beat = |f₁ − f₂|. Doppler shift f′ = f(v ± v_o)/(v ∓ v_s), with signs chosen so relative approach raises the observed frequency and recession lowers it.

Sound intensity from a point source I = P/(4πr²); level β = 10 log(I/I₀) dB with I₀ = 1×10⁻¹² W/m². A +10 dB increase is ten times the intensity; doubling distance reduces level about 6 dB.

Reflection obeys θ_i = θ_r; refraction obeys Snell's law n₁ sin θ₁ = n₂ sin θ₂ with n = c/v. Total internal reflection occurs only from denser to less dense medium beyond sin θ_c = n₂/n₁.

Mirrors and lenses: 1/f = 1/d_o + 1/d_i and m = −d_i/d_o. A positive image distance means a real image (same side as object for mirrors, opposite side for lenses); converging elements have f > 0.

Interference: double-slit maxima at d sin θ = mλ with fringe spacing Δy = λL/d on a distant screen; single-slit minima at a sin θ = mλ; thin films show 2nt = (m + ½)λ when one reflection reverses phase.

Polarization: unpolarized light through a polarizer loses half its intensity; a second analyzer transmits I = I₀ cos²θ (Malus's law). Reflection is fully polarized at Brewster's angle tan θ_p = n₂/n₁.

Electromagnetic waves travel at c = 3.00×10⁸ m/s in vacuum with c = fλ, are transverse with E ⊥ B ⊥ propagation. Spectrum order: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma.

Coulomb's law: F = k|q₁q₂|/r² with k = 8.99×10⁹ N·m²/C². Force acts along the joining line, like charges repel, and net force on a charge is the vector superposition of all individual forces.

Field and potential: E = F/q = kQ/r² for a point charge; V = kQ/r is a scalar; work moving charge is W = qΔV; for a uniform field E = −ΔV/Δx. Field lines point toward decreasing potential and are perpendicular to equipotentials.

Capacitance: C = Q/V; parallel plates C = ε₀A/d; stored energy U = ½CV² = Q²/(2C). Capacitors add directly in parallel and reciprocally in series, 1/C_eq = Σ1/C_i.

DC circuits: V = IR; P = IV = I²R = V²/R. Resistors add directly in series and reciprocally in parallel; an ammeter goes in series with low resistance, a voltmeter in parallel with high resistance.

Kirchhoff's rules: currents into a junction sum to zero; potential changes around any closed loop sum to zero. Terminal voltage of a real source is V = ε − Ir, where r is internal resistance.

Magnetic force and fields: F = qvB sin θ, perpendicular to both v and B, so magnetic force does no work; F = BIL sin θ on a current-carrying wire. Long straight wire B = μ₀I/(2πr); solenoid B = μ₀nI; μ₀ = 4π×10⁻⁷ T·m/A.

Electromagnetic induction: flux Φ = BA cos θ; induced emf ε = −N dΦ/dt. Lenz's law states the induced current opposes the flux change that created it; motional emf on a moving rod is ε = Bℓv.

AC and transformers: V_s/V_p = N_s/N_p, and an ideal transformer has P_p = P_s. RMS values relate by V_rms = V_peak/√2; an RC circuit charges with time constant τ = RC.

Calorimetry: Q = mcΔT for temperature change and Q = mL for a phase change, which occurs at constant temperature. In an insulated system, heat lost by warmer objects equals heat gained by cooler ones.

First law of thermodynamics: ΔU = Q − W, where W is work done by the gas. W = PΔV at constant pressure; isothermal work is nRT ln(V₂/V₁); adiabatic means Q = 0; isochoric means W = 0.

Ideal gas: PV = nRT with R = 8.31 J/(mol·K), or PV = NkT with k = 1.38×10⁻²³ J/K; temperature must be in kelvin. Average translational KE per molecule = (3/2)kT and v_rms = √(3RT/M).

Second law and entropy: ΔS = Q_rev/T; entropy of an isolated system never decreases, and heat flows spontaneously from hot to cold. No engine between reservoirs exceeds the Carnot limit e = 1 − T_C/T_H (kelvin).

Heat engines and refrigerators: e = W/Q_H = 1 − Q_C/Q_H; a refrigerator's coefficient of performance is Q_C/W. Irreversibility such as friction always lowers real efficiency below Carnot.

Photoelectric effect: K_max = hf − φ, threshold frequency f₀ = φ/h, and stopping potential satisfies eV_s = K_max. Light intensity controls the number of emitted electrons; frequency controls their maximum kinetic energy.

Photons and matter waves: E = hf = hc/λ with h = 6.63×10⁻³⁴ J·s; de Broglie wavelength λ = h/p = h/(mv), which becomes measurable for electrons and other low-mass particles.

Atomic and nuclear physics: hydrogen levels E_n = −13.6 eV/n²; mass defect converts to binding energy through E = mc²; radioactive decay follows N = N₀e^(−λt) with half-life T½ = ln2/λ.

Standard model and conservation laws: quarks and leptons are the fermion building blocks, force carriers are bosons. Nuclear reactions conserve electric charge, nucleon number, and total mass-energy; alpha, beta, and gamma decay each change the nucleus differently.