ABHYAS · CLASS 11 AND 12 STUDENTS (CBSE AND STATE BOARDS) AND JEE/NEET BEGINNERS PRACTISING PHYSICS NUMERICALS

Class 11–12 Physics कक्षा 11–12 भौतिकी

Unlimited Class 11–12 physics numericals with hints and worked steps: projectiles, inclined planes, circular motion, work–energy, rotational energy, gravitation and orbits, SHM, ideal gases and thermodynamics, waves, Coulomb’s law, capacitors, circuits, magnetic force, electromagnetic induction, lenses and prisms, photons and de Broglie waves.

Motion in a plane

Projectile motion: time, height and rangeFor a projectile launched at speed u and angle θ (no air resistance): T = 2u sinθ/g, H = u² sin²θ/(2g), R = u² sin2θ/g. Horizontal launch from height h: t = √(2h/g), range = u·t.

Laws of motion

Newton's laws on a frictionless inclineOn a smooth incline of angle θ: acceleration down the slope a = g sinθ, normal force N = mg cosθ. Speed after sliding a length L from rest: v = √(2gL sinθ).

Motion in a plane / laws of motion

Uniform circular motionCentripetal acceleration a = v²/r = ω²r, force F = mv²/r. On a flat curve, friction provides it: v_max = √(μrg).

Work, energy and power

Work–energy theorem and spring energyNet work done = change in kinetic energy: W = ½mv² − ½mu². Work by a force at angle θ: W = Fs cosθ. Spring potential energy U = ½kx².

System of particles and rotational motion

Moment of inertia and rotational kinetic energyRotational KE = ½Iω². Ring: I = MR²; disc: ½MR²; solid sphere: (2/5)MR²; hollow sphere: (2/3)MR². Rolling without slipping: KE = ½Mv²(1 + I/MR²).

Gravitation

Gravitation: g with height, orbital and escape speedg at height h: g′ = g R²/(R + h)². Orbital speed at radius r: v = √(GM/r) = √(gR²/r). Escape speed ve = √(2gR). Use R(Earth) = 6400 km, g = 9.8 m/s².

Oscillations

Simple harmonic motion: spring and pendulumSpring–mass: T = 2π√(m/k). Simple pendulum (small swings): T = 2π√(L/g). Frequency f = 1/T.

Kinetic theory / thermal properties

Ideal gas equation (pV = nRT)pV = nRT with R = 8.314 J mol⁻¹ K⁻¹; use SI units (Pa, m³, K). 1 L = 10⁻³ m³; T(K) = t(°C) + 273.

Thermodynamics

First law of thermodynamics and engine efficiencyΔU = Q − W (Q: heat supplied to the gas, W: work done by the gas). Isobaric work W = pΔV. Carnot efficiency η = 1 − T_cold/T_hot (kelvin).

Waves

Waves: strings and air columnsSpeed on a string v = √(T/μ). Fundamental of a string fixed at both ends or an open pipe: f = v/2L; closed pipe: f = v/4L.

Electric charges and fields

Coulomb's lawF = k q₁q₂ / r² with k = 9 × 10⁹ N m² C⁻². 1 μC = 10⁻⁶ C. Like charges repel, unlike attract.

Electrostatic potential and capacitance

Capacitors: combinations and stored energyParallel: C = C₁ + C₂. Series: 1/C = 1/C₁ + 1/C₂. Charge Q = CV; energy U = ½CV².

Current electricity

Current electricity: emf, internal resistance, Wheatstone bridgeI = E/(R + r); terminal voltage V = E − Ir. Wheatstone bridge is balanced when P/Q = R/S (no current through the galvanometer).

Moving charges and magnetism

Magnetic force on a moving charge and a wireForce on a charge: F = qvB sinθ. On a wire: F = BIL sinθ. A charged particle moving perpendicular to B goes in a circle of radius r = mv/(qB).

Electromagnetic induction

Electromagnetic induction: induced emfFaraday's law: ε = −N dΦ/dt with Φ = BA cosθ. Motional emf of a rod: ε = Blv. Self-induction: ε = −L dI/dt.

Ray optics

Lens maker's formula and prismLens maker's formula: 1/f = (n − 1)(1/R₁ − 1/R₂). For an equiconvex lens (R₁ = R, R₂ = −R): f = R/[2(n − 1)]. Prism at minimum deviation: n = sin((A + δm)/2) / sin(A/2).

Dual nature of radiation and matter

Photons, photoelectric effect and de Broglie wavelengthPhoton energy E = hc/λ ≈ 1240/λ(nm) eV. Photoelectric: K_max = E − φ. Electron accelerated through V volts: λ ≈ 1.227/√V nm = 1227/√V pm.