Tamil Nadu HSE First Year Physics cheat sheet

Fundamental SI base quantities: length (metre), mass (kilogram), time (second), electric current (ampere), temperature (kelvin), amount of substance (mole), luminous intensity (candela).

Dimensional analysis: any physical quantity Q has dimensions expressed as [M^a L^b T^c A^d K^e mol^f cd^g]; only quantities with identical dimensions can be added or equated.

Dimensional formula of common quantities: force [MLT^-2], energy/work [ML^2T^-2], power [ML^2T^-3], pressure [ML^-1T^-2], momentum [MLT^-1], angular momentum [ML^2T^-1].

Principle of homogeneity: every term in a valid physical equation must have the same dimensions; this checks correctness but cannot determine dimensionless constants.

Significant figures: multiplication/division retains the least number of significant figures; addition/subtraction retains the least number of decimal places.

Error propagation: for Z = A+B, dZ = dA + dB (absolute); for Z = AB or A/B, fractional error dZ/Z = dA/A + dB/B; for Z = A^n, fractional error = n(dA/A).

Percentage error = (mean absolute error / mean value) x 100; mean absolute error = (1/n) sum |a_i - a_mean|.

Vernier calliper: least count = 1 main scale division - 1 vernier scale division; if n vernier divisions equal (n-1) MSD, LC = 1 MSD / n.

Screw gauge: least count = pitch / number of circular scale divisions; zero error is positive if circular scale zero lies above the reference line, negative if below.

Equations of uniformly accelerated motion: v = u + at; s = ut + (1/2)at^2; v^2 = u^2 + 2as; displacement in nth second = u + a(2n-1)/2.

Projectile motion: time of flight T = 2u sin(theta)/g; maximum height H = u^2 sin^2(theta)/(2g); horizontal range R = u^2 sin(2theta)/g; maximum range at theta = 45 degrees.

Relative velocity: v_AB = v_A - v_B; for two bodies moving in the same direction, relative speed is the difference of speeds.

Average velocity = total displacement / total time; average speed = total path length / total time; average speed >= magnitude of average velocity.

Distance-time graph slope gives speed; velocity-time graph slope gives acceleration and area under the curve gives displacement.

Circular motion: angular velocity omega = d(theta)/dt; tangential acceleration a_t = d|v|/dt; radial acceleration a_r = v^2/r = r omega^2.

Uniform circular motion has constant speed but varying velocity direction, so the motion is accelerated; period T = 2 pi r / v.

Equations of motion are valid only for constant acceleration; for variable acceleration use calculus: v = dx/dt and a = dv/dt = v dv/dx.

Scalars have magnitude only; vectors have magnitude and direction and add by triangle or parallelogram law: R = sqrt(A^2 + B^2 + 2AB cos theta).

Newton's first law: a body continues in uniform motion or at rest unless acted upon by a net external force; this defines inertial frames.

Newton's second law: F_net = dp/dt = ma when mass is constant; the acceleration is along the net force direction.

Newton's third law: forces occur in equal and opposite pairs acting on different bodies; action and reaction never cancel on the same body.

Friction: limiting friction f_max = mu_s N; kinetic friction f_k = mu_k N, with mu_k < mu_s; static friction adjusts from zero up to f_max.

Angle of repose: tan(theta) = mu_s; a block stays at rest on an incline only if tan(theta) <= mu_s.

Banking of roads: for zero friction, tan(theta) = v^2/(rg); with friction, maximum safe speed is v = sqrt(rg(tan theta + mu)/(1 - mu tan theta)).

Impulse J = F_avg * delta_t = change in momentum; this is useful when force acts for a short time.

Apparent weight in a lift: N = m(g + a) when accelerating upward, N = m(g - a) when accelerating downward, N = 0 in free fall.

Momentum conservation: if net external force is zero, total momentum of an isolated system remains constant; used in collisions and recoil problems.

Work W = F s cos(theta); work is positive when theta < 90 degrees, zero when theta = 90 degrees, negative when theta > 90 degrees.

Work-energy theorem: net work done by all forces equals change in kinetic energy, W_net = (1/2)mv^2 - (1/2)mu^2.

Kinetic energy KE = (1/2)mv^2 = p^2/(2m); potential energy of a spring U = (1/2)kx^2.

Gravitational potential energy near Earth U = mgh; conservative forces have path-independent work and a defined potential energy.

Power P = W/t = F v cos(theta); SI unit watt; 1 horsepower = 746 watt.

Conservation of mechanical energy: if only conservative forces act, KE + PE = constant; with friction, mechanical energy decreases by the work done against friction.

Collisions: momentum is conserved in all collisions; kinetic energy is conserved only in perfectly elastic collisions.

Coefficient of restitution e = relative velocity of separation / relative velocity of approach; e = 1 for elastic, e = 0 for perfectly inelastic collisions.

In an elastic collision of equal masses, the bodies exchange velocities; a lighter body rebounds with nearly its own speed when struck by a heavy one.

Centre of mass: R = (sum m_i r_i)/(sum m_i); it moves as if all external forces act at that point.

Torque tau = r x F; magnitude tau = r F sin(theta); torque is zero when force passes through the axis.

Angular momentum L = I omega; conservation of angular momentum: if external torque is zero, I1 omega1 = I2 omega2.

Moment of inertia of a thin rod about its centre: I = ML^2/12; about one end: I = ML^2/3.

Moment of inertia of a solid sphere about a diameter: I = (2/5)MR^2; of a hollow sphere: I = (2/3)MR^2; of a disc about its axis: I = (1/2)MR^2.

Parallel-axis theorem: I = I_cm + Md^2; perpendicular-axis theorem for a planar lamina: I_z = I_x + I_y.

Rolling without slipping: v_cm = R omega; total KE = (1/2)Mv^2 + (1/2)I omega^2 = (1/2)Mv^2(1 + I/(MR^2)).

Equilibrium of a rigid body requires both net force zero and net torque zero; for a body to topple, the line of action of weight must pass outside the base.

Radius of gyration k relates to moment of inertia by I = Mk^2; it depends on mass distribution and axis, not on the total mass alone.

Newton's law of gravitation: F = G m1 m2 / r^2; G = 6.67 x 10^-11 N m^2 kg^-2.

Acceleration due to gravity at height h: g_h = g (R/(R+h))^2; at depth d: g_d = g(1 - d/R); at the centre of Earth g = 0.

Variation of g with latitude: g_phi = g - R omega^2 cos^2(phi); at the equator the reduction is maximum, at the poles it is zero.

Orbital velocity of a satellite near Earth: v_o = sqrt(GM/R) = sqrt(gR) approximately 7.9 km/s.

Escape velocity: v_e = sqrt(2GM/R) = sqrt(2gR) approximately 11.2 km/s; independent of the mass of the projected body.

Kepler's laws: orbits are ellipses with the Sun at one focus; equal areas in equal times (areal velocity constant); T^2 is proportional to a^3.

Total energy of an orbiting satellite E = -GMm/(2r); kinetic energy = GMm/(2r); potential energy = -GMm/r.

Geostationary satellite: period 24 hours, equatorial orbit at about 36,000 km altitude, moves west to east.

Weightlessness in an orbiting satellite arises because the satellite and everything in it fall with the same centripetal acceleration.

Stress = force/area; strain = change in dimension / original dimension; Young's modulus Y = stress/strain for longitudinal deformation.

Hooke's law: stress is proportional to strain within the elastic limit; the slope of the stress-strain graph gives Young's modulus.

Elastic potential energy per unit volume = (1/2) x stress x strain = (1/2)Y(strain)^2.

Bulk modulus K = -P/(delta V/V); compressibility is the reciprocal of bulk modulus; shear modulus involves tangential stress.

Poisson's ratio = lateral strain / longitudinal strain; theoretical range -1 to 0.5, typical metals around 0.3.

Surface tension S = force/length; surface energy = surface tension x change in area; excess pressure inside a soap bubble = 4S/r, inside a liquid drop or bubble = 2S/r.

Capillary rise h = 2S cos(theta)/(r rho g); liquid rises when contact angle is acute, is depressed when the angle is obtuse.

Bernoulli's theorem: P + (1/2)rho v^2 + rho g h = constant along a streamline for steady, incompressible, non-viscous flow.

Viscous force F = eta A dv/dx (Stokes law: F = 6 pi eta r v); terminal velocity v_t = 2r^2(rho - sigma)g/(9 eta).

First law of thermodynamics: delta Q = delta U + delta W; sign convention matters - work done by the system is positive.

Work done in an isobaric process W = P delta V; in an isothermal process W = nRT ln(V2/V1); in an adiabatic process W = (P1V1 - P2V2)/(gamma - 1).

Adiabatic relation: PV^gamma = constant; also TV^(gamma-1) = constant and P^(1-gamma) T^gamma = constant.

For a monatomic gas gamma = 5/3; for a diatomic gas gamma = 7/5; gamma = Cp/Cv and Cp - Cv = R.

Efficiency of a Carnot engine eta = 1 - T2/T1, with temperatures in kelvin; no engine between the same reservoirs can exceed this.

Coefficient of performance of a refrigerator = Q2/W = T2/(T1 - T2).

Second law: heat cannot flow spontaneously from a colder to a hotter body without external work; entropy of an isolated system never decreases.

Calorimetry: Q = mc delta theta; latent heat Q = mL; principle of mixtures: heat lost = heat gained.

Thermal expansion: length L = L0(1 + alpha delta T); area coefficient about 2 alpha; volume coefficient about 3 alpha for isotropic solids.

Ideal gas equation: PV = nRT; in terms of molecules PV = (1/3)Nm v_rms^2.

Pressure P = (1/3)(Nm/V) v_rms^2; molecular speeds: v_rms = sqrt(3RT/M), v_avg = sqrt(8RT/(pi M)), v_mp = sqrt(2RT/M).

Ordering of speeds: v_mp < v_avg < v_rms; ratio v_rms : v_avg : v_mp = sqrt(3) : sqrt(8/pi) : sqrt(2).

Average translational kinetic energy per molecule = (3/2)kT; total internal energy of a monatomic ideal gas U = (3/2)nRT.

Degrees of freedom: monatomic f = 3; diatomic f = 5 at ordinary temperature; polyatomic f = 6; equipartition gives (1/2)kT per degree of freedom.

Mean free path lambda = 1/(sqrt(2) pi n d^2), where n is number density and d is molecular diameter; it increases as pressure falls.

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

Dalton's law of partial pressures: total pressure of a non-reacting gas mixture equals the sum of partial pressures.

Real gases deviate from ideal behaviour at high pressure and low temperature; van der Waals equation: (P + a n^2/V^2)(V - nb) = nRT.

SHM: acceleration a = -omega^2 x; displacement x = A sin(omega t + phi); velocity v = omega sqrt(A^2 - x^2); maximum speed = A omega.

Time period of a simple pendulum T = 2 pi sqrt(L/g); of a spring-mass system T = 2 pi sqrt(m/k).

Total energy in SHM E = (1/2)m omega^2 A^2; kinetic and potential energies vary but their sum stays constant for undamped SHM.

Wave equation: v = f lambda; for a stretched string v = sqrt(T/mu) where mu is linear density.

Speed of sound in a gas v = sqrt(gamma P/rho) = sqrt(gamma RT/M); it increases with temperature and is independent of pressure at constant temperature.

Standing waves on a string fixed at both ends: frequencies f_n = n v/(2L); in a pipe closed at one end only odd harmonics occur, f_n = (2n-1)v/(4L).

Beats: beat frequency = |f1 - f2|; produced when two waves of nearly equal frequency superpose.

Doppler effect for sound: apparent frequency f' = f (v +/- v_o)/(v -/+ v_s), with upper signs for approach; the effect depends on the medium.

Resonance occurs when driving frequency equals the natural frequency, giving maximum amplitude; damping reduces amplitude and slightly lowers the natural frequency.

Before using a vernier calliper or screw gauge, record the zero error and apply it with the correct sign to every reading.

Least count of vernier calliper = 1 MSD / number of vernier divisions; of screw gauge = pitch / number of circular scale divisions.

For a simple pendulum, time at least 20 oscillations and divide; measure length from the point of suspension to the centre of the bob.

For a metre bridge, make all connections tight and tap the jockey gently; insert the key only while taking a reading to avoid heating.

In Ohm's law experiments, connect the ammeter in series and the voltmeter in parallel with the resistor, observing correct polarity.

Take at least three to five readings over the full range, plot a graph, and find the slope rather than relying on a single reading.

Record every measurement with its unit and the correct number of significant figures; report the result as mean value plus or minus mean absolute error.

In resonance column experiments, use a tuning fork of known frequency and locate resonance by the loudest sound, measuring from the water level.

In optics experiments, keep the optical bench aligned, avoid parallax while reading positions, and use a distant object to find the focal length of a concave mirror.