CBSE Class 11 Physics

Chapters: 10

1. Physical World and Measurement

Units and Measurements

Dimensions and Dimensional Analysis – The dimensions of a quantity show how it is built from base quantities: mass [M], length [L], time [T] (and current [A], temperature [K], amount [mol], luminous intensity [cd]). Force = [M L T⁻²]. In a correct equation every term has the same dimensions (principle of homogeneity). We use this to check equations, to convert units from one system to another, and to find how one quantity depends on others. It cannot give number constants like 2π, and it cannot handle sums or trig and log functions.
Significant Figures and Errors in Measurement – No measurement is perfect. Significant figures are the digits we trust plus the first doubtful digit. Errors tell how far a reading may be from the true value: absolute error Δa, relative error Δa/a and percentage error (Δa/a) × 100. When we add or subtract, absolute errors add. When we multiply or divide, percentage errors add. For a power aⁿ, the percentage error becomes n times.
Units and Measurement (Class 11) – To measure means to compare a quantity with a fixed, agreed amount called a unit. Result = number × unit. The world now uses the SI system with 7 base units: metre, kilogram, second, ampere, kelvin, mole and candela. Every other unit (like newton or joule) is a derived unit made by multiplying or dividing base units. Prefixes like kilo (10³) and milli (10⁻³) make very big or very small numbers easy.

2. Kinematics

Motion in a Straight Line · Motion in a Plane

Motion in a Straight Line (Class 11) – To describe motion we first choose a frame of reference: an origin, a direction and a clock. Position x changes with time t. Velocity v = dx/dt is the slope of the x–t graph; acceleration a = dv/dt is the slope of the v–t graph, and the area under the v–t graph is the displacement. For constant a: v = u + at, x = ut + ½at², v² = u² + 2as.
Projectile Motion and Uniform Circular Motion (Class 11) – In a plane, r = r₀ + v₀t + ½at² and v = v₀ + at, applied separately along x and y. A projectile has constant horizontal velocity u cos θ and a vertical velocity that changes by g each second, so its path is a parabola: y = x tan θ − gx²/(2u²cos²θ). T = 2u sin θ/g, H = u² sin²θ/2g, R = u² sin 2θ/g (maximum at 45°). In uniform circular motion speed is constant but the velocity turns, giving a centripetal acceleration a = v²/r = ω²r towards the centre.
Scalars and Vectors for Motion in a Plane (Class 11) – A scalar has only size; a vector has size and direction. Vectors are equal if their size and direction match. Multiplying by a number changes the length (a negative number flips it). Vectors add tail-to-head (triangle or parallelogram law); A − B = A + (−B). Any vector in a plane is A = Ax î + Ay ĵ with Ax = A cos θ, Ay = A sin θ. A·B = AB cos θ is a scalar; A×B has size AB sin θ and is perpendicular to both.

3. Laws of Motion

Laws of Motion

Centripetal Force, Car on a Level Road and on a Banked Road – A body moving in a circle is always changing direction, so it needs a net force towards the centre: the centripetal force F = mv²/r. It is not a new kind of force; tension, gravity, friction or a part of the normal force supplies it. On a level road only friction supplies it, so vmax = √(μs r g). On a road banked at θ, a part of the normal force helps: with no friction the ideal speed is v₀ = √(r g tanθ), and with friction vmax = √[r g (μs + tanθ)/(1 − μs tanθ)].
Inertia, First Law, Momentum, Second Law and Impulse – A body keeps its state of rest or uniform motion unless a net external force acts on it (first law). Inertia is this laziness to change, and mass measures it. Momentum p = mv. The rate of change of momentum equals the net force: F = dp/dt, which gives F = ma when mass is constant (second law). Impulse J = F × Δt = Δp; a longer stopping time means a smaller force.
Friction: Static, Kinetic and Rolling Friction, Laws and Lubrication – Friction is the force that opposes relative motion (or its start) between surfaces in contact. Static friction adjusts itself up to a maximum, the limiting friction fs,max = μs N. Once sliding starts, kinetic friction fk = μk N acts, and μk < μs. Friction depends on the normal force and the nature of the surfaces, not on the area of contact. Rolling friction is much smaller than sliding friction; lubricants and ball bearings reduce friction.
Third Law, Conservation of Momentum and Equilibrium of Forces – Forces always come in pairs: if A pushes B, B pushes A with an equal and opposite force at the same instant (third law). The pair acts on different bodies. For a system with no net external force, the total linear momentum stays constant, which explains recoil, rockets and collisions. A particle is in equilibrium when all forces on it add to zero; three concurrent forces in equilibrium form a closed triangle.

4. Work, Energy and Power

Work, Energy and Power

Elastic and Inelastic Collisions in 1D and 2D – In every collision, total momentum is conserved (no outside force during the short hit). In an elastic collision kinetic energy is also conserved. In an inelastic collision some kinetic energy becomes heat, sound or dent energy; if the bodies stick together it is perfectly inelastic. In 1D, elastic collision gives v₁ = (m₁ − m₂)u₁/(m₁ + m₂) and v₂ = 2m₁u₁/(m₁ + m₂) when body 2 starts at rest. In 2D, momentum is conserved separately along x and y.
Motion in a Vertical Circle – A ball on a string moving in a vertical circle speeds up at the bottom and slows down at the top, because gravity does work on it. At every point the net force towards the centre must be mv²/r. At the bottom T = mg + mv²/r (largest); at the top T = mv²/r − mg (smallest). The string stays tight at the top only if v_top ≥ √(gr). Using energy conservation, this needs u ≥ √(5gr) at the bottom.
Potential Energy, Spring Energy and Conservative Forces – Potential energy U is energy stored because of position or shape. Near the Earth U = mgh; in a stretched or squeezed spring U = ½kx². A force is conservative if its work depends only on the start and end points, not on the path (gravity, spring force). Then F = −dU/dx and mechanical energy K + U stays constant. Friction and air drag are non-conservative: they turn mechanical energy into heat.
Work, Kinetic Energy, Work–Energy Theorem and Power – Work is done when a force moves something along its direction: W = F·s = F s cos θ. For a changing force, work is the area under the F–x graph. A moving body has kinetic energy K = ½mv². The work–energy theorem says: net work done on a body = change in its kinetic energy. Power is how fast work is done: P = W/t = F·v.

5. Motion of System of Particles and Rigid Body

System of Particles and Rotational Motion

Centre of Mass: Two Particles, Rigid Body and Uniform Rod – The centre of mass is the one point that moves as if all the mass of a system were packed there. For two particles on a line, x_cm = (m₁x₁ + m₂x₂)/(m₁ + m₂). It lies on the line joining them, closer to the heavier one. For many particles, x_cm = Σmx/Σm (same for y and z). For a uniform rod of length L, the centre of mass is at L/2, its middle. Internal forces cannot move the centre of mass; only an outside force can: M·a_cm = F_ext.
Moment of Inertia, Radius of Gyration and Rotational Motion – Every idea of straight-line motion has a turning twin: θ for x, ω for v, α for a, I for m, τ for F, L = Iω for p. Moment of inertia I = Σmr² tells how hard it is to change a body's spin; it grows fast when mass sits far from the axis. Radius of gyration k is the distance at which all mass could sit to give the same I: I = Mk². Standard values: ring MR², disc ½MR², solid sphere ⅖MR², rod about centre ML²/12. With constant α: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ, and τ = Iα.
Torque, Angular Momentum and Equilibrium of Rigid Bodies – Torque is the turning effect of a force: τ = r × F, size rF sinθ, unit N m. Angular momentum is the turning version of momentum: L = r × p; for a body spinning about a fixed axis L = Iω. Torque changes angular momentum: τ = dL/dt. If the outside torque is zero, L stays constant, so pulling mass in makes a body spin faster. A rigid body is in equilibrium when the total force is zero (no sliding) and the total torque about any point is zero (no turning).

6. Gravitation

Gravitation

Acceleration Due to Gravity and Its Variation with Height and Depth – The acceleration of a freely falling body near the Earth is g = GM/R² ≈ 9.8 m/s². It does not depend on the falling body's mass. Going up to a height h, g falls: g_h = g R²/(R + h)², which is about g(1 − 2h/R) for small h. Going down to a depth d, g also falls: g_d = g(1 − d/R). At the centre of the Earth, g = 0. So g is largest at the surface.
Escape Speed, Orbital Velocity and Energy of an Orbiting Satellite – Throw something fast enough sideways and it keeps falling around the Earth without landing: that speed is the orbital velocity, v₀ = √(GM/r), about 7.9 km/s just above the surface. Throw it faster, at the escape speed vₑ = √(2GM/R) = √(2gR) ≈ 11.2 km/s, and it leaves Earth for ever. vₑ = √2 × v₀. A satellite in a circular orbit has KE = GMm/2r, PE = −GMm/r and total energy E = −GMm/2r. The total is negative, so the satellite is bound. Higher orbits are slower and take longer: T = 2π√(r³/GM).
Gravitational Potential Energy and Gravitational Potential – Gravitational potential energy (U) of a mass m at distance r from the centre of the Earth is U = −GMm/r, taking U = 0 at infinity. It is negative because gravity attracts: you must do work to pull the mass away to infinity. Near the ground, the change in U for a small lift h is mgh. The work to lift a mass from the surface to height h is GMm(1/R − 1/(R + h)). Gravitational potential V is the PE per kilogram: V = −GM/r, in J/kg. Potential is a scalar, so potentials of many masses just add.
Kepler's Laws of Planetary Motion – Kepler gave three rules for how planets move. 1) Each planet moves on an ellipse with the Sun at one focus. 2) The line from the Sun to the planet sweeps equal areas in equal times, so the planet moves faster when it is near the Sun. 3) The square of the time for one round (T²) is proportional to the cube of the semi-major axis (a³). The second law is really conservation of angular momentum. The third law follows from Newton's law of gravitation.
Newton's Universal Law of Gravitation – Every mass in the universe pulls every other mass. The pull between two point masses m₁ and m₂ a distance r apart is F = G m₁ m₂ / r². It acts along the line joining them. The two bodies pull each other with equal and opposite forces. G = 6.67 × 10⁻¹¹ N m² kg⁻² is the same everywhere, so it is called the universal gravitational constant. When many masses pull one body, the forces add as vectors (superposition).

7. Properties of Bulk Matter

Mechanical Properties of Solids · Mechanical Properties of Fluids · Thermal Properties of Matter

Mechanical Properties of Solids: Stress, Strain and Elasticity – When you pull, push or twist a solid, it changes shape a little. If it comes back when you let go, it is elastic. Stress is the restoring force per area (F/A). Strain is the fractional change in size (like ΔL/L). Up to the elastic limit, stress is proportional to strain (Hooke's law), and the ratio is a modulus: Young's modulus Y for stretching, bulk modulus B for squeezing all round, shear modulus G for sliding faces. A stretched wire also gets thinner (Poisson's ratio), and it stores energy ½ × stress × strain × volume.
Pressure in Fluids and Pascal's Law – A fluid (liquid or gas) pushes on every surface it touches. Pressure is this normal force per area, P = F/A. Because of gravity, the fluid above a point has weight, so pressure grows with depth: P = P₀ + ρgh. Points at the same depth in a still liquid have the same pressure, whatever the vessel's shape. Pascal's law says an extra pressure applied to an enclosed fluid reaches every point equally. The hydraulic lift and brakes use this: a small force on a small piston becomes a big force on a big piston, F = f × A/a.
Surface Tension, Angle of Contact and Capillary Rise – Molecules at a liquid's surface are pulled inward by their neighbours, so the surface behaves like a stretched skin. The extra energy stored per unit area of surface is the surface energy, and it equals the surface tension S (force per unit length, N/m). Where a liquid meets a solid, the angle of contact θ tells whether it wets the solid (θ < 90°, like water on glass) or not (θ > 90°, like mercury). Curved surfaces have extra pressure inside: 2S/r for a drop or air bubble in liquid, 4S/r for a soap bubble. In a thin tube the liquid rises (or falls) by h = 2S cosθ / (rρg).
Viscosity, Stokes' Law and Bernoulli's Theorem – A moving fluid is like a stack of layers sliding over each other. The friction between layers is viscosity: F = ηA(dv/dx). A small ball falling through a fluid feels a drag F = 6πηrv (Stokes' law); when drag plus buoyancy balance the weight, it falls at a steady terminal velocity. Slow flow is streamline; above a critical velocity (Reynolds number about 2000) it becomes turbulent. For steady streamline flow, Av is constant (continuity) and P + ½ρv² + ρgh is constant (Bernoulli). So fast fluid has low pressure. This explains Torricelli's efflux speed √(2gh), the venturimeter, and the lift on wings and spinning balls.
Heat Transfer: Conduction, Convection and Radiation – Heat moves in three ways. In conduction, heat passes from particle to particle while the particles stay in place; the rate through a slab is H = kA(T₁ − T₂)/L, where k is thermal conductivity. In convection, the fluid itself moves: hot fluid is lighter and rises, cool fluid sinks. In radiation, heat travels as electromagnetic waves and needs no medium. A blackbody absorbs all radiation and is the best emitter. Its peak wavelength falls as temperature rises (Wien: λmT = b), and its total emitted power grows as T⁴ (Stefan: P = σAT⁴). Newton's law of cooling says a body cools at a rate proportional to its temperature excess over the surroundings.
Thermal Expansion, Specific Heat, Calorimetry and Latent Heat – Temperature tells how hot a body is; heat is energy that flows because of a temperature difference. Most things expand when heated: ΔL = αLΔT, ΔA = βAΔT, ΔV = γVΔT, with β = 2α and γ = 3α. Water is an exception between 0 °C and 4 °C, where it shrinks on heating, so it is densest at 4 °C. The heat needed to warm a body is Q = mcΔT, where c is the specific heat; gases have two, Cp > Cv, with Cp − Cv = R per mole. In calorimetry, heat lost by hot bodies equals heat gained by cold ones. During melting or boiling the temperature stays constant and heat Q = mL goes into changing the state.

8. Thermodynamics

Thermodynamics

First Law of Thermodynamics: System, Heat, Work and ΔU – Thermodynamics tracks energy. The part we study is the system; the rest is the surroundings; the wall between them is the boundary. A system can be open, closed or isolated. Internal energy U is the total energy stored in the system. It changes only in two ways: by heat q or by work w. The first law says ΔU = q + w (IUPAC signs: q and w are positive when energy goes INTO the system). U is a state function; q and w are path functions. For expansion against a constant outside pressure, w = −p_ext ΔV; for a reversible isothermal expansion of an ideal gas, w = −2.303 nRT log(V₂/V₁).
First Law of Thermodynamics – Internal energy U is the total energy of the molecules inside a system. It changes in only two ways: by heat Q (energy that flows because of a temperature difference) and by work W (energy moved by a force, like a moving piston). First law: ΔQ = ΔU + ΔW. Heat given to a gas partly raises its internal energy and partly lets it do work. Work by a gas at constant pressure is PΔV. For an ideal gas Cp − Cv = R.
Thermodynamic Processes: Isothermal, Adiabatic and More – A thermodynamic process takes a gas from one state to another; on a P–V graph it is a path. Isothermal: T fixed, PV = constant, W = nRT ln(V₂/V₁), ΔU = 0. Adiabatic: no heat in or out, PV^γ = constant, W = nR(T₁ − T₂)/(γ − 1), the gas cools when it expands. Isobaric: P fixed, W = PΔV. Isochoric: V fixed, W = 0. Reversible processes go slowly through equilibrium states; real, fast ones are irreversible. In a cyclic process the gas returns to its start, ΔU = 0 and net work = area of the loop.
Second Law of Thermodynamics, Heat Engines and Refrigerators – The first law says energy is conserved; the second law says which way heat and energy can go. Heat flows by itself only from hot to cold (Clausius). No engine can turn all the heat it takes into work; some must be thrown into a colder body (Kelvin–Planck). A heat engine takes Q₁ from a hot source, does work W and rejects Q₂: efficiency η = W/Q₁ = 1 − Q₂/Q₁. A refrigerator uses work W to move Q₂ from cold to hot: COP α = Q₂/W. The best possible engine, the Carnot engine, has η = 1 − T₂/T₁.
Thermal Equilibrium and the Zeroth Law – Two bodies in contact swap heat until their temperatures are equal. Then they are in thermal equilibrium and nothing changes any more. Zeroth law: if A and B are each in equilibrium with C, then A and B are in equilibrium with each other. This is why a thermometer works. A gas in equilibrium is described by state variables P, V, T and n, which are linked by an equation of state. For an ideal gas it is PV = nRT.

9. Behaviour of Perfect Gases and Kinetic Theory of Gases

Kinetic Theory

Degrees of Freedom, Equipartition of Energy and Specific Heats – A degree of freedom is one independent way in which a molecule can move and store energy. A single atom can only move along x, y and z (f = 3). A dumbbell molecule like O₂ can also spin about two axes (f = 5), and when hot it can vibrate (2 more). The law of equipartition of energy says that in thermal balance each degree of freedom holds, on average, ½kT of energy (a vibration holds kT because it has both kinetic and potential energy). So one mole has U = (f/2)RT, giving Cv = (f/2)R, Cp = Cv + R and γ = 1 + 2/f. For solids, each atom vibrates in 3 directions, giving C = 3R.
Perfect Gas Equation PV = nRT and Work in Compressing a Gas – A gas is made of countless tiny molecules flying about. Their hits on the walls make pressure. For a low-density gas, three simple laws hold: at fixed temperature, P × V stays constant (Boyle); at fixed pressure, V grows in step with kelvin temperature (Charles); at the same P and T, equal volumes hold equal numbers of molecules (Avogadro). Put together they give the perfect gas equation PV = nRT = N k T. One mole holds Avogadro's number, 6.022 × 10²³, of particles. Pushing a piston in does work on the gas; the work equals the area under the P–V graph, and at fixed temperature W = nRT ln(V₁/V₂).
Kinetic Theory of Gases: Pressure, Temperature and rms Speed – Kinetic theory explains gas behaviour by picturing a gas as tiny, fast, randomly moving molecules that bounce elastically and do not pull on each other. Each hit on a wall reverses the molecule's velocity and hands the wall momentum 2mvₓ. Adding the hits of all molecules gives the pressure P = ⅓ n m v̄² = ⅓ ρ v̄². Comparing with PV = N k T shows that the average kinetic energy of a molecule is (3/2) k T: temperature is a measure of the average kinetic energy of the molecules. The root mean square speed is v_rms = √(3RT/M) = √(3kT/m), so lighter gases move faster at the same temperature.
Mean Free Path of Gas Molecules – Gas molecules are fast, yet a smell spreads slowly because each molecule keeps bumping into others and changing direction. The average distance a molecule travels between two collisions is its mean free path λ. A molecule of diameter d hits any other molecule whose centre comes within d of its path, so it sweeps a collision tube of cross-section πd². Counting molecules in that tube gives λ = 1/(√2 n π d²) = k_BT/(√2 π d² P). The mean free path is shorter when the gas is crowded (large n or P) or the molecules are big, and longer at low pressure and high temperature. For air at room conditions λ is about 0.1 micrometre.

10. Oscillations and Waves

Oscillations · Waves

Simple Harmonic Motion (SHM) – A motion that repeats after a fixed time is periodic. If the object goes to and fro about a middle point and the force pulling it back is proportional to its distance from the middle (F = −kx), the motion is simple harmonic. Its position is x = A sin(ωt + φ), with ω = 2π/T = 2πf. Speed is largest in the middle, acceleration is largest at the ends, and total energy ½kA² stays constant.
Oscillations of a Spring and a Simple Pendulum – A block on a spring feels a pull back F = −kx, where k is the force constant (stiffness). It does SHM with T = 2π√(m/k). A simple pendulum, for small swings, feels a pull back mg sinθ ≈ mgθ and does SHM with T = 2π√(L/g). The pendulum’s period does not depend on the bob’s mass or (for small swings) on the amplitude.
Beats – When two sounds of slightly different frequencies f₁ and f₂ are heard together, they add by superposition. The loudness rises and falls regularly. Each rise is one beat. The number of beats per second is the beat frequency, f_beat = |f₁ − f₂|. Beats are used to tune musical instruments and to find an unknown frequency.
Progressive Waves: Types, Speed and Equation – A wave carries energy from place to place without carrying the matter along. In a transverse wave the particles move at right angles to the wave; in a longitudinal wave they move along it. Speed v = fλ. On a string v = √(T/μ); for sound in a gas v = √(γP/ρ). A wave moving along +x is y = A sin(kx − ωt), with k = 2π/λ and ω = 2πf.
Superposition, Reflection and Standing Waves – When waves overlap, their displacements add (superposition). A wave reflected from a fixed end comes back upside down; from a free end it comes back upright. A wave and its reflection make a standing wave with nodes (no motion) and antinodes (most motion). A string fixed at both ends allows fₙ = n·v/2L (all harmonics). An open pipe allows fₙ = n·v/2L; a pipe closed at one end allows only odd harmonics, fₙ = n·v/4L (n = 1, 3, 5…).