Class 11 Physics

The foundation of physics, covering the international system of units, measurement errors, significant figures, and the powerful concept of dimensional analysis.

A comprehensive analysis of rectilinear motion, including position, displacement, velocity, acceleration, the kinematic equations, and relative velocity, tailored for board and competitive exam preparation.

An in-depth analysis of two-dimensional motion using vector algebra, covering projectile motion and uniform circular motion for competitive exam preparation.

A deep dive into Newton's three laws of motion, the concepts of inertia, momentum, impulse, friction, and the dynamics of circular motion.

Chapter 4 solved motion by tracking forces instant by instant. This chapter offers a shortcut so powerful it becomes the preferred tool for the rest of physics: follow the energy instead. It opens with the scalar product $\vec{A}\cdot\vec{B} = AB\cos\theta$, then defines work $W = \vec{F}\cdot\vec{d}$ — and immediately shows that work can be positive, negative or zero, so a force perpendicular to the motion does none at all. From there comes kinetic energy $K = \frac{1}{2}mv^2$ and the work-energy theorem $W_{net} = \Delta K$, proved for constant and then for variable forces, where work becomes the area under the force-displacement graph. Conservative forces earn a potential energy with $F = -\frac{dV}{dx}$, giving gravitational $V = mgh$ and spring $V = \frac{1}{2}kx^2$, and their sum with kinetic energy is conserved: $K + V =$ constant. Power $P = \frac{dW}{dt} = \vec{F}\cdot\vec{v}$ measures how fast the work is done, and the chapter closes with collisions, where momentum is always conserved but kinetic energy only sometimes is. Work, Energy and Power is high-weightage in both JEE Main and NEET, and its energy method quietly solves problems that would be brutal with forces alone.

Up to now every body has been a point. This chapter removes that assumption and asks what changes when a body has size, shape and the ability to spin. It begins with the centre of mass — the single point that moves as though all the mass and all the external force acted there, so that an exploding shell's fragments scatter while their centre of mass sails on undisturbed — and with the linear momentum of a system, whose conservation follows directly. It then builds the machinery of rotation: the vector product, angular velocity with the relation $\vec{v} = \vec{\omega} \times \vec{r}$, torque $\vec{\tau} = \vec{r} \times \vec{F}$, and angular momentum $\vec{l} = \vec{r} \times \vec{p}$, with the rotational form of Newton's second law $\frac{d\vec{L}}{dt} = \vec{\tau}_{ext}$ and the conservation law it implies. The equilibrium of a rigid body supplies the two conditions every ladder, bar and lever problem rests on, and the principle of moments turns them into a tool. Moment of inertia $I = \sum m_i r_i^2$ then plays the role mass plays in straight-line motion, and once the perpendicular and parallel axis theorems are in hand every standard shape becomes tractable. The chapter closes with the full kinematics and dynamics of rotation about a fixed axis — $\omega = \omega_0 + \alpha t$, $\tau = I\alpha$, $KE = \frac{1}{2}I\omega^2$, $P = \tau\omega$ — the conservation of angular momentum that makes a skater speed up as she pulls her arms in, and rolling motion, where translation and rotation finally meet in the condition $v_{cm} = R\omega$. Two topics trimmed from the rationalised syllabus, the axis theorems and rolling motion, are restored in full because JEE Main, JEE Advanced and NEET ask about them every single year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.

One force, written in one line, that holds an apple to the ground and the Earth to the Sun. The chapter opens with Kepler's three laws — the empirical pattern in the planets' motion that Newton then explained — and shows that the law of areas is simply the conservation of angular momentum for a central force. It then states the universal law of gravitation, $F = G\frac{m_1 m_2}{r^2}$, adds the superposition principle and the two shell theorems that let an entire planet be treated as a point mass, and describes how Cavendish actually weighed $G$. From there everything follows: acceleration due to gravity, $g = \frac{G M_E}{R_E^2}$, which lets us weigh the Earth itself, and how $g$ varies with altitude, with depth, with latitude as the Earth spins, and with the planet's oblate shape. Then the energy picture — gravitational potential energy $U = -\frac{G M m}{r}$, why it is negative, the gravitational potential $V = -\frac{G M}{r}$ that belongs to the field alone, and the escape speed $v_e = \sqrt{2 g R_E} \approx 11.2$ km/s that follows from setting the total energy to zero. The chapter closes on satellites: orbital speed and period, the energy triple $K = -E$ and $U = 2E$ with the binding energy that would free a satellite, and the geostationary and polar orbits that carry our communications and weather instruments. Topics the rationalised syllabus trimmed — the latitude variation of $g$, geostationary and polar satellites, and weightlessness — are restored in full, because Boards, JEE Main and NEET ask about them every single year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.

Everything you have solved so far assumed rigid bodies. Nothing is rigid. This chapter measures how much real materials give, and how engineers turn that measurement into cables that hold and bridges that stand. It begins with elastic behaviour and the atomic picture of a solid as a lattice of tiny springs, then defines the two quantities that make deformation comparable across objects of any size — stress, the restoring force per unit area, and strain, the fractional change in dimension — in the three pairs that run through the whole subject: longitudinal, shearing and hydraulic. Hooke's law makes stress proportional to strain for small deformations, and the stress-strain curve shows exactly where that proportionality dies, marking the elastic limit, the plastic region, the ultimate tensile strength and fracture. Three moduli follow: Young's modulus $Y = \frac{FL}{A\,\Delta L}$ for stretching, the shear modulus $G$ for sliding and twisting, and the bulk modulus $B = -\frac{\Delta p}{\Delta V/V}$ for squeezing from every side, with compressibility as its reciprocal. Poisson's ratio links the sideways contraction to the lengthways stretch, and the relations $Y = 3B(1 - 2\sigma) = 2G(1 + \sigma)$ tie all four constants together. The elastic potential energy $U = \frac{1}{2}F\,\Delta L$ and the strain energy density $u = \frac{1}{2}\sigma\varepsilon$ account for the work stored, and the chapter closes on applications — sizing crane ropes, why an I-section girder is shaped as it is, the sag of a loaded beam, and why no mountain on Earth can be much taller than 10 km. Topics the rationalised syllabus trimmed — thermal stress, elastic hysteresis and fatigue, and the relations between the elastic constants — are restored in full, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.

A fluid is anything that cannot sustain a shearing stress — which is exactly the property Chapter 8 said solids possess — and that one sentence generates this entire chapter. It opens with pressure, a scalar despite being built from a force, and Pascal's law, which transmits an applied pressure undiminished through an enclosed fluid and so lets a car be lifted by a foot on a pedal. Pressure grows with depth as $P = P_a + \rho g h$, giving the hydrostatic paradox, the mercury barometer and the distinction between gauge and absolute pressure. Archimedes' principle then explains why anything floats: the upthrust equals the weight of fluid displaced, so a floating body sinks to exactly the fraction $\frac{\rho_{\text{body}}}{\rho_{\text{fluid}}}$ of its volume. Set the fluid moving and conservation of mass becomes the equation of continuity $A_1v_1 = A_2v_2$, while conservation of energy becomes Bernoulli's principle $P + \frac{1}{2}\rho v^2 + \rho g h =$ constant — from which follow the speed of efflux $v = \sqrt{2gh}$, the Venturi meter, the swerve of a spun cricket ball and the lift on an aircraft wing. Real fluids resist through viscosity, giving Stokes' law and the terminal velocity $v_t \propto r^2$ that keeps dust airborne and raindrops survivable, Poiseuille's law with its startling fourth power of the radius, and the Reynolds number that decides whether a flow is smooth or turbulent. The chapter closes on surface tension: surface energy, the angle of contact, the excess pressure $\frac{2S}{r}$ inside a drop against $\frac{4S}{r}$ inside a bubble, and the capillary rise $h = \frac{2S\cos\theta}{r\rho g}$ that carries water to the top of a tree. Topics the rationalised syllabus trimmed — Archimedes' principle and floatation, the Venturi meter, Poiseuille's law and the Reynolds number — are restored in full, because Boards, JEE Main and NEET ask about them every single year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.

Heat is not a substance a body contains — it is energy in transit, and it flows for exactly one reason: a difference in temperature. The chapter starts there, with the zeroth law of thermodynamics that makes temperature measurable at all, then builds the absolute scale from the ideal-gas thermometer and the triple point of water at $273.16$ K. Matter responds to heat first by growing: thermal expansion, with $\Delta L = \alpha L\,\Delta T$ and the ratio $\alpha : \beta : \gamma = 1 : 2 : 3$ that ties linear, areal and volume expansion together — the physics behind expansion joints, shrink fits, bimetallic thermostats, and the anomalous behaviour of water that lets a lake freeze from the top down and its fish survive. Then it absorbs: specific heat capacity $\Delta Q = ms\,\Delta T$, water's exceptionally large value and the climate that follows from it, and calorimetry, where heat lost equals heat gained. At a change of state the temperature stops rising altogether and the energy goes into latent heat instead, which is why steam scalds far worse than boiling water. The second half is transport. Conduction carries heat through a solid, $H = KA\frac{\Delta T}{L}$, and becomes almost easy once written as a thermal resistance $R = \frac{L}{KA}$ that adds in series and in parallel exactly like an electrical one. Convection carries it by moving the fluid itself, from a pan of water to the sea breeze and the monsoon. Radiation needs no medium at all: the blackbody spectrum, Wien's law $\lambda_m T = b$ that reads a star's temperature from its colour, the Stefan-Boltzmann law $H = \sigma A e T^4$ whose fourth power is unforgiving, and Kirchhoff's law, that a good absorber is a good emitter. The chapter closes on Newton's law of cooling — an approximation with a stated range, derived rather than asserted — and the greenhouse effect that keeps the planet about 33°C warmer than it would otherwise be. Topics the rationalised syllabus trimmed — areal expansion and the $1:2:3$ ratio, bimetallic strips, thermal resistance in series and parallel, Kirchhoff's law and Prevost's theory, and the greenhouse effect — are restored in full, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.

Chapter 10 measured how matter responds to heat. This chapter asks the harder question: what can you do with it? It opens by making the vocabulary precise — systems open, closed and isolated; state variables, extensive and intensive; and the zeroth law, which is what makes temperature a consistent property in the first place. Then the two ways energy crosses a boundary: heat and work, both path functions, against internal energy $U$, which is a state function — a distinction that decides more exam questions than any formula. The first law, $\Delta Q = \Delta U + \Delta W$, is conservation of energy with that distinction built in, and it forbids the perpetual motion machine of the first kind. For a gas it forces two specific heats rather than one, giving Mayer's relation $C_p - C_v = R$ and the ratio $\gamma = \frac{C_p}{C_v}$. The $P$-$V$ diagram then makes everything visual: the work done is the area under the curve, so it depends on the path, and a closed loop encloses the net work of a cycle. Four processes follow — isothermal ($\Delta U = 0$, $W = nRT\ln\frac{V_2}{V_1}$), adiabatic ($\Delta Q = 0$, $PV^\gamma$ constant, an adiabat always steeper than an isotherm), isobaric and isochoric (where no work is done at all). The second law then rules out what the first law allows: the Kelvin-Planck statement forbids a perfect engine, the Clausius statement forbids a perfect refrigerator, and between them they give time a direction. Heat engines convert heat to work with efficiency $\eta = 1 - \frac{Q_2}{Q_1}$; refrigerators and heat pumps run the same cycle backwards with a coefficient of performance that is routinely greater than one; and the Carnot engine sets the ceiling for all of them at $\eta = 1 - \frac{T_2}{T_1}$, a limit that depends on nothing but two temperatures. Carnot's theorem proves no engine can beat it, by an argument that couples a rival engine to a reversed Carnot engine and watches the pair break the second law. Topics the rationalised syllabus trimmed — the coefficient of performance, heat pumps, work as the area under a $P$-$V$ curve, and the values of $\gamma$ every adiabatic problem needs — are restored in full, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.

The previous chapter treated a gas as a black box obeying laws about pressure, volume and temperature. This chapter opens the box. Starting from a single idea — that a gas is an enormous number of tiny molecules in ceaseless random motion — it derives the gas laws instead of assuming them, and along the way explains what temperature actually is. It begins with the evidence that molecules exist at all, from Dalton's laws of proportion and Avogadro's hypothesis to Brownian motion seen under a microscope, and with the scale of things: molecules an angstrom across, spaced tens of angstroms apart in a gas, travelling thousands of angstroms between collisions. The ideal gas equation is then set out in all four of its forms — $PV = \mu RT$, $PV = Nk_BT$, $P = nk_BT$ and $P = \frac{\rho RT}{M_0}$ — with Boyle's law, Charles' law, the pressure law and Dalton's law of partial pressures falling out as special cases, and with an honest account of where real gases depart from the model and why. The heart of the chapter is the derivation of $$P = \frac{1}{3}nm\overline{v^2} = \frac{1}{3}\rho\,\overline{v^2}$$ from nothing but elastic collisions and Newton's laws, and its comparison with the gas equation, which yields the result the whole subject rests on: the average translational kinetic energy of a molecule is $\frac{3}{2}k_BT$, independent of pressure, volume and the nature of the gas. Temperature, it turns out, is molecular kinetic energy. From there come the root mean square speed $v_{rms} = \sqrt{\frac{3RT}{M_0}}$ and the full Maxwell distribution of molecular speeds, with the most probable and average speeds and the fixed ratio between all three. Counting degrees of freedom and applying the law of equipartition of energy — $\frac{1}{2}k_BT$ for every quadratic term, so a vibrational mode counts twice — then predicts the molar specific heats of gases from molecular structure alone, giving $C_v = \frac{f}{2}R$, $C_p - C_v = R$ and $\gamma = 1 + \frac{2}{f}$, and the Dulong-Petit value $3R$ for solids. The chapter closes with the mean free path $l = \frac{1}{\sqrt{2}n\pi d^2}$, which explains why a gas whose molecules move faster than sound still takes minutes to diffuse across a room, and with Graham's law for the rates at which different gases do it. Topics the rationalised syllabus trimmed — the Maxwell speed distribution, the formulas for the most probable and average speeds, Brownian motion, the pressure law and Graham's law — are restored in full, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.

Almost everything in nature that is stable repeats: a pendulum, a plucked string, an atom in a crystal, the current in a radio circuit, the pressure in your ear as a sound arrives. This chapter is about the simplest and by far the most important kind of repetition. It begins by separating periodic motion from oscillatory motion and fixing the four quantities that describe both — period $T$, frequency $\nu = \frac{1}{T}$, displacement measured from the mean position, and amplitude — together with the angular frequency $\omega = 2\pi\nu$, whose confusion with $\nu$ costs more marks in this chapter than any other single slip. Then simple harmonic motion, defined by $$x(t) = A\cos(\omega t + \phi)$$ and viewed a second way as the shadow of uniform circular motion on a diameter, the reference circle that turns hard timing questions into easy geometry. Differentiating twice gives $v = -\omega A\sin(\omega t+\phi)$ and $a = -\omega^2 x$ — the last being the real definition of SHM, since any motion whose acceleration is proportional to displacement and directed back towards the centre is simple harmonic. Multiplying by the mass turns that into the force law $F = -kx$, so every linear restoring force gives $\omega = \sqrt{\frac{k}{m}}$ and $T = 2\pi\sqrt{\frac{m}{k}}$ — a period independent of amplitude, and independent of $g$ even for a vertical spring, provided displacement is measured from the new equilibrium. Spring combinations follow, parallel adding stiffness and series adding compliance. The energy is $E = \frac{1}{2}kA^2$, constant in total but sloshing between kinetic and potential at twice the frequency of the motion, in a parabolic well that is the reason harmonic motion appears everywhere in physics. The simple pendulum gives $T = 2\pi\sqrt{\frac{L}{g}}$ under the small-angle approximation, with the second's pendulum and the lift, car and liquid variants worked through. Finally the two topics that describe real oscillators rather than ideal ones: damped oscillations, where a resistive force $-bv$ makes the amplitude decay as $e^{-bt/2m}$ and the energy twice as fast, and forced oscillations and resonance, where a driven system responds most violently when the driving frequency matches its natural one — the effect behind a tuned radio, a pumped swing, a shattered wine glass and a collapsed bridge. Topics the rationalised syllabus trimmed — damped oscillations, forced oscillations and resonance, the series spring combination and equivalent-stiffness rules, and the second's pendulum with the effective-gravity variants — are restored in full, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.

The previous chapter studied a single particle oscillating in place. This one lets the oscillation travel. A wave carries energy and momentum from one point to another without carrying the medium with it — the cork bobs, the ripple moves on — and that one sentence explains everything from a plucked string to a radio signal. The chapter separates transverse waves, where the medium oscillates across the direction of travel, from longitudinal waves, where it oscillates along it, and explains why sound in air must be the second kind. The travelling wave is then written down once and used everywhere: $$y(x,t) = a\sin(kx - \omega t + \phi)$$ with $a$ the amplitude, $k = \frac{2\pi}{\lambda}$ the angular wave number, $\omega = 2\pi\nu$ the angular frequency, and the sign between $kx$ and $\omega t$ deciding which way the wave goes. Holding the phase constant gives the wave speed $v = \nu\lambda = \frac{\omega}{k}$, a speed set by the medium and not by the source — so carrying a note into water changes its wavelength but never its frequency. Two speeds are then derived: $v = \sqrt{\frac{T}{\mu}}$ for a transverse wave on a stretched string, and $v = \sqrt{\frac{\gamma P}{\rho}}$ for sound in a gas, where Newton's formula assumed isothermal compressions and came out 15% low until the Laplace correction made them adiabatic. The principle of superposition — displacements add algebraically and the waves then carry on unchanged — produces interference, and, once a wave meets a boundary and reflects, the standing wave $y = 2a\sin kx\cos\omega t$, which does not travel at all. Boundary conditions then quantise everything: a string fixed at both ends supports $\nu_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$ and all harmonics; an open pipe supports all of them too; and a closed pipe supports only the odd ones, so its first overtone is its third harmonic — the most reliable trap in the chapter. Two close frequencies give beats at $\lvert\nu_1-\nu_2\rvert$, and relative motion between source and observer gives the Doppler effect, whose whole difficulty is a sign convention. Topics the rationalised syllabus trimmed — the Doppler effect in full, with the reflecting-surface and moving-medium cases and the sonic boom, and the overtone naming that exam questions turn on — are restored here, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.