NYSTCE Physics cheat sheet

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

Accuracy is closeness to an accepted value; precision is repeatability among measurements. Calibration reduces systematic error, while repeated trials reduce random error.

For multiplication or division, keep the result to the least number of significant figures in the inputs; for addition or subtraction, keep the least number of decimal places.

Percent error = |measured − accepted| / |accepted| × 100%; percent difference compares two measured values using their average as the denominator.

A valid physical equation must be dimensionally homogeneous; check that both sides reduce to the same combination of base dimensions.

On a linear graph, slope has units of y per x and area under the curve has units of y times x; linearize nonlinear data before extracting a slope.

A falsifiable hypothesis makes predictions that could be contradicted by observation; controlled variables are held constant while the independent variable is changed.

Uncertainty propagation: for sums and differences add absolute uncertainties; for products and quotients add relative (percent) uncertainties.

Constant-acceleration kinematics: v = v₀ + at, Δx = v₀t + ½at², and v² = v₀² + 2aΔx; these apply only when acceleration is constant.

Newton's second law: F_net = ma. Acceleration is parallel to the net external force and inversely proportional to mass.

Static friction satisfies f_s ≤ μ_sN and adjusts to prevent sliding; kinetic friction is f_k = μ_kN and opposes sliding motion.

Work-energy theorem: W_net = ΔK = ½mv_f² − ½mv_i². For conservative forces, work equals the negative change in potential energy.

Impulse equals momentum change: J = F_avgΔt = Δp. Total momentum is conserved when the net external force on the system is zero.

Uniform circular motion requires centripetal acceleration a_c = v²/r directed toward the center; the net inward force has magnitude mv²/r.

Gravitation: F = Gm₁m₂/r²; near a spherical body g = GM/r²; circular orbital speed is v = √(GM/r), independent of the orbiting mass.

Rotational dynamics: τ_net = Iα and L = Iω. Rolling without slipping relates linear and angular speed by v = ωr.

For ideal projectile motion, horizontal and vertical motions are independent; for level ground and no air resistance, range = v₀² sin(2θ)/g.

First law of thermodynamics with work done by the system: ΔU = Q − W. Internal energy is a state function; heat and work are path-dependent transfers.

Ideal gas law: PV = nRT, where T must be absolute temperature. At constant T, PV is constant; at constant V, P/T is constant.

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

Sensible heat is Q = mcΔT; during a phase change at constant temperature, latent heat is Q = mL.

Isothermal process: T constant and ΔU = 0 for an ideal gas; adiabatic process: Q = 0, so ΔU = −W; for a reversible adiabatic ideal gas, PV^γ is constant.

Second law: isolated-system entropy does not decrease; heat flows spontaneously from hotter to colder objects.

Carnot efficiency for a heat engine is η = 1 − T_c/T_h with absolute temperatures; no real engine between the same reservoirs exceeds it.

Thermal conduction rate is P = kAΔT/L, where k is thermal conductivity, A is cross-sectional area, and L is thickness.

Wave relation: v = fλ. Frequency is set by the source; wave speed depends on the medium, and wavelength adjusts when speed changes.

Standing waves on a string fixed at both ends: f_n = n v/(2L). An open pipe has f_n = n v/(2L); a pipe closed at one end has odd harmonics f_n = n v/(4L).

Sound intensity level in decibels: β = 10 log₁₀(I/I₀), where I₀ = 1 × 10⁻¹² W/m². A 10 dB increase means ten times the intensity.

Doppler shift: observed frequency rises when source and observer approach and falls when they recede; the shift depends on relative motion and wave speed.

Snell's law: n₁ sin θ₁ = n₂ sin θ₂, with angles measured from the normal. Total internal reflection requires n₁ > n₂ and θ₁ > θ_c, where sin θ_c = n₂/n₁.

Thin lens or mirror: 1/f = 1/d_o + 1/d_i, with magnification m = −d_i/d_o; follow a consistent sign convention for real and virtual images.

Double-slit constructive interference: d sin θ = mλ; single-slit destructive minima: a sin θ = mλ, where m is a nonzero integer.

Lens power is P = 1/f in diopters (m⁻¹). Polarization confirms light is transverse; Brewster's angle satisfies tan θ_B = n₂/n₁.

Coulomb's law: F = k|q₁q₂|/r²; electric field is E = F/q, and the field of a point charge is E = kq/r².

Electric potential from a point charge is V = kq/r; potential energy of a charge in a potential is U = qV. Potential is a scalar, while field is a vector.

Ohm's law for an ohmic conductor: V = IR. Resistance from geometry is R = ρL/A, where ρ is resistivity.

Series resistors add: R_eq = R₁ + R₂ + …; parallel resistors add reciprocals: 1/R_eq = 1/R₁ + 1/R₂ + … .

Electrical power: P = IV = I²R = V²/R. Kirchhoff's junction rule conserves charge; the loop rule conserves energy per unit charge.

Magnetic force on a moving charge: F = qv × B; force magnitude is qvB sin θ. The force is perpendicular to velocity, so magnetic forces do no work.

Long straight wire: B = μ₀I/(2πr); ideal solenoid interior: B = μ₀nI, where n is turns per unit length.

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

Capacitance: C = Q/V; parallel-plate capacitor C = ε₀A/d; stored energy U = ½CV² = Q²/(2C).

Relativity postulates: physical laws are the same in all inertial frames, and light speed in vacuum is invariant. Moving clocks run slow and lengths contract along the motion direction.

Lorentz factor: γ = 1/√(1 − v²/c²). Time dilation Δt = γΔτ and length contraction L = L₀/γ.

Relativistic energy: E = γmc²; rest energy E₀ = mc²; photon energy E = hf = hc/λ and photon momentum p = h/λ.

Photoelectric effect: K_max = hf − φ, where φ is the work function. Emission occurs only when hf exceeds φ; intensity affects current, not maximum kinetic energy.

de Broglie wavelength: λ = h/p. Heisenberg uncertainty principle: ΔxΔp ≥ ħ/2.

Hydrogen Bohr levels: E_n = −13.6 eV/n². Photon emission occurs when an electron drops to a lower energy level, with energy equal to the level difference.

Radioactive decay: N = N₀(1/2)^(t/T₁/₂), where T₁/₂ is half-life. Activity decreases by half each half-life.

Mass defect and binding energy: E_b = Δm c². Alpha decay emits helium-4 nuclei; beta-minus decay converts a neutron to a proton and emits an electron and antineutrino.

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

Accuracy is closeness to an accepted value; precision is repeatability among measurements. Calibration reduces systematic error, while repeated trials reduce random error.

For multiplication or division, keep the result to the least number of significant figures in the inputs; for addition or subtraction, keep the least number of decimal places.

Percent error = |measured − accepted| / |accepted| × 100%; percent difference compares two measured values using their average as the denominator.

A valid physical equation must be dimensionally homogeneous; check that both sides reduce to the same combination of base dimensions.

On a linear graph, slope has units of y per x and area under the curve has units of y times x; linearize nonlinear data before extracting a slope.

A falsifiable hypothesis makes predictions that could be contradicted by observation; controlled variables are held constant while the independent variable is changed.

Uncertainty propagation: for sums and differences add absolute uncertainties; for products and quotients add relative (percent) uncertainties.

Constant-acceleration kinematics: v = v₀ + at, Δx = v₀t + ½at², and v² = v₀² + 2aΔx; these apply only when acceleration is constant.

Newton's second law: F_net = ma. Acceleration is parallel to the net external force and inversely proportional to mass.

Static friction satisfies f_s ≤ μ_sN and adjusts to prevent sliding; kinetic friction is f_k = μ_kN and opposes sliding motion.

Work-energy theorem: W_net = ΔK = ½mv_f² − ½mv_i². For conservative forces, work equals the negative change in potential energy.

Impulse equals momentum change: J = F_avgΔt = Δp. Total momentum is conserved when the net external force on the system is zero.

Uniform circular motion requires centripetal acceleration a_c = v²/r directed toward the center; the net inward force has magnitude mv²/r.

Gravitation: F = Gm₁m₂/r²; near a spherical body g = GM/r²; circular orbital speed is v = √(GM/r), independent of the orbiting mass.

Rotational dynamics: τ_net = Iα and L = Iω. Rolling without slipping relates linear and angular speed by v = ωr.

For ideal projectile motion, horizontal and vertical motions are independent; for level ground and no air resistance, range = v₀² sin(2θ)/g.

First law of thermodynamics with work done by the system: ΔU = Q − W. Internal energy is a state function; heat and work are path-dependent transfers.

Ideal gas law: PV = nRT, where T must be absolute temperature. At constant T, PV is constant; at constant V, P/T is constant.

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

Sensible heat is Q = mcΔT; during a phase change at constant temperature, latent heat is Q = mL.

Isothermal process: T constant and ΔU = 0 for an ideal gas; adiabatic process: Q = 0, so ΔU = −W; for a reversible adiabatic ideal gas, PV^γ is constant.

Second law: isolated-system entropy does not decrease; heat flows spontaneously from hotter to colder objects.

Carnot efficiency for a heat engine is η = 1 − T_c/T_h with absolute temperatures; no real engine between the same reservoirs exceeds it.

Thermal conduction rate is P = kAΔT/L, where k is thermal conductivity, A is cross-sectional area, and L is thickness.

Wave relation: v = fλ. Frequency is set by the source; wave speed depends on the medium, and wavelength adjusts when speed changes.

Standing waves on a string fixed at both ends: f_n = n v/(2L). An open pipe has f_n = n v/(2L); a pipe closed at one end has odd harmonics f_n = n v/(4L).

Sound intensity level in decibels: β = 10 log₁₀(I/I₀), where I₀ = 1 × 10⁻¹² W/m². A 10 dB increase means ten times the intensity.

Doppler shift: observed frequency rises when source and observer approach and falls when they recede; the shift depends on relative motion and wave speed.

Snell's law: n₁ sin θ₁ = n₂ sin θ₂, with angles measured from the normal. Total internal reflection requires n₁ > n₂ and θ₁ > θ_c, where sin θ_c = n₂/n₁.

Thin lens or mirror: 1/f = 1/d_o + 1/d_i, with magnification m = −d_i/d_o; follow a consistent sign convention for real and virtual images.

Double-slit constructive interference: d sin θ = mλ; single-slit destructive minima: a sin θ = mλ, where m is a nonzero integer.

Lens power is P = 1/f in diopters (m⁻¹). Polarization confirms light is transverse; Brewster's angle satisfies tan θ_B = n₂/n₁.

Coulomb's law: F = k|q₁q₂|/r²; electric field is E = F/q, and the field of a point charge is E = kq/r².

Electric potential from a point charge is V = kq/r; potential energy of a charge in a potential is U = qV. Potential is a scalar, while field is a vector.

Ohm's law for an ohmic conductor: V = IR. Resistance from geometry is R = ρL/A, where ρ is resistivity.

Series resistors add: R_eq = R₁ + R₂ + …; parallel resistors add reciprocals: 1/R_eq = 1/R₁ + 1/R₂ + … .

Electrical power: P = IV = I²R = V²/R. Kirchhoff's junction rule conserves charge; the loop rule conserves energy per unit charge.

Magnetic force on a moving charge: F = qv × B; force magnitude is qvB sin θ. The force is perpendicular to velocity, so magnetic forces do no work.

Long straight wire: B = μ₀I/(2πr); ideal solenoid interior: B = μ₀nI, where n is turns per unit length.

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

Capacitance: C = Q/V; parallel-plate capacitor C = ε₀A/d; stored energy U = ½CV² = Q²/(2C).

Relativity postulates: physical laws are the same in all inertial frames, and light speed in vacuum is invariant. Moving clocks run slow and lengths contract along the motion direction.

Lorentz factor: γ = 1/√(1 − v²/c²). Time dilation Δt = γΔτ and length contraction L = L₀/γ.

Relativistic energy: E = γmc²; rest energy E₀ = mc²; photon energy E = hf = hc/λ and photon momentum p = h/λ.

Photoelectric effect: K_max = hf − φ, where φ is the work function. Emission occurs only when hf exceeds φ; intensity affects current, not maximum kinetic energy.

de Broglie wavelength: λ = h/p. Heisenberg uncertainty principle: ΔxΔp ≥ ħ/2.

Hydrogen Bohr levels: E_n = −13.6 eV/n². Photon emission occurs when an electron drops to a lower energy level, with energy equal to the level difference.

Radioactive decay: N = N₀(1/2)^(t/T₁/₂), where T₁/₂ is half-life. Activity decreases by half each half-life.

Mass defect and binding energy: E_b = Δm c². Alpha decay emits helium-4 nuclei; beta-minus decay converts a neutron to a proton and emits an electron and antineutrino.