Maharashtra HSC Physics cheat sheet

Moment of inertia I = Σmᵢrᵢ²; for a rigid body about a given axis it depends on mass distribution and axis position, not on angular speed.

Standard results: ring about central axis MR²; disc ½MR²; solid sphere (2/5)MR²; hollow sphere (2/3)MR²; rod about centre ML²/12, about end ML²/3.

Parallel-axis theorem: I = I_cm + Md², where d is the perpendicular distance between the parallel axes.

Perpendicular-axis theorem (planar lamina only): I_z = I_x + I_y for mutually perpendicular axes through the same point in the plane.

Radius of gyration K satisfies I = MK², so K = √(I/M); it has units of length.

Rotational analogues: τ = Iα, L = Iω, KE_rot = ½Iω²; rolling body total KE = ½Mv²(1 + K²/R²).

Rolling without slipping: v = Rω and a = Rα; friction is static and does no work, so mechanical energy is conserved.

For a body rolling down an incline, a = g sinθ/(1 + K²/R²); the sphere beats the disc, which beats the ring, independent of mass and radius.

Angular momentum is conserved when net external torque is zero: I₁ω₁ = I₂ω₂ for an isolated system.

Pressure at depth h: P = P₀ + ρgh; gauge pressure omits P₀, and pressure is the same at all points on a horizontal level in a static fluid.

Pascal's law: pressure change applied to an enclosed fluid is transmitted undiminished; hydraulic lift gives F₂ = F₁(A₂/A₁).

Buoyant force equals weight of displaced fluid (Archimedes); a body floats when its average density is less than the fluid's.

Continuity equation: A₁v₁ = A₂v₂ for incompressible flow - a narrower pipe gives faster flow.

Bernoulli's theorem: P + ½ρv² + ρgh = constant along a streamline for steady, non-viscous, incompressible, irrotational flow.

Torricelli's result: efflux speed from a small hole at depth h is v = √(2gh), the same as free fall through h.

Venturimeter uses A₁v₁ = A₂v₂ with Bernoulli to give flow rate from the pressure difference between wide and narrow sections.

Viscous force: F = ηA(dv/dx); Stokes' law F = 6πηrv; terminal velocity v_t = 2r²(ρ − σ)g/(9η), so v_t ∝ r².

Poiseuille's law: volume flow rate Q = πPr⁴/(8ηL); halving the radius cuts flow to one-sixteenth.

Surface tension: T = F/L; excess pressure is 2T/R for a soap bubble in air, 4T/R for a bubble in air film, and 2T/R for a liquid drop.

Capillary rise: h = 2T cosθ/(rρg); a non-wetting liquid (θ > 90°) is depressed in the tube.

Ideal gas law: PV = nRT = Nk_BT, with k_B = R/N_A ≈ 1.38 × 10⁻²³ J/K.

Pressure from molecular motion: P = (1/3)(N/V)m v̄² = (1/3)ρ v̄²_rms.

Kinetic energy per molecule = (3/2)k_BT; per mole = (3/2)RT, independent of gas identity at a given T.

Root-mean-square speed v_rms = √(3RT/M) = √(3k_BT/m); it rises as √T and falls as √M.

Mean free path λ = 1/(√2 πnd²), where n is number density and d is molecular diameter.

Degrees of freedom: monatomic 3, diatomic 5 (at ordinary T), polyatomic 6; C_V = (f/2)R and C_P = C_V + R.

Ratio γ = C_P/C_V: 5/3 monatomic, 7/5 diatomic; adiabatic relation PV^γ = constant.

Stefan–Boltzmann law: power radiated P = σeAT⁴, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴; net loss is σeA(T⁴ − T₀⁴).

Wien's displacement law: λ_max T = b ≈ 2.9 × 10⁻³ m K, so hotter bodies peak at shorter wavelengths.

Newton's law of cooling (small excess temperature): rate of cooling ∝ (θ − θ₀).

First law: ΔQ = ΔU + ΔW, with ΔW = ∫P dV; work done by the gas is positive in expansion.

State functions (U, P, V, T) depend only on state; heat and work are path functions.

Isothermal (ΔT = 0): ΔU = 0, so Q = W = nRT ln(V₂/V₁); work is done at the expense of absorbed heat.

Adiabatic (Q = 0): ΔU = −W, with PV^γ = constant and W = nR(T₁ − T₂)/(γ − 1).

Isochoric: W = 0, so Q = ΔU = nC_VΔT. Isobaric: Q = nC_PΔT and W = PΔV = nRΔT.

Second law (Kelvin–Planck): no engine converts heat entirely into work in a cycle; (Clausius): heat cannot flow spontaneously from cold to hot unaided.

Efficiency η = W/Q₁ = 1 − Q₂/Q₁; Carnot η = 1 − T₂/T₁ (absolute temperatures), the maximum possible between the reservoirs.

Coefficient of performance of a refrigerator: COP = Q₂/W = T₂/(T₁ − T₂) for the reversible case.

Entropy change ΔS = ΔQ_rev/T; entropy of an isolated system never decreases, so reversible processes have ΔS = 0.

SHM condition: restoring force F = −kx, so acceleration a = −ω²x and the motion is sinusoidal about the mean position.

Displacement x = A sin(ωt + φ); velocity v = Aω cos(ωt + φ), maximum at the mean position.

Acceleration magnitude is maximum at the extremes (ω²A) and zero at the mean position; v and a differ in phase by π/2 and π respectively.

Period of a spring–mass system: T = 2π√(m/k), independent of amplitude and of gravity.

Simple pendulum: T = 2π√(L/g) for small angles; the period is independent of mass and amplitude.

A particle in uniform circular motion has its projection on a diameter executing SHM of amplitude equal to the radius and ω equal to the angular speed.

Energy: KE = ½mω²(A² − x²), PE = ½mω²x², total E = ½mω²A² = ½kA²; average KE equals average PE over a cycle.

For a floating cylinder of cross-section A and mass m in liquid of density ρ, T = 2π√(m/(Aρg)).

Free oscillations occur at natural frequency; forced oscillations at driving frequency; resonance when the two coincide and damping is small.

Damped motion: amplitude decays as e^(−bt/2m); light damping lowers and broadens the resonance peak.

Superposition: the resultant displacement at a point is the vector sum of the displacements due to each wave.

Path difference δ = nλ gives constructive interference; δ = (2n + 1)λ/2 gives destructive interference for coherent sources in phase.

Phase difference φ = (2π/λ) × path difference; intensity of two coherent waves I = I₁ + I₂ + 2√(I₁I₂)cosφ.

Stationary waves: nodes are points of zero amplitude, antinodes of maximum; the node–antinode spacing is λ/4.

String fixed at both ends: fundamental f = v/2L with v = √(T/μ); harmonics are all integer multiples (f_n = nv/2L).

Stretched string law: f ∝ (1/L)√(T/μ), so frequency rises with tension and falls with linear density.

Open pipe: f_n = nv/2L (all harmonics); closed pipe: f_n = (2n − 1)v/4L (odd harmonics only, fundamental v/4L).

Beats: beat frequency = |f₁ − f₂|; audible only when the frequency difference is small.

Doppler effect (sound): apparent frequency f' = f(v ± v_o)/(v ∓ v_s), using upper signs for approach; no shift for motion perpendicular to the line joining source and observer.

Free and forced vibrations differ in frequency: free vibration occurs at natural frequency, forced at the driving frequency.

Huygens' principle: every point on a wavefront acts as a secondary source; the envelope of secondary wavelets gives the new wavefront.

Young's double slit: fringe width W = λD/d; bright fringes at path difference nλ, dark at (2n − 1)λ/2.

In YDSE in a medium of refractive index μ, the fringe width shrinks to λD/(μd) because the wavelength becomes λ/μ.

Coherent sources are needed for a stable interference pattern; independent lamps do not produce sustained fringes.

Diffraction at a single slit: minima at a sinθ = nλ; central maximum width is 2λD/a, twice the width of secondary maxima.

Plane diffraction grating: (a + b) sinθ = nλ gives the directions of principal maxima.

Resolving power of a telescope = D/(1.22λ); larger aperture gives finer resolution.

Polarisation by reflection: Brewster's law tan θ_p = μ, and the reflected and refracted rays are then perpendicular.

Malus' law: intensity through an analyser is I = I₀cos²θ for polarised light incident on it.

Light is transverse because it can be polarised; interference and diffraction show its wave nature.

Coulomb's law: F = (1/4πε₀)q₁q₂/r², with 1/4πε₀ ≈ 9 × 10⁹ N m² C⁻²; in a medium the force is F/ε_r.

Electric field E = F/q₀; for a point charge E = (1/4πε₀)q/r², directed radially; field lines never cross.

Electric dipole moment p = q(2a); axial field 2kp/r³, equatorial field kp/r³; torque in a field τ = pE sinθ.

Gauss's law: Φ = ∮E·dA = q_enclosed/ε₀; field inside a conductor in electrostatic equilibrium is zero.

Field of an infinite charged sheet σ/(2ε₀); just outside a conductor σ/ε₀; inside a uniformly charged shell, zero.

Potential is scalar: V = (1/4πε₀)q/r; potential energy of two charges U = (1/4πε₀)q₁q₂/r.

E = −dV/dr; potential is constant inside a charged conductor and on its surface, so E is zero there.

Capacitance C = Q/V; parallel plate C = ε₀A/d, or ε₀ε_rA/d with a dielectric of relative permittivity ε_r.

Capacitors in series add reciprocals; in parallel they add directly. Energy stored U = ½CV² = ½QV = Q²/2C.

Dielectric slab inserted with battery disconnected: Q fixed, C rises, V falls, energy falls; with battery connected V is fixed and energy rises.

Ohm's law V = IR holds for metallic conductors at constant temperature; the V–I graph is a straight line through the origin.

Drift velocity v_d = I/(nAe); current density J = nev_d = σE; typical drift speeds are far below the signal speed.

Resistivity ρ = RA/L; for a wire R ∝ L/A, so doubling length doubles R and doubling diameter quarters it.

Temperature dependence: R = R₀(1 + αΔT) for metals; semiconductors have negative α.

Series: R = R₁ + R₂ + …; parallel: 1/R = 1/R₁ + 1/R₂ + …, and parallel resistance is less than the smallest branch.

Kirchhoff's laws: junction rule ΣI = 0 (charge conservation); loop rule ΣΔV = 0 (energy conservation).

Wheatstone bridge balances when P/Q = R/S; the galvanometer then reads zero and no current flows through it.

EMF and internal resistance: I = ε/(R + r), terminal voltage V = ε − Ir; terminal voltage equals ε only when no current is drawn.

Power P = VI = I²R = V²/R; maximum power transfer to a load occurs when load resistance equals internal resistance.

Potentiometer compares EMFs by null deflection: ε₁/ε₂ = l₁/l₂, and it draws no current at balance.

Force on a moving charge F = qvB sinθ, perpendicular to both v and B; a charge moving parallel to B feels no force.

Force on a current-carrying conductor F = BIL sinθ; direction given by Fleming's left-hand rule.

Biot–Savart law: dB = (μ₀/4π)(I dl sinθ)/r².

Field of a long straight wire B = μ₀I/2πr; at the centre of a circular coil B = μ₀NI/2R; inside a long solenoid B = μ₀nI.

Ampere's circuital law: ∮B·dl = μ₀I_enclosed; use symmetry to find fields of straight wires and solenoids.

Force between parallel wires F/L = μ₀I₁I₂/2πd; currents in the same direction attract, opposite currents repel.

Torque on a current loop τ = NIAB sinθ; the magnetic moment is m = NIA, and a moving-coil galvanometer uses τ ∝ I.

Bar magnet: field lines run from N to S outside; magnetic moment m = pole strength × magnetic length; axial field μ₀2m/4πr³.

Earth's magnetism uses declination (angle between geographic and magnetic meridian) and dip (angle of the field with the horizontal).

Diamagnetic substances are weakly repelled, paramagnetic weakly attracted, ferromagnetic strongly attracted and retain magnetism; Curie point marks loss of ferromagnetism.

Faraday's law: induced EMF ε = −N dΦ/dt; Lenz's law fixes the direction so the induced current opposes the change producing it.

Magnetic flux Φ = BA cosθ; flux changes by changing B, A, or the orientation of the coil.

Motional EMF: ε = Bvl for a rod moving perpendicular to B, or ε = Bvl sinθ in general.

Self-inductance ε = −L dI/dt; energy stored in an inductor U = ½LI².

Mutual inductance: ε₂ = −M dI₁/dt; for two coaxial coils M = μ₀N₁N₂A/l.

AC: I = I₀ sinωt; rms value I_rms = I₀/√2, average over a full cycle is zero.

Reactance: X_L = ωL (rises with frequency), X_C = 1/ωC (falls with frequency); impedance Z = √(R² + (X_L − X_C)²).

Series LCR resonance at ω₀ = 1/√(LC); at resonance Z = R is minimum, current is maximum, and the circuit is purely resistive.

Power in AC: P = V_rms I_rms cosφ, where cosφ = R/Z; a purely inductive or capacitive circuit dissipates no average power.

Transformer: V_s/V_p = N_s/N_p = I_p/I_s for an ideal transformer; a step-up transformer raises voltage and lowers current.

Photon energy E = hν = hc/λ; h ≈ 6.63 × 10⁻³⁴ J s, and 1 eV = 1.6 × 10⁻¹⁹ J.

Photoelectric effect: emission occurs only above a threshold frequency ν₀, instantaneously, and photocurrent depends on intensity.

Einstein's equation: hν = φ₀ + K_max, with work function φ₀ = hν₀; stopping potential e V₀ = K_max.

Maximum kinetic energy depends on frequency, not intensity; intensity controls the number of photoelectrons.

Photon momentum p = h/λ = E/c; photons have zero rest mass.

de Broglie wavelength λ = h/p = h/mv = h/√(2mK) for a particle of kinetic energy K.

Electron accelerated through V has λ = h/√(2meV) ≈ 12.27/√V Å, so faster electrons have shorter wavelength.

Davisson–Germer electron diffraction confirms the wave nature of matter.

Stopping potential versus frequency is a straight line of slope h/e, with x-intercept at ν₀.

Rutherford scattering: most alpha particles pass undeviated, a few deflect strongly, implying a tiny dense positive nucleus.

Bohr's postulates: quantised orbits with mvr = nh/2π and radiation only on transitions between stationary orbits.

Bohr radius r_n = 0.529 n²/Z Å; orbit radius grows as n², so higher orbits are much larger.

Energy levels of hydrogen: E_n = −13.6 Z²/n² eV; ionisation energy from the ground state is 13.6 eV.

Speed in nth orbit v_n = (2.18 × 10⁶ m/s) Z/n, so v ∝ 1/n.

Spectral series: Lyman (n → 1, ultraviolet), Balmer (n → 2, visible), Paschen (n → 3, infrared).

Rydberg formula: 1/λ = R(1/n₁² − 1/n₂²), R ≈ 1.097 × 10⁷ m⁻¹.

Photon energy for a transition is hν = E₂ − E₁; the smallest energy gap gives the longest wavelength.

Bohr theory fails for multi-electron atoms and cannot explain fine structure or relative intensities.

Nuclear radius R = R₀A^(1/3), R₀ ≈ 1.2 fm; nuclear density is nearly constant for all nuclei.

Notation ᴬ_ZX: Z protons, A − Z neutrons; isotopes share Z, isobars share A, isotones share neutron number.

1 u = 931.5 MeV/c²; mass defect Δm converts to binding energy E_b = Δm c².

Binding energy per nucleon peaks near A ≈ 56 (iron), so fusion of light nuclei and fission of heavy nuclei both release energy.

Radioactive decay law N = N₀e^(−λt); half-life T₁/₂ = 0.693/λ; activity A = λN.

Decay modes: alpha emission lowers A by 4 and Z by 2; beta-minus raises Z by 1 with A unchanged; gamma changes neither.

After n half-lives the undecayed fraction is (1/2)ⁿ; mean life τ = 1/λ = 1.44 T₁/₂.

Nuclear forces are short-range (a few fm), charge-independent, and strongly attractive between nucleons.

Mass–energy: E = mc²; total energy released in a reaction equals the mass defect times c².

Intrinsic semiconductors: electrons in the conduction band equal holes in the valence band; conductivity rises with temperature (negative temperature coefficient).

Doping: pentavalent donors give n-type (majority electrons), trivalent acceptors give p-type (majority holes); the crystal stays electrically neutral.

In a p–n junction, diffusion creates a depletion layer with a built-in potential barrier; forward bias lowers the barrier, reverse bias raises it.

Diode I–V: negligible current below the knee voltage (about 0.7 V for Si, 0.3 V for Ge); reverse breakdown occurs at a large reverse voltage.

Half-wave rectifier gives output only in one half-cycle; full-wave rectifier uses both halves, and a filter capacitor smooths the output.

Zener diode operates in reverse breakdown as a voltage regulator, holding a nearly constant voltage across the load.

Transistor action needs the emitter–base junction forward biased and the collector–base junction reverse biased.

Current gains: α = I_C/I_E and β = I_C/I_B, related by β = α/(1 − α).

In CE configuration the output is 180° out of phase with the input, and voltage gain is high.

Logic gates: AND outputs 1 only if all inputs are 1; OR if any input is 1; NOT inverts; NAND and NOR are universal gates.

Angular kinematics: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ for constant α.

Torque τ = rF sinθ = r × F; it is zero when the force line passes through the axis.

Equilibrium of a rigid body requires both net force and net torque to vanish.

For a flywheel, stored energy ½Iω² explains why a large rim mass smooths engine speed.

Moment of inertia of a thin rod about a point at distance h from centre is M(L²/12 + h²).

Bulk modulus K = −V dP/dV; compressibility is 1/K.

Shear modulus η_s = shear stress/shear strain, applicable to solids, not to ideal fluids.

Streamline flow is steady and laminar; turbulent flow has irregular, unpredictable eddies above a critical Reynolds number.

Reynolds number Re = ρvD/η distinguishes laminar (low Re) from turbulent (high Re) flow.

Aerofoil lift and the curved-ball swing follow from the Bernoulli pressure difference between faster and slower flow regions.

Boyle's law PV = constant at fixed T; Charles' law V/T = constant at fixed P; Gay-Lussac's law P/T = constant at fixed V.

Dalton's law of partial pressures: total pressure is the sum of partial pressures of non-reacting gases.

Average speed v_avg = √(8RT/πM) and most probable speed v_mp = √(2RT/M); v_rms > v_avg > v_mp.

Specific heat of gases: C_V = (f/2)R per mole; for a monatomic gas C_V = 3R/2 and C_P = 5R/2.

A black body absorbs all incident radiation; its spectrum depends only on temperature, not on material.

Zeroth law defines temperature and justifies the thermometer; bodies in thermal equilibrium share the same temperature.

Sign convention: heat absorbed by the system is positive, work done by the system is positive.

For a cyclic process ΔU = 0, so net heat absorbed equals net work done.

In a PV diagram, work done equals the area enclosed by the cycle; clockwise cycles give positive net work.

A reversible process is quasi-static with no dissipation; real processes are irreversible and raise total entropy.

SHM displacement can be written x = A cos(ωt + φ); the choice of sine or cosine only changes the phase constant.

ω = 2π/T = 2πf; the phase changes by 2π in one period.

A simple pendulum has T = 2π√(L/g), so measuring T and L gives g = 4π²L/T².

For a body of moment of inertia I about the pivot, T = 2π√(I/mgd), the physical pendulum result.

In SHM, KE is maximum at the mean position, PE maximum at the extremes, and both are equal at x = A/√2.

Two identical waves travelling in opposite directions superpose to give a stationary wave; the wave speed and wavelength stay the same.

In a stationary wave, all particles between two adjacent nodes vibrate in phase; adjacent loops are in opposite phase.

Reflection at a rigid boundary gives a phase change of π (node at the boundary); at a free boundary there is no phase change (antinode).

Fundamental frequency of a stretched string f = (1/2L)√(T/μ); the string's overtones are all harmonics.

Sound travels faster in solids than in gases; speed in a gas v = √(γRT/M), independent of pressure at fixed T.

Refractive index μ = c/v = λ_vacuum/λ_medium; frequency is unchanged on refraction.

Optical path difference uses μ × geometrical path; a thin film shows a phase change of π on reflection at a denser medium.

For thin-film interference, constructive reflection needs 2μt cos r = (2n − 1)λ/2 when one reflection is at a denser medium.

Diffraction limits resolution: two point sources are just resolved when the central maximum of one falls on the first minimum of the other.

Unpolarised light through a polariser loses half its intensity; a second polariser at 90° extinguishes the beam.

Electric flux through a closed surface depends only on enclosed charge, not on the surface shape or outside charges.

Field just outside a charged spherical conductor is σ/ε₀; inside the conductor it is zero.

Capacitance of an isolated sphere C = 4πε₀R.

Energy density of an electric field u = ½ε₀E².

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

Current I = Q/t = nAev_d; conventional current is opposite to electron drift.

Resistance R = ρL/A; resistivity depends on material and temperature, not on the sample's shape.

Mobility μ = v_d/E; conductivity σ = neμ.

A cell's EMF is the potential difference across it when no current flows; internal resistance causes the terminal voltage to drop under load.

Ammeter is connected in series with low resistance; voltmeter in parallel with high resistance.

A moving charge in a magnetic field follows a circular path of radius r = mv/qB; the field does no work on it.

A charged particle entering B at an angle follows a helical path with pitch v cosθ × T.

Magnetic field at the centre of a semicircular arc B = μ₀I/4R.

Tangent galvanometer: B = (μ₀n/2R)I, and the deflection satisfies I ∝ tanθ.

Relative permeability μ_r = μ/μ₀; for ferromagnets it is large and depends on the applied field (hysteresis).

Eddy currents are induced in bulk conductors; they cause energy loss in transformer cores, reduced by lamination.

Induced EMF in a rotating coil ε = NBAω sinωt, giving peak EMF NBAω.

Average value of AC over a half cycle is 2I₀/π; rms is I₀/√2, so form factor is 1.11.

Quality factor Q = ω₀L/R = (1/R)√(L/C); a higher Q gives a sharper resonance.

Choke coil controls AC current with little power loss because a pure inductor dissipates no average power.

Intensity of light sets the number of photons per second; frequency sets each photon's energy.

Photoelectric current saturates at high voltage when all emitted electrons reach the anode.

The photoelectric effect is explained only by the photon picture; wave theory cannot account for the threshold frequency.

de Broglie wavelength of a thermal neutron at room temperature is of order 10⁻¹⁰ m, comparable to atomic spacing.

Interference and diffraction of electrons demonstrate wave behaviour, complementing their particle behaviour.

Bohr's angular momentum quantisation mvr = nh/2π gives discrete orbits and stable atoms.

Total energy of the nth orbit E_n = −ke²Z/2r_n, so kinetic energy is minus the total energy.

Angular frequency of the electron in the nth orbit falls as 1/n³.

Absorption spectra are dark lines on a bright background at the same wavelengths as the element's emission lines.

The Bohr model explains the hydrogen spectrum but not the Zeeman effect or multi-electron spectra.

Atomic mass unit 1 u = 1.66 × 10⁻²⁷ kg; proton mass 1.00728 u, neutron 1.00867 u.

Binding energy of deuteron is about 2.22 MeV; of helium-4 about 28.3 MeV.

Activity unit: 1 becquerel = 1 decay per second; 1 curie = 3.7 × 10¹⁰ Bq.

Beta-plus decay converts a proton to a neutron, lowering Z by 1 and emitting a positron and neutrino.

In nuclear fission of U-235, about 200 MeV is released per fission, mostly as kinetic energy of fragments.

Energy bands: full valence band and empty conduction band separated by a large gap gives an insulator; a small gap (about 1 eV) gives a semiconductor.

At absolute zero an intrinsic semiconductor behaves as an insulator; thermal energy creates electron–hole pairs above it.

Dynamic resistance of a diode r_d = ΔV/ΔI, small in forward bias and very large in reverse bias.

Rectifier ripple factor measures residual AC; a larger filter capacitor reduces ripple.

Transistor as a switch is driven between cut-off (no collector current) and saturation (maximum collector current).