MSCE Physics cheat sheet

SI base units: meter (length), kilogram (mass), second (time), ampere (current), kelvin (temperature), mole (amount), candela (luminous intensity).

Dimensional analysis: both sides of any valid equation must have identical base-unit dimensions; use it to catch errors and to derive relations up to a dimensionless constant.

Significant figures: multiplication/division keeps the least number of sig figs; addition/subtraction keeps the least number of decimal places; exact conversion factors are unlimited.

Random error scatters symmetrically and is reduced by averaging; systematic error shifts all readings the same way and is corrected by calibration, not repetition.

For independent random uncertainties: sums/differences add absolute uncertainties; products/quotients add fractional (percent) uncertainties; powers multiply fractional uncertainty by the exponent.

Precision describes repeatability of a measurement; accuracy describes closeness to the accepted value. A calibrated instrument can be precise but inaccurate.

Scientific models are approximations with stated domains of validity; a model failing outside its domain does not falsify it inside that domain.

A hypothesis must be falsifiable and testable; a theory is a well-tested explanatory framework, not a guess, and a law is a concise relational statement often with unresolved scope.

Constant-acceleration kinematics: v = v0 + at; x = x0 + v0t + ½at²; v² = v0² + 2a(x − x0); average velocity = (v0 + v)/2 only when a is constant.

Projectile motion: horizontal and vertical motions are independent; horizontal velocity is constant (no air resistance), vertical acceleration is g ≈ 9.8 m/s² downward.

Newton's second law: ΣF = ma is a vector equation; acceleration is along the net force. Newton's third-law pairs act on different bodies and never cancel each other.

Friction: static friction satisfies fs ≤ μsN and adjusts to the applied force; kinetic friction equals μkN and opposes relative sliding.

Circular motion requires centripetal acceleration a = v²/r directed toward the center; the centripetal force is a net force supplied by tension, gravity, friction, or normal force.

Inclined planes: the component of weight along the incline is mg sin θ and the normal force is mg cos θ for a frictionless surface at angle θ.

Momentum form of Newton's second law: F_net = Δp/Δt, valid even when mass changes; impulse J = FΔt = Δp.

Free-body diagrams show only forces acting on the object of interest, drawn from the object's center; net force, not individual forces, sets acceleration.

Work: W = Fd cos θ; work is zero when force is perpendicular to displacement (e.g., centripetal force, normal force on level ground).

Work-energy theorem: W_net = ΔKE = ½mv² − ½mv0²; conservative forces have potential energy, nonconservative forces (friction) change mechanical energy.

Mechanical energy conservation: KE + PE is constant when only conservative forces do work; gravitational PE = mgh near Earth, elastic PE = ½kx².

Power: P = W/t = Fv cos θ; average power differs from instantaneous power when velocity varies.

Momentum conservation: total momentum of an isolated system is constant; it is a vector law applied separately to each axis.

Collisions: elastic collisions conserve kinetic energy and momentum; perfectly inelastic collisions stick together and maximize kinetic-energy loss while conserving momentum.

Coefficient of restitution e = (relative separation speed)/(relative approach speed); e = 1 for elastic, e = 0 for perfectly inelastic.

Impulse-momentum theorem: J = ∫F dt = Δp; a longer collision time with the same impulse reduces the peak force (airbags, crumple zones).

Wave relation: v = fλ; frequency is set by the source and does not change when a wave crosses into a new medium, but speed and wavelength do.

Standing waves on a string fixed at both ends: λn = 2L/n and fn = nv/(2L); open-open pipes use the same series, closed-open pipes use odd harmonics fn = nv/(4L).

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

Doppler effect: observed frequency rises when source and observer approach, falls when they recede; the shift depends on relative motion along the line of sight.

Snell's law: n1 sin θ1 = n2 sin θ2 with n = c/v; light bends toward the normal entering a denser medium.

Total internal reflection occurs only going from higher to lower index when the incidence angle exceeds θc = arcsin(n2/n1).

Thin lens/mirror equation: 1/f = 1/do + 1/di with magnification m = −di/do; real images have positive di for lenses, virtual images have negative di.

Young's double slit: bright fringes at d sin θ = mλ; single-slit minima at a sin θ = mλ (m ≥ 1).

Coulomb's law: F = kq1q2/r² with k ≈ 8.99×10⁹ N·m²/C²; the force is along the line joining charges, repulsive for like signs.

Electric field E = F/q and point-charge field E = kq/r²; field lines point away from positive charge and toward negative charge.

Ohm's law V = IR for ohmic conductors at constant temperature; resistance R = ρL/A rises with length and falls with cross-sectional area.

Series: same current, resistances add, voltage divides. Parallel: same voltage, conductances add (1/R_eq = Σ1/Ri), current divides.

Power in circuits: P = IV = I²R = V²/R; power dissipated in a resistor is always positive regardless of current direction.

Kirchhoff's rules: junction rule (charge conservation, ΣI = 0) and loop rule (energy conservation, ΣV = 0 around a closed loop).

Magnetic force on a moving charge F = qvB sin θ, perpendicular to both v and B; on a current-carrying wire F = BIL sin θ.

Faraday's law: induced emf ε = −N ΔΦ/Δt; Lenz's law says the induced current opposes the change in flux that produced it.

Temperature scales: T(K) = T(°C) + 273.15; absolute zero is 0 K, where molecular translational kinetic energy is minimal.

Linear expansion: ΔL = αL0ΔT; area and volume expansions use 2α and 3α for isotropic solids.

Specific heat: Q = mcΔT; latent heat for phase change: Q = mL, occurring at constant temperature.

Ideal gas law: PV = nRT with R ≈ 8.314 J/(mol·K); also PV = NkT with k ≈ 1.38×10⁻²³ J/K.

Average translational kinetic energy per molecule: KE = (3/2)kT; root-mean-square speed v_rms = √(3kT/m) = √(3RT/M).

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

Thermodynamic processes: isothermal (ΔT = 0, ΔU = 0), adiabatic (Q = 0), isobaric (W = PΔV), isochoric (W = 0).

Second law: entropy of an isolated system never decreases; heat flows spontaneously from hot to cold, and no engine exceeds Carnot efficiency e = 1 − Tc/Th.

Time dilation: Δt = γΔt0 and length contraction L = L0/γ with γ = 1/√(1 − v²/c²); proper time is measured in the frame where events occur at one place.

Mass-energy equivalence: E = mc²; total energy E = γmc² and rest energy E0 = mc²; relativistic momentum p = γmv.

Photoelectric effect: photon energy E = hf; maximum kinetic energy K_max = hf − φ, where φ is the work function; no emission below threshold frequency regardless of intensity.

Photon momentum p = h/λ; de Broglie wavelength λ = h/p applies to all matter and is observable for electrons and small particles.

Bohr model: En = −13.6 eV/n² for hydrogen; photon energy equals the energy-level difference, with spectral lines converging at the series limit.

Half-life: N = N0(1/2)^(t/T½); activity A = λN with λ = ln2/T½; decay is random per nucleus but statistically predictable for large samples.

Nuclear binding energy: Δm = mass defect, BE = Δmc²; binding energy per nucleon peaks near iron-56, so fusion releases energy for light nuclei and fission for heavy nuclei.

Radioactive decay types: alpha (−2 charge, −4 mass), beta-minus (n → p + e⁻ + antineutrino), gamma (photon, no change in Z or A).

Lab safety hierarchy: engineering controls (fume hoods, shields) first, then administrative controls (procedures, supervision), then personal protective equipment (goggles, gloves, aprons).

Laser classes: Class 1 is safe under normal use; Class 2 visible lasers rely on the blink reflex; Class 3B and 4 require eye protection matched to the wavelength and controlled access.

Electrical safety: use GFCI-protected outlets near water, inspect cords for damage, never exceed rated current, and de-energize before adjusting circuits.

Formative assessment informs instruction during learning (exit tickets, predict-observe-explain, whiteboards); summative assessment certifies learning at the end of a unit.

Inquiry labs should isolate one independent variable, hold controls constant, and include repeated trials to estimate uncertainty.

Common student misconceptions: heavier objects fall faster, force is needed to sustain motion, current is used up in a circuit, and seasons come from Earth-Sun distance.

A valid rubric states observable criteria and performance levels; inter-rater reliability improves with anchor examples and shared scoring practice.

Chemical and radiation handling: label all containers, know the location of eyewash and shower stations, and follow local rules for radioactive source storage and dosimetry.