Bagrut Physics 5 Units cheat sheet

Kinematics (constant a): v = v₀ + at, x = x₀ + v₀t + ½at², v² = v₀² + 2aΔx, and Δx = ½(v₀ + v)t.

Newton's second law is ΣF = ma in an inertial frame; friction satisfies f_s ≤ μ_sN and f_k = μ_kN, and the normal force on an incline is N = mg·cosθ.

Momentum p = mv is conserved in any collision; impulse J = FΔt = Δp. Elastic collisions conserve kinetic energy, inelastic ones do not, and perfectly inelastic collisions merge the bodies.

Work-energy theorem: W_net = ΔKE, with KE = ½mv². For a conservative force, W = −ΔU, and mechanical energy is conserved when only conservative forces do work.

Uniform circular motion: a_c = v²/r = ω²r, directed toward the center; the period is T = 2πr/v. For a conical pendulum, tanθ = v²/(rg).

Rotational dynamics: τ = Iα, L = Iω is conserved when the net external torque is zero, and KE_rot = ½Iω². For a solid disk, I = ½MR²; for a hoop, I = MR²; for a point mass at distance r, I = mr².

Rolling without slipping: v = ωR and a = αR, and the contact point is instantaneously at rest. Total KE = ½mv² + ½Iω².

Gravitational potential energy near Earth is U = mgh; in general U = −GMm/r, and the escape speed from a body of mass M and radius R is v_esc = √(2GM/R).

Newton's law of gravitation: F = GMm/r², attractive and along the line joining the centers; G ≈ 6.67×10⁻¹¹ N·m²/kg².

For a circular orbit, GMm/r² = mv²/r gives v = √(GM/r), so the orbital speed decreases with radius; the period satisfies T² = (4π²/GM)r³ (Kepler's third law).

Kepler's second law (equal areas in equal times) is equivalent to conservation of angular momentum, so the orbital speed is greatest at periapsis.

Geostationary orbit requires a period of 24 h, an equatorial plane, and prograde motion; its radius is about 4.2×10⁷ m (altitude about 3.6×10⁷ m).

Satellite total energy: E = −GMm/(2r) for a circular orbit, with KE = +GMm/(2r) and U = −GMm/r, so E = −KE.

g at altitude h is g = GM/(R+h)²; at Earth's surface g ≈ 9.8 m/s². Weight is mg and apparent weight changes with the acceleration of the support.

Escape speed from Earth's surface is v_esc = √(2GM/R) ≈ 11.2 km/s; escape means total energy ≥ 0.

Apparent weightlessness in orbit means free fall: the only force is gravity, and the normal force is zero, not that gravity is absent.

Coulomb's law: F = kq₁q₂/r² with k ≈ 9×10⁹ N·m²/C²; the force is along the line of centers and obeys superposition.

Electric field E = F/q₀; for a point charge E = kq/r². Uniform field between parallel plates: E = V/d. Potential V = kq/r, and potential energy U = qV.

A conductor in electrostatic equilibrium has E = 0 inside, charge on the surface, and the surface is an equipotential; field lines are perpendicular to conductors.

Ohm's law V = IR holds for ohmic resistors at constant temperature; R = ρL/A, and resistivity increases with temperature in metals.

Series: R_eq = ΣR, the same current flows through all elements, and voltages add. Parallel: 1/R_eq = Σ1/R, the voltage is common, and currents add.

Kirchhoff's rules: the junction rule ΣI_in = ΣI_out (charge conservation); the loop rule ΣΔV = 0 (energy conservation).

Capacitance C = Q/V; for a parallel-plate capacitor C = ε₀A/d; ε₀ ≈ 8.85×10⁻¹² F/m. Stored energy U = ½CV² = Q²/(2C).

Capacitors in parallel add (C_eq = ΣC); in series 1/C_eq = Σ1/C. With the battery disconnected, Q is fixed; with the battery connected, V is fixed.

RC charging: q(t) = Cε(1 − e^(−t/RC)), with time constant τ = RC; the current is i = (ε/R)e^(−t/RC).

Magnetic force on a moving charge: F = qvB·sinθ, perpendicular to both v and B; the radius of circular motion is r = mv/(qB).

Force on a current-carrying wire: F = BIL·sinθ. A long straight wire creates B = μ₀I/(2πr), and inside a solenoid B = μ₀nI.

Magnetic flux Φ = BA·cosθ, where θ is the angle between B and the normal to the area; the unit is the weber (T·m²).

Faraday's law: ε = −N·ΔΦ/Δt; the minus sign is Lenz's law - the induced current opposes the change in flux.

Motional emf for a rod of length L moving at speed v perpendicular to B: ε = BLv.

A changing current in one coil induces an emf in a neighboring coil (mutual induction); self-induction opposes any change in the coil's own current.

A charged particle moving parallel to B experiences no magnetic force; the magnetic force does no work because it is always perpendicular to v.

SHM: x = A·cos(ωt + φ), v = −Aω·sin(ωt + φ), a = −ω²x; the maximum speed is Aω and the maximum acceleration is Aω².

For a mass-spring system, ω = √(k/m) and T = 2π√(m/k); for a simple pendulum, T = 2π√(L/g), independent of mass and small amplitude.

Wave relation: v = fλ. On a string, v = √(F/μ); for sound in air, v ≈ 343 m/s at 20 °C and increases with temperature.

Superposition: constructive interference occurs when the path difference is nλ, destructive when it is (n + ½)λ; beats occur at f_beat = |f₁ − f₂|.

Standing waves on a string fixed at both ends: f_n = nv/(2L). For a pipe open at both ends, the same formula applies; for a pipe closed at one end, only odd harmonics occur: f_n = nv/(4L), n odd.

Doppler effect: f' = f(v ± v_o)/(v ∓ v_s), where the upper sign in the numerator applies when the observer approaches the source; the observed frequency rises on approach.

Energy in a wave is proportional to the square of the amplitude; intensity follows the inverse-square law for a point source in three dimensions.

Resonance occurs when the driving frequency equals a natural frequency; at resonance the amplitude is maximal and damping broadens the response curve.

Reflection: the angle of incidence equals the angle of reflection, both measured from the normal. Refraction obeys Snell's law: n₁sinθ₁ = n₂sinθ₂, with n = c/v.

Total internal reflection occurs only when light travels from a denser to a less dense medium and θ₁ exceeds the critical angle sinθ_c = n₂/n₁.

Thin lens / mirror equation: 1/f = 1/d_o + 1/d_i, with magnification m = −d_i/d_o = h_i/h_o; a real image has d_i > 0, and a virtual image has d_i < 0.

A converging lens has f > 0; a diverging lens has f < 0. For a converging lens, an object beyond 2f gives a real, inverted, diminished image; an object inside f gives a virtual, upright, magnified image.

Young's double slit: 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: a·sinθ = mλ (m ≠ 0); the central maximum is twice as wide as the side maxima.

Diffraction grating: d·sinθ = mλ gives sharp maxima; the maximum order satisfies m ≤ d/λ.

Thin-film interference: the path difference is 2nt. A phase flip of π occurs on reflection at a medium of higher n. For a film in air, constructive reflection needs 2nt = (m + ½)λ for a soap film and 2nt = mλ for an anti-reflection coating on glass.

Light passing through a small aperture spreads; the resolving power of a circular aperture is limited by θ_min ≈ 1.22λ/D.

Ideal gas law: PV = nRT, with R = 8.31 J/(mol·K); T is in kelvin. Also PV = NkT, with k = 1.38×10⁻²³ J/K.

Kinetic theory: average translational KE per molecule = (3/2)kT; the root-mean-square speed is v_rms = √(3RT/M) = √(3kT/m).

First law: ΔU = Q − W, where W is work done by the gas. For a monatomic ideal gas, U = (3/2)nRT.

Isothermal: T constant, ΔU = 0, and W = nRT·ln(V₂/V₁). Adiabatic: Q = 0, so ΔU = −W. Isochoric: W = 0, so Q = ΔU. Isobaric: W = PΔV and Q = nC_pΔT.

For an ideal gas, C_p − C_v = R; monatomic C_v = (3/2)R, diatomic C_v = (5/2)R at moderate temperature.

Efficiency of a heat engine: η = W/Q_H = 1 − Q_C/Q_H; the Carnot limit is η_max = 1 − T_C/T_H (kelvin).

Second law: entropy of an isolated system never decreases; heat flows spontaneously from hot to cold, and no engine exceeds the Carnot efficiency.

Conduction rate: P = kAΔT/L. The internal energy of an ideal gas depends only on T, so U is unchanged in an isothermal process.

Photon energy E = hf = hc/λ, with h = 6.63×10⁻³⁴ J·s and hc ≈ 1240 eV·nm.

Photoelectric effect: K_max = hf − W, where W is the work function. The stopping potential satisfies eV_stop = K_max; the threshold frequency is f₀ = W/h. Intensity controls the current, and frequency controls K_max.

Bohr model: E_n = −13.6 eV/n² for hydrogen; the transition energy is ΔE = hf = E_i − E_f. The ionization energy from n = 1 is 13.6 eV.

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

Uncertainty principle: Δx·Δp ≥ ħ/2, with ħ = h/2π; it bounds simultaneous knowledge, not measurement clumsiness.

Nuclear notation: ᴬ_ZX, with A = Z + N. Alpha decay reduces A by 4 and Z by 2; beta-minus raises Z by 1 with A unchanged; gamma emission changes neither.

Radioactive decay: N = N₀e^(−λt), with half-life T½ = ln2/λ ≈ 0.693/λ; activity A = λN, measured in becquerel (1 Bq = 1 decay/s).

Mass-energy: E = mc²; the mass defect Δm of a nucleus corresponds to binding energy Δm·c², and 1 u ≈ 931.5 MeV/c².

Nuclear binding energy per nucleon peaks near iron (A ≈ 56); fusion releases energy for light nuclei and fission for heavy nuclei.

For repeated readings, report the mean and the standard deviation of the mean; for a single reading of an analog scale, the uncertainty is typically half the smallest division.

Absolute uncertainty carries units; relative (fractional) uncertainty is Δx/x and is often given as a percentage.

Addition/subtraction: absolute uncertainties add: Δ(a ± b) = Δa + Δb. Multiplication/division: relative uncertainties add: Δ(ab)/ab = Δa/a + Δb/b.

Powers: if y = xⁿ, then Δy/y = |n|·Δx/x; for y = kxⁿ, a log-log plot has slope n, and a linearized plot (e.g., T² vs. L) should pass through the origin if the relation is proportional.

A best-fit line is drawn through the data, not point-to-point; the slope's uncertainty comes from the spread of extreme acceptable lines.

A valid experiment isolates one independent variable, controls the others, and repeats measurements; systematic errors shift all readings in one direction, while random errors scatter them.

Significant figures: the result carries no more significant figures than the least precise input; report the uncertainty to one or two significant figures and match the value's decimal place.

Graph axes carry quantity and unit; the slope and intercept carry units derived from the axes, and extrapolation beyond the data range is unreliable.