PAU Physics (Castilla y León) cheat sheet

Newton's law of gravitation: F = G·m₁·m₂/r², attractive and always along the line joining the centres; G = 6.67×10⁻¹¹ N·m²·kg⁻².

Field strength g = F/m = G·M/r², directed towards the centre of the mass; outside a sphere it acts as if all mass were at the centre (shell theorem).

Gravitational potential V = −G·M/r (zero at infinity, always negative); potential energy U = −G·M·m/r. Field is the negative gradient: g = −dV/dr.

Escape velocity from a body of mass M and radius R: v_e = √(2GM/R) = √(2gR); independent of the escaping body's mass and direction (ignoring drag and rotation).

Circular orbit: GM·m/r² = m·v²/r gives v = √(GM/r); period T = 2πr/v so T² = (4π²/GM)·r³ (Kepler's third law for circular orbits).

Kepler's laws: elliptical orbits with the Sun at one focus; equal areas in equal times (conservation of angular momentum); T² ∝ a³ for all planets of the same central body.

Total mechanical energy in a circular orbit: E = −GM·m/(2r); kinetic energy is half the magnitude of the potential energy (virial relation).

Geostationary orbit requires T = 24 h, equatorial plane, and prograde motion; radius follows from r = (GM·T²/4π²)^(1/3), about 4.2×10⁷ m from Earth's centre.

Satellites in the same circular orbit have equal speeds regardless of mass; a heavier satellite needs more energy to be placed there but moves at the same v.

Coulomb's law: F = k·|q₁·q₂|/r² with k = 1/(4πε₀) ≈ 9×10⁹ N·m²·C⁻²; force is repulsive for like signs, attractive for opposite signs.

Field of a point charge: E = k·|Q|/r², pointing away from positive charge and towards negative charge; superposition applies vectorially.

Electric potential of a point charge: V = k·Q/r (scalar, sign of Q retained); potential energy of a charge q in it: U = q·V.

Uniform field between parallel plates: E = V/d, directed from the positive to the negative plate; force on charge q is qE, independent of position.

Gauss's law: flux of E through a closed surface = Q_enclosed/ε₀; gives E = σ/ε₀ just outside a conductor and E = 0 inside a conductor in equilibrium.

Work done by the field equals minus the change in potential energy; W = q·(V_A − V_B) for moving q from A to B, path-independent because the field is conservative.

Equipotential surfaces are perpendicular to field lines; no work is done moving a charge along an equipotential.

Capacitance C = Q/V; parallel-plate capacitor C = ε₀·ε_r·A/d; stored energy U = ½·C·V² = Q²/(2C).

Motion of a charge entering a uniform E perpendicular to its velocity is parabolic, like projectile motion, with acceleration a = qE/m.

Force on a moving charge: F = q·v×B, magnitude q·v·B·sinθ; the magnetic force does no work because it is always perpendicular to v.

Force on a straight current-carrying wire: F = I·L×B, magnitude B·I·L·sinθ; direction from the right-hand rule (or Fleming's left-hand rule).

Field of a long straight wire: B = μ₀·I/(2πr); inside a long solenoid: B = μ₀·n·I with n the turns per unit length.

Field at the centre of a circular loop of N turns and radius R: B = μ₀·N·I/(2R), perpendicular to the plane of the loop.

Magnetic flux Φ = B·A·cosθ with θ the angle between B and the surface normal; unit weber (Wb).

Faraday's law: induced emf ε = −N·dΦ/dt; the minus sign is Lenz's law - the induced current opposes the change in flux that produced it.

Motional emf for a rod of length L moving at speed v perpendicular to B: ε = B·L·v; if the circuit has resistance R, current I = BLv/R.

Self-inductance: ε = −L·dI/dt; energy stored in an inductor U = ½·L·I².

Charged particle moving perpendicular to a uniform B follows a circle of radius r = m·v/(|q|·B); the period T = 2πm/(|q|·B) is independent of speed.

Wave speed: v = λ·f = λ/T; for a stretched string v = √(T/μ) with μ the linear mass density.

Harmonic wave: y(x,t) = A·sin(ωt − kx + φ), with k = 2π/λ and ω = 2πf; the sign of kx sets the propagation direction.

Sound in air is a longitudinal pressure wave; speed in air ≈ 340 m/s at 20 °C and increases with temperature; in a solid v = √(E/ρ).

Sound intensity level: β = 10·log(I/I₀) dB with I₀ = 10⁻¹² W/m²; intensity I = P/(4πr²) for a point source, so β falls by 6 dB per doubling of distance.

Doppler effect for sound: f' = f·(v ± v_o)/(v ∓ v_s), upper signs for approach; the observed frequency rises on approach and falls on receding.

Standing waves on a string fixed at both ends: λ_n = 2L/n, f_n = n·v/(2L); a pipe open at both ends has the same set; a pipe closed at one end has only odd harmonics f_n = n·v/(4L), n odd.

Interference is constructive when path difference = n·λ and destructive when = (n + ½)·λ, for coherent sources in phase.

Diffraction grating: d·sinθ = n·λ; double-slit: d·sinθ = n·λ for bright fringes, with fringe spacing Δy = λ·D/d for small angles.

Energy of a wave is proportional to the square of its amplitude; intensity is power per unit area perpendicular to propagation.

Reflection: angle of incidence equals angle of reflection, both measured from the normal; for a plane mirror the image is virtual, same size, and as far behind as the object is in front.

Snell's law: n₁·sinθ₁ = n₂·sinθ₂; refractive index n = c/v; light bends towards the normal entering a denser medium.

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

Thin lens and spherical mirror equation: 1/f = 1/s' − 1/s (lens, Cartesian sign convention) or 1/f = 1/v + 1/u (real-is-positive convention); magnification m = y'/y = −s'/s.

Lens maker's equation: 1/f = (n − 1)·(1/R₁ − 1/R₂) for a thin lens in air, with radii signed by convention.

Power of a lens P = 1/f in dioptres (m⁻¹); converging lens f > 0, diverging lens f < 0.

Magnifying glass: angular magnification M = 25 cm/f (image at infinity) or 1 + 25/f (image at near point), with f in cm.

Compound microscope: M ≈ (L/f_o)·(25/f_e) for tube length L; astronomical telescope: M = f_o/f_e, so a long objective and short eyepiece give high magnification.

Dispersion: refractive index depends on wavelength, so a prism separates white light; violet bends more than red in glass.

Lorentz factor γ = 1/√(1 − v²/c²); time dilation Δt = γ·Δt₀ and length contraction L = L₀/γ, both for the proper frame quantities.

Relativistic mass-energy: E = m·c²; total energy E = γ·m₀·c²; kinetic energy K = (γ − 1)·m₀·c²; rest energy E₀ = m₀·c².

Photon energy E = h·f = h·c/λ, with h = 6.63×10⁻³⁴ J·s; photon momentum p = h/λ.

Photoelectric effect: h·f = W₀ + K_max, with W₀ the work function; no emission below the threshold frequency f₀ = W₀/h regardless of intensity.

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

Bohr model: E_n = −13.6 eV/n² for hydrogen; photon emitted when an electron falls from n_i to n_f: h·f = E_i − E_f.

Heisenberg uncertainty principle: Δx·Δp ≥ ħ/2 with ħ = h/(2π); applies to conjugate pairs such as energy and time.

Radioactive decay: N = N₀·e^(−λt); half-life T½ = ln2/λ; activity A = λ·N measured in becquerels.

Mass defect and binding energy: Δm = Z·m_p + (A − Z)·m_n − m_nucleus; E_b = Δm·c², with 1 u = 931.5 MeV/c².

Absolute uncertainty of a single reading is usually half the smallest scale division; for a digital instrument it is the last-digit resolution.

For sums and differences, absolute uncertainties add: if z = x ± y then Δz = Δx + Δy.

For products and quotients, relative uncertainties add: if z = x·y or z = x/y then Δz/z = Δx/x + Δy/y.

For a power z = xⁿ, the relative uncertainty is |n|·(Δx/x); for z = k·x the relative uncertainty is unchanged.

Report a result as (value ± Δvalue) with the uncertainty to one significant figure (two if the leading digit is 1) and the value rounded to the same decimal place.

A graph should be plotted as a straight line where possible; the slope of the linearised graph gives the physical constant, and its uncertainty comes from the steepest and shallowest acceptable lines.

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

Percentage error = |measured − accepted|/accepted × 100 %; percentage difference compares two experimental values using their mean in the denominator.

Significant figures: keep the number of significant figures justified by the least precise measurement; do not round intermediate results.

Newton's law of gravitation: F = G·m₁·m₂/r², attractive and always along the line joining the centres; G = 6.67×10⁻¹¹ N·m²·kg⁻².

Field strength g = F/m = G·M/r², directed towards the centre of the mass; outside a sphere it acts as if all mass were at the centre (shell theorem).

Gravitational potential V = −G·M/r (zero at infinity, always negative); potential energy U = −G·M·m/r. Field is the negative gradient: g = −dV/dr.

Escape velocity from a body of mass M and radius R: v_e = √(2GM/R) = √(2gR); independent of the escaping body's mass and direction (ignoring drag and rotation).

Circular orbit: GM·m/r² = m·v²/r gives v = √(GM/r); period T = 2πr/v so T² = (4π²/GM)·r³ (Kepler's third law for circular orbits).

Kepler's laws: elliptical orbits with the Sun at one focus; equal areas in equal times (conservation of angular momentum); T² ∝ a³ for all planets of the same central body.

Total mechanical energy in a circular orbit: E = −GM·m/(2r); kinetic energy is half the magnitude of the potential energy (virial relation).

Geostationary orbit requires T = 24 h, equatorial plane, and prograde motion; radius follows from r = (GM·T²/4π²)^(1/3), about 4.2×10⁷ m from Earth's centre.

Satellites in the same circular orbit have equal speeds regardless of mass; a heavier satellite needs more energy to be placed there but moves at the same v.

Coulomb's law: F = k·|q₁·q₂|/r² with k = 1/(4πε₀) ≈ 9×10⁹ N·m²·C⁻²; force is repulsive for like signs, attractive for opposite signs.

Field of a point charge: E = k·|Q|/r², pointing away from positive charge and towards negative charge; superposition applies vectorially.

Electric potential of a point charge: V = k·Q/r (scalar, sign of Q retained); potential energy of a charge q in it: U = q·V.

Uniform field between parallel plates: E = V/d, directed from the positive to the negative plate; force on charge q is qE, independent of position.

Gauss's law: flux of E through a closed surface = Q_enclosed/ε₀; gives E = σ/ε₀ just outside a conductor and E = 0 inside a conductor in equilibrium.

Work done by the field equals minus the change in potential energy; W = q·(V_A − V_B) for moving q from A to B, path-independent because the field is conservative.

Equipotential surfaces are perpendicular to field lines; no work is done moving a charge along an equipotential.

Capacitance C = Q/V; parallel-plate capacitor C = ε₀·ε_r·A/d; stored energy U = ½·C·V² = Q²/(2C).

Motion of a charge entering a uniform E perpendicular to its velocity is parabolic, like projectile motion, with acceleration a = qE/m.

Force on a moving charge: F = q·v×B, magnitude q·v·B·sinθ; the magnetic force does no work because it is always perpendicular to v.

Force on a straight current-carrying wire: F = I·L×B, magnitude B·I·L·sinθ; direction from the right-hand rule (or Fleming's left-hand rule).

Field of a long straight wire: B = μ₀·I/(2πr); inside a long solenoid: B = μ₀·n·I with n the turns per unit length.

Field at the centre of a circular loop of N turns and radius R: B = μ₀·N·I/(2R), perpendicular to the plane of the loop.

Magnetic flux Φ = B·A·cosθ with θ the angle between B and the surface normal; unit weber (Wb).

Faraday's law: induced emf ε = −N·dΦ/dt; the minus sign is Lenz's law - the induced current opposes the change in flux that produced it.

Motional emf for a rod of length L moving at speed v perpendicular to B: ε = B·L·v; if the circuit has resistance R, current I = BLv/R.

Self-inductance: ε = −L·dI/dt; energy stored in an inductor U = ½·L·I².

Charged particle moving perpendicular to a uniform B follows a circle of radius r = m·v/(|q|·B); the period T = 2πm/(|q|·B) is independent of speed.

Wave speed: v = λ·f = λ/T; for a stretched string v = √(T/μ) with μ the linear mass density.

Harmonic wave: y(x,t) = A·sin(ωt − kx + φ), with k = 2π/λ and ω = 2πf; the sign of kx sets the propagation direction.

Sound in air is a longitudinal pressure wave; speed in air ≈ 340 m/s at 20 °C and increases with temperature; in a solid v = √(E/ρ).

Sound intensity level: β = 10·log(I/I₀) dB with I₀ = 10⁻¹² W/m²; intensity I = P/(4πr²) for a point source, so β falls by 6 dB per doubling of distance.

Doppler effect for sound: f' = f·(v ± v_o)/(v ∓ v_s), upper signs for approach; the observed frequency rises on approach and falls on receding.

Standing waves on a string fixed at both ends: λ_n = 2L/n, f_n = n·v/(2L); a pipe open at both ends has the same set; a pipe closed at one end has only odd harmonics f_n = n·v/(4L), n odd.

Interference is constructive when path difference = n·λ and destructive when = (n + ½)·λ, for coherent sources in phase.

Diffraction grating: d·sinθ = n·λ; double-slit: d·sinθ = n·λ for bright fringes, with fringe spacing Δy = λ·D/d for small angles.

Energy of a wave is proportional to the square of its amplitude; intensity is power per unit area perpendicular to propagation.

Reflection: angle of incidence equals angle of reflection, both measured from the normal; for a plane mirror the image is virtual, same size, and as far behind as the object is in front.

Snell's law: n₁·sinθ₁ = n₂·sinθ₂; refractive index n = c/v; light bends towards the normal entering a denser medium.

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

Thin lens and spherical mirror equation: 1/f = 1/s' − 1/s (lens, Cartesian sign convention) or 1/f = 1/v + 1/u (real-is-positive convention); magnification m = y'/y = −s'/s.

Lens maker's equation: 1/f = (n − 1)·(1/R₁ − 1/R₂) for a thin lens in air, with radii signed by convention.

Power of a lens P = 1/f in dioptres (m⁻¹); converging lens f > 0, diverging lens f < 0.

Magnifying glass: angular magnification M = 25 cm/f (image at infinity) or 1 + 25/f (image at near point), with f in cm.

Compound microscope: M ≈ (L/f_o)·(25/f_e) for tube length L; astronomical telescope: M = f_o/f_e, so a long objective and short eyepiece give high magnification.

Dispersion: refractive index depends on wavelength, so a prism separates white light; violet bends more than red in glass.

Lorentz factor γ = 1/√(1 − v²/c²); time dilation Δt = γ·Δt₀ and length contraction L = L₀/γ, both for the proper frame quantities.

Relativistic mass-energy: E = m·c²; total energy E = γ·m₀·c²; kinetic energy K = (γ − 1)·m₀·c²; rest energy E₀ = m₀·c².

Photon energy E = h·f = h·c/λ, with h = 6.63×10⁻³⁴ J·s; photon momentum p = h/λ.

Photoelectric effect: h·f = W₀ + K_max, with W₀ the work function; no emission below the threshold frequency f₀ = W₀/h regardless of intensity.

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

Bohr model: E_n = −13.6 eV/n² for hydrogen; photon emitted when an electron falls from n_i to n_f: h·f = E_i − E_f.

Heisenberg uncertainty principle: Δx·Δp ≥ ħ/2 with ħ = h/(2π); applies to conjugate pairs such as energy and time.

Radioactive decay: N = N₀·e^(−λt); half-life T½ = ln2/λ; activity A = λ·N measured in becquerels.

Mass defect and binding energy: Δm = Z·m_p + (A − Z)·m_n − m_nucleus; E_b = Δm·c², with 1 u = 931.5 MeV/c².

Absolute uncertainty of a single reading is usually half the smallest scale division; for a digital instrument it is the last-digit resolution.

For sums and differences, absolute uncertainties add: if z = x ± y then Δz = Δx + Δy.

For products and quotients, relative uncertainties add: if z = x·y or z = x/y then Δz/z = Δx/x + Δy/y.

For a power z = xⁿ, the relative uncertainty is |n|·(Δx/x); for z = k·x the relative uncertainty is unchanged.

Report a result as (value ± Δvalue) with the uncertainty to one significant figure (two if the leading digit is 1) and the value rounded to the same decimal place.

A graph should be plotted as a straight line where possible; the slope of the linearised graph gives the physical constant, and its uncertainty comes from the steepest and shallowest acceptable lines.

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

Percentage error = |measured − accepted|/accepted × 100 %; percentage difference compares two experimental values using their mean in the denominator.

Significant figures: keep the number of significant figures justified by the least precise measurement; do not round intermediate results.

Newton's law of gravitation: F = G·m₁·m₂/r², attractive and always along the line joining the centres; G = 6.67×10⁻¹¹ N·m²·kg⁻².

Field strength g = F/m = G·M/r², directed towards the centre of the mass; outside a sphere it acts as if all mass were at the centre (shell theorem).

Gravitational potential V = −G·M/r (zero at infinity, always negative); potential energy U = −G·M·m/r. Field is the negative gradient: g = −dV/dr.

Escape velocity from a body of mass M and radius R: v_e = √(2GM/R) = √(2gR); independent of the escaping body's mass and direction (ignoring drag and rotation).

Circular orbit: GM·m/r² = m·v²/r gives v = √(GM/r); period T = 2πr/v so T² = (4π²/GM)·r³ (Kepler's third law for circular orbits).

Kepler's laws: elliptical orbits with the Sun at one focus; equal areas in equal times (conservation of angular momentum); T² ∝ a³ for all planets of the same central body.

Total mechanical energy in a circular orbit: E = −GM·m/(2r); kinetic energy is half the magnitude of the potential energy (virial relation).

Geostationary orbit requires T = 24 h, equatorial plane, and prograde motion; radius follows from r = (GM·T²/4π²)^(1/3), about 4.2×10⁷ m from Earth's centre.

Satellites in the same circular orbit have equal speeds regardless of mass; a heavier satellite needs more energy to be placed there but moves at the same v.

Coulomb's law: F = k·|q₁·q₂|/r² with k = 1/(4πε₀) ≈ 9×10⁹ N·m²·C⁻²; force is repulsive for like signs, attractive for opposite signs.

Field of a point charge: E = k·|Q|/r², pointing away from positive charge and towards negative charge; superposition applies vectorially.

Electric potential of a point charge: V = k·Q/r (scalar, sign of Q retained); potential energy of a charge q in it: U = q·V.

Uniform field between parallel plates: E = V/d, directed from the positive to the negative plate; force on charge q is qE, independent of position.

Gauss's law: flux of E through a closed surface = Q_enclosed/ε₀; gives E = σ/ε₀ just outside a conductor and E = 0 inside a conductor in equilibrium.

Work done by the field equals minus the change in potential energy; W = q·(V_A − V_B) for moving q from A to B, path-independent because the field is conservative.

Equipotential surfaces are perpendicular to field lines; no work is done moving a charge along an equipotential.

Capacitance C = Q/V; parallel-plate capacitor C = ε₀·ε_r·A/d; stored energy U = ½·C·V² = Q²/(2C).

Motion of a charge entering a uniform E perpendicular to its velocity is parabolic, like projectile motion, with acceleration a = qE/m.

Force on a moving charge: F = q·v×B, magnitude q·v·B·sinθ; the magnetic force does no work because it is always perpendicular to v.

Force on a straight current-carrying wire: F = I·L×B, magnitude B·I·L·sinθ; direction from the right-hand rule (or Fleming's left-hand rule).

Field of a long straight wire: B = μ₀·I/(2πr); inside a long solenoid: B = μ₀·n·I with n the turns per unit length.

Field at the centre of a circular loop of N turns and radius R: B = μ₀·N·I/(2R), perpendicular to the plane of the loop.

Magnetic flux Φ = B·A·cosθ with θ the angle between B and the surface normal; unit weber (Wb).

Faraday's law: induced emf ε = −N·dΦ/dt; the minus sign is Lenz's law - the induced current opposes the change in flux that produced it.

Motional emf for a rod of length L moving at speed v perpendicular to B: ε = B·L·v; if the circuit has resistance R, current I = BLv/R.

Self-inductance: ε = −L·dI/dt; energy stored in an inductor U = ½·L·I².

Charged particle moving perpendicular to a uniform B follows a circle of radius r = m·v/(|q|·B); the period T = 2πm/(|q|·B) is independent of speed.

Wave speed: v = λ·f = λ/T; for a stretched string v = √(T/μ) with μ the linear mass density.

Harmonic wave: y(x,t) = A·sin(ωt − kx + φ), with k = 2π/λ and ω = 2πf; the sign of kx sets the propagation direction.

Sound in air is a longitudinal pressure wave; speed in air ≈ 340 m/s at 20 °C and increases with temperature; in a solid v = √(E/ρ).

Sound intensity level: β = 10·log(I/I₀) dB with I₀ = 10⁻¹² W/m²; intensity I = P/(4πr²) for a point source, so β falls by 6 dB per doubling of distance.

Doppler effect for sound: f' = f·(v ± v_o)/(v ∓ v_s), upper signs for approach; the observed frequency rises on approach and falls on receding.

Standing waves on a string fixed at both ends: λ_n = 2L/n, f_n = n·v/(2L); a pipe open at both ends has the same set; a pipe closed at one end has only odd harmonics f_n = n·v/(4L), n odd.

Interference is constructive when path difference = n·λ and destructive when = (n + ½)·λ, for coherent sources in phase.

Diffraction grating: d·sinθ = n·λ; double-slit: d·sinθ = n·λ for bright fringes, with fringe spacing Δy = λ·D/d for small angles.

Energy of a wave is proportional to the square of its amplitude; intensity is power per unit area perpendicular to propagation.

Reflection: angle of incidence equals angle of reflection, both measured from the normal; for a plane mirror the image is virtual, same size, and as far behind as the object is in front.

Snell's law: n₁·sinθ₁ = n₂·sinθ₂; refractive index n = c/v; light bends towards the normal entering a denser medium.

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

Thin lens and spherical mirror equation: 1/f = 1/s' − 1/s (lens, Cartesian sign convention) or 1/f = 1/v + 1/u (real-is-positive convention); magnification m = y'/y = −s'/s.

Lens maker's equation: 1/f = (n − 1)·(1/R₁ − 1/R₂) for a thin lens in air, with radii signed by convention.

Power of a lens P = 1/f in dioptres (m⁻¹); converging lens f > 0, diverging lens f < 0.

Magnifying glass: angular magnification M = 25 cm/f (image at infinity) or 1 + 25/f (image at near point), with f in cm.

Compound microscope: M ≈ (L/f_o)·(25/f_e) for tube length L; astronomical telescope: M = f_o/f_e, so a long objective and short eyepiece give high magnification.

Dispersion: refractive index depends on wavelength, so a prism separates white light; violet bends more than red in glass.

Lorentz factor γ = 1/√(1 − v²/c²); time dilation Δt = γ·Δt₀ and length contraction L = L₀/γ, both for the proper frame quantities.

Relativistic mass-energy: E = m·c²; total energy E = γ·m₀·c²; kinetic energy K = (γ − 1)·m₀·c²; rest energy E₀ = m₀·c².

Photon energy E = h·f = h·c/λ, with h = 6.63×10⁻³⁴ J·s; photon momentum p = h/λ.

Photoelectric effect: h·f = W₀ + K_max, with W₀ the work function; no emission below the threshold frequency f₀ = W₀/h regardless of intensity.

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

Bohr model: E_n = −13.6 eV/n² for hydrogen; photon emitted when an electron falls from n_i to n_f: h·f = E_i − E_f.

Heisenberg uncertainty principle: Δx·Δp ≥ ħ/2 with ħ = h/(2π); applies to conjugate pairs such as energy and time.

Radioactive decay: N = N₀·e^(−λt); half-life T½ = ln2/λ; activity A = λ·N measured in becquerels.

Mass defect and binding energy: Δm = Z·m_p + (A − Z)·m_n − m_nucleus; E_b = Δm·c², with 1 u = 931.5 MeV/c².

Absolute uncertainty of a single reading is usually half the smallest scale division; for a digital instrument it is the last-digit resolution.

For sums and differences, absolute uncertainties add: if z = x ± y then Δz = Δx + Δy.

For products and quotients, relative uncertainties add: if z = x·y or z = x/y then Δz/z = Δx/x + Δy/y.

For a power z = xⁿ, the relative uncertainty is |n|·(Δx/x); for z = k·x the relative uncertainty is unchanged.

Report a result as (value ± Δvalue) with the uncertainty to one significant figure (two if the leading digit is 1) and the value rounded to the same decimal place.

A graph should be plotted as a straight line where possible; the slope of the linearised graph gives the physical constant, and its uncertainty comes from the steepest and shallowest acceptable lines.

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

Percentage error = |measured − accepted|/accepted × 100 %; percentage difference compares two experimental values using their mean in the denominator.

Significant figures: keep the number of significant figures justified by the least precise measurement; do not round intermediate results.