}]]}]}]}] 30:T1fd2,

### Mechanics #### 1. Kinematics - **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 n-th second) - **Projectile Motion:** - Time of Flight: $T = \frac{2u \sin\theta}{g}$ - Max Height: $H = \frac{u^2 \sin^2\theta}{2g}$ - Range: $R = \frac{u^2 \sin(2\theta)}{g}$ #### 2. Newton's Laws of Motion - **First Law:** Inertia - **Second Law:** $F = ma$ - **Third Law:** Action-Reaction - **Impulse:** $J = F \Delta t = \Delta p$ - **Conservation of Momentum:** $\sum p_{initial} = \sum p_{final}$ #### 3. Work, Energy & Power - **Work Done:** $W = F \cdot s = Fs \cos\theta$ - **Kinetic Energy:** $KE = \frac{1}{2}mv^2$ - **Potential Energy (Gravity):** $PE = mgh$ - **Work-Energy Theorem:** $W_{net} = \Delta KE$ - **Power:** $P = \frac{W}{t} = F \cdot v$ #### 4. Rotational Motion - **Angular Displacement:** $\theta$ - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ - **Torque:** $\tau = r \times F = I\alpha$ - **Moment of Inertia:** $I = \sum mr^2$ - **Rotational KE:** $KE_{rot} = \frac{1}{2}I\omega^2$ - **Angular Momentum:** $L = I\omega = r \times p$ #### 5. Gravitation - **Newton's Law of Gravitation:** $F = \frac{Gm_1m_2}{r^2}$ - **Gravitational Potential Energy:** $U = -\frac{Gm_1m_2}{r}$ - **Escape Velocity:** $v_e = \sqrt{\frac{2GM}{R}}$ - **Orbital Velocity:** $v_o = \sqrt{\frac{GM}{r}}$ ### Thermodynamics #### 1. Heat & Temperature - **Heat Transfer:** - Conduction: $\frac{dQ}{dt} = -KA\frac{dT}{dx}$ - Convection - Radiation: $P = \epsilon \sigma A T^4$ (Stefan-Boltzmann Law) - **Specific Heat Capacity:** $Q = mc\Delta T$ - **Latent Heat:** $Q = mL$ #### 2. Laws of Thermodynamics - **Zeroth Law:** Thermal Equilibrium - **First Law:** $\Delta U = Q - W$ - **Second Law:** Entropy always increases for an isolated system. - Efficiency of Heat Engine: $\eta = 1 - \frac{T_C}{T_H}$ - **Third Law:** Entropy of a perfect crystal at absolute zero is zero. #### 3. Kinetic Theory of Gases - **Ideal Gas Equation:** $PV = nRT = Nk_BT$ - **Average KE per molecule:** $KE_{avg} = \frac{3}{2}k_BT$ - **RMS Speed:** $v_{rms} = \sqrt{\frac{3RT}{M}}$ - **Degrees of Freedom (f):** - Monatomic: f=3 - Diatomic: f=5 (at normal temp) - Polyatomic: f=6 - **Internal Energy:** $U = \frac{f}{2}nRT$ ### Waves and Optics #### 1. Wave Motion - **Wave Equation:** $y(x,t) = A \sin(kx - \omega t + \phi)$ - **Wave Speed:** $v = f\lambda = \frac{\omega}{k}$ - **Speed of Sound:** $v = \sqrt{\frac{B}{\rho}}$ (Solids/Liquids), $v = \sqrt{\frac{\gamma P}{\rho}}$ (Gases) - **Standing Waves:** - String (fixed ends): $L = \frac{n\lambda}{2}$, $f_n = \frac{nv}{2L}$ - Open Organ Pipe: $L = \frac{n\lambda}{2}$, $f_n = \frac{nv}{2L}$ - Closed Organ Pipe: $L = \frac{(2n-1)\lambda}{4}$, $f_n = \frac{(2n-1)v}{4L}$ - **Doppler Effect:** $f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$ #### 2. Ray Optics - **Reflection:** Angle of incidence = Angle of reflection - **Refraction (Snell's Law):** $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - **Lens/Mirror Formula:** $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ - **Magnification:** $m = -\frac{v}{u} = \frac{h_i}{h_o}$ - **Power of Lens:** $P = \frac{1}{f}$ (in diopters, f in meters) - **Total Internal Reflection:** $\sin C = \frac{n_2}{n_1}$ #### 3. Wave Optics - **Young's Double Slit Experiment (YDSE):** - Path Difference: $\Delta x = d \sin\theta$ - Bright Fringes (Constructive): $\Delta x = n\lambda$ - Dark Fringes (Destructive): $\Delta x = (n + \frac{1}{2})\lambda$ - Fringe Width: $\beta = \frac{\lambda D}{d}$ - **Diffra