Class 11 Physics Formulas

Cheatsheet Content

### Units and Measurements - **Fundamental Units (SI):** - Length: meter (m) - Mass: kilogram (kg) - Time: second (s) - Electric Current: ampere (A) - Temperature: kelvin (K) - Luminous Intensity: candela (cd) - Amount of Substance: mole (mol) - **Dimensional Analysis:** Checks consistency of equations. - **Significant Figures:** Rules for precision in measurements. - **Error Analysis:** - Absolute Error: $\Delta A = |A_{mean} - A_i|$ - Relative Error: $\frac{\Delta A_{mean}}{A_{mean}}$ - Percentage Error: $\frac{\Delta A_{mean}}{A_{mean}} \times 100\%$ ### Kinematics - **Displacement:** $\vec{s}$ (vector) - **Velocity:** $\vec{v} = \frac{d\vec{s}}{dt}$ - **Acceleration:** $\vec{a} = \frac{d\vec{v}}{dt}$ - **Equations of Motion (Constant Acceleration):** - $v = u + at$ - $s = ut + \frac{1}{2}at^2$ - $v^2 = u^2 + 2as$ - $s_n = u + \frac{a}{2}(2n-1)$ (Displacement in nth second) - **Projectile Motion:** - Horizontal Range: $R = \frac{u^2 \sin(2\theta)}{g}$ - Maximum Height: $H = \frac{u^2 \sin^2\theta}{2g}$ - Time of Flight: $T = \frac{2u \sin\theta}{g}$ ### Laws of Motion - **Newton's First Law:** Inertia. - **Newton's Second Law:** $\vec{F} = m\vec{a}$ - **Newton's Third Law:** To every action, there is an equal and opposite reaction. - **Impulse:** $\vec{J} = \vec{F}_{avg} \Delta t = \Delta \vec{p}$ - **Momentum:** $\vec{p} = m\vec{v}$ - **Conservation of Momentum:** In an isolated system, $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$ - **Friction:** - Static Friction: $f_s \le \mu_s N$ - Kinetic Friction: $f_k = \mu_k N$ - **Centripetal Force:** $F_c = \frac{mv^2}{r} = m\omega^2 r$ ### Work, Energy and Power - **Work Done:** $W = \vec{F} \cdot \vec{s} = Fs \cos\theta$ - **Kinetic Energy:** $KE = \frac{1}{2}mv^2$ - **Potential Energy (Gravitational):** $PE = mgh$ - **Work-Energy Theorem:** $W_{net} = \Delta KE$ - **Power:** $P = \frac{W}{t} = \vec{F} \cdot \vec{v}$ - **Conservation of Mechanical Energy:** $KE_i + PE_i = KE_f + PE_f$ (for conservative forces) - **Elastic Collision (1D):** - $v_1' = \frac{(m_1-m_2)v_1 + 2m_2v_2}{m_1+m_2}$ - $v_2' = \frac{2m_1v_1 + (m_2-m_1)v_2}{m_1+m_2}$ ### System of Particles and Rotational Motion - **Centre of Mass:** - For two particles: $x_{CM} = \frac{m_1x_1 + m_2x_2}{m_1+m_2}$ - For n particles: $x_{CM} = \frac{\sum m_ix_i}{\sum m_i}$ - **Rotational Kinematics (Constant Angular Acceleration):** - $\omega = \omega_0 + \alpha t$ - $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha\theta$ - **Relationship between Linear and Angular:** - $v = r\omega$ - $a_t = r\alpha$ (tangential acceleration) - $a_c = r\omega^2 = v^2/r$ (centripetal acceleration) - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ - **Moment of Inertia:** $I = \sum m_i r_i^2$ - **Parallel Axis Theorem:** $I = I_{CM} + Md^2$ - **Perpendicular Axis Theorem (for planar bodies):** $I_z = I_x + I_y$ - **Angular Momentum:** $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ - **Conservation of Angular Momentum:** If $\vec{\tau}_{net} = 0$, then $\vec{L} = \text{constant}$ - **Rotational Kinetic Energy:** $KE_{rot} = \frac{1}{2}I\omega^2$ ### Gravitation - **Newton's Law of Gravitation:** $F = G \frac{m_1 m_2}{r^2}$ - **Acceleration due to gravity:** $g = G \frac{M}{R^2}$ - **Variation of g with altitude:** $g' = g (1 - \frac{2h}{R})$ (for $h \ll R$) - **Variation of g with depth:** $g' = g (1 - \frac{d}{R})$ - **Gravitational Potential Energy:** $U = -G \frac{M m}{r}$ - **Gravitational Potential:** $V = -G \frac{M}{r}$ - **Escape Velocity:** $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$ - **Orbital Velocity:** $v_o = \sqrt{\frac{GM}{r}}$ - **Time Period of Satellite:** $T = 2\pi \sqrt{\frac{r^3}{GM}}$ - **Kepler's Laws:** 1. Orbits are ellipses. 2. Equal areas in equal times. 3. $T^2 \propto r^3$ ### Properties of Bulk Matter - **Elasticity:** - Stress: $\sigma = \frac{F}{A}$ - Strain: $\epsilon = \frac{\Delta L}{L}$ (longitudinal), $\frac{\Delta V}{V}$ (volume), $\phi$ (shear) - Young's Modulus: $Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{FL}{A\Delta L}$ - Bulk Modulus: $B = \frac{\text{Normal Stress}}{\text{Volume Strain}} = \frac{-P}{\Delta V/V}$ - Shear Modulus: $G = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{\phi}$ - **Fluid Mechanics:** - Density: $\rho = \frac{m}{V}$ - Pressure: $P = \frac{F}{A}$ - Pascal's Law: Pressure applied to an enclosed fluid is transmitted undiminished. - Hydrostatic Pressure: $P = \rho g h$ - Archimedes' Principle: Buoyant force $F_B = \rho_{fluid} V_{submerged} g$ - Equation of Continuity: $A_1v_1 = A_2v_2$ - Bernoulli's Principle: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ - **Surface Tension:** - $S = \frac{F}{L}$ (force per unit length) - Work done in increasing surface area: $W = S \Delta A$ - Excess pressure in a liquid drop: $\Delta P = \frac{2S}{R}$ - Excess pressure in a soap bubble: $\Delta P = \frac{4S}{R}$ - Capillary Rise: $h = \frac{2S \cos\theta}{\rho g r}$ - **Viscosity:** - Stokes' Law: $F_v = 6\pi \eta r v$ - Poiseuille's Formula: $Q = \frac{\pi PR^4}{8\eta L}$ - Reynolds Number: $Re = \frac{\rho v D}{\eta}$ ### Thermal Properties of Matter - **Heat Capacity:** $C = \frac{\Delta Q}{\Delta T}$ - **Specific Heat Capacity:** $c = \frac{C}{m}$ - **Latent Heat:** $Q = mL$ - **Thermal Expansion:** - Linear: $\Delta L = L_0 \alpha \Delta T$ - Area: $\Delta A = A_0 \beta \Delta T$ where $\beta = 2\alpha$ - Volume: $\Delta V = V_0 \gamma \Delta T$ where $\gamma = 3\alpha$ - **Heat Transfer:** - Conduction: $\frac{dQ}{dt} = -KA \frac{dT}{dx}$ - Convection: Involves mass transfer. - Radiation (Stefan-Boltzmann Law): $P = \epsilon \sigma A T^4$ - Newton's Law of Cooling: $\frac{dT}{dt} = -k(T - T_s)$ ### Thermodynamics - **First Law of Thermodynamics:** $\Delta U = Q - W$ - **Work Done by Gas:** $W = \int P dV$ - **Specific Heats of Gases:** - $C_P - C_V = R$ (Mayer's relation) - Ratio of specific heats: $\gamma = \frac{C_P}{C_V}$ - **Isothermal Process:** $PV = \text{constant}$, $W = nRT \ln(\frac{V_f}{V_i})$ - **Adiabatic Process:** $PV^\gamma = \text{constant}$, $T V^{\gamma-1} = \text{constant}$ - $W = \frac{nR(T_i - T_f)}{\gamma - 1}$ - **Isobaric Process:** $P = \text{constant}$, $W = P\Delta V$ - **Isochoric Process:** $V = \text{constant}$, $W = 0$ ### Kinetic Theory - **Ideal Gas Equation:** $PV = nRT = NkT$ - **Average Kinetic Energy of a Gas Molecule:** $KE_{avg} = \frac{3}{2}kT$ - **Root Mean Square Speed:** $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ - **Degrees of Freedom (f):** - Monatomic: $f=3$ - Diatomic: $f=5$ (at moderate temp) - Polyatomic: $f=6$ (at moderate temp) - **Law of Equipartition of Energy:** Energy per degree of freedom = $\frac{1}{2}kT$ - **Internal Energy:** $U = \frac{f}{2}nRT$ ### Oscillations - **Simple Harmonic Motion (SHM):** - Displacement: $x(t) = A \sin(\omega t + \phi)$ - Velocity: $v(t) = A\omega \cos(\omega t + \phi)$ - Acceleration: $a(t) = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (for spring-mass system) - **Time Period:** $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$ - **Time Period of Simple Pendulum:** $T = 2\pi \sqrt{\frac{L}{g}}$ - **Kinetic Energy:** $KE = \frac{1}{2}m\omega^2 (A^2 - x^2)$ - **Potential Energy:** $PE = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 x^2$ - **Total Energy:** $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$ ### Waves - **Wave Speed:** $v = f\lambda$ - **Equation of a Plane Progressive Wave:** $y(x,t) = A \sin(kx - \omega t + \phi)$ - Wave number: $k = \frac{2\pi}{\lambda}$ - Angular frequency: $\omega = 2\pi f$ - **Speed of Transverse Wave on a String:** $v = \sqrt{\frac{T}{\mu}}$ (Tension T, mass per unit length $\mu$) - **Speed of Longitudinal Wave (Sound):** - In a fluid: $v = \sqrt{\frac{B}{\rho}}$ (Bulk Modulus B, density $\rho$) - In a solid rod: $v = \sqrt{\frac{Y}{\rho}}$ (Young's Modulus Y, density $\rho$) - **Doppler Effect:** $f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$ - Observer moving towards source (+ $v_o$) - Source moving towards observer (- $v_s$) - **Standing Waves:** - **Open Organ Pipe:** $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1,2,3,...$) - **Closed Organ Pipe:** $\lambda_n = \frac{4L}{n}$, $f_n = \frac{nv}{4L}$ ($n=1,3,5,...$) - **String Fixed at Both Ends:** $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1,2,3,...$) - **Beats:** $f_{beat} = |f_1 - f_2|$