Physics 1st Paper (Class 11-12

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### Physical World & Measurement - **Science & Physics:** - **Science:** Systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. - **Physics:** The natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. - **Physical Quantities:** - **Fundamental Quantities:** Independent quantities that cannot be expressed in terms of other physical quantities (e.g., Length, Mass, Time). - **Derived Quantities:** Quantities that can be expressed in terms of fundamental quantities (e.g., Velocity, Force, Energy). - **Units of Measurement:** - **SI Units (International System of Units):** - 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) - **Supplementary Units:** Radian (rad) for plane angle, Steradian (sr) for solid angle. - **Dimensions and Dimensional Analysis:** - **Dimension:** The nature of a physical quantity (e.g., Length [L], Mass [M], Time [T]). - **Dimensional Formula:** An expression showing how and which fundamental units are involved in a physical quantity (e.g., Velocity $[LT^{-1}]$, Force $[MLT^{-2}]$). - **Principle of Homogeneity of Dimensions:** Only quantities of the same dimensions can be added, subtracted, or equated. Useful for checking formula correctness and deriving relationships. - **Measurement Errors:** - **Absolute Error ($\Delta x$):** The magnitude of the difference between the true value and the measured value. - **Relative Error:** $\frac{\text{Absolute Error}}{\text{True Value or Mean Value}}$ - **Percentage Error:** Relative Error $\times 100\%$ - **Types of Errors:** - **Systematic Errors:** Errors due to faulty instruments, imperfect experimental techniques, or personal bias. Can be minimized. - **Random Errors:** Errors due to unpredictable fluctuations in experimental conditions or personal judgment. Cannot be eliminated but can be minimized by taking many readings. - **Gross Errors:** Errors due to carelessness of the observer. - **Significant Figures:** Digits in a measurement that carry meaning contributing to its precision. Rules for counting and operations. ### Vectors - **Scalar vs. Vector Quantities:** - **Scalar:** Has only magnitude (e.g., mass, speed, time, distance, temperature). - **Vector:** Has both magnitude and direction (e.g., displacement, velocity, acceleration, force, momentum). - **Representation of Vectors:** Geometrically by directed line segments (arrows), analytically by components. - **Types of Vectors:** - **Unit Vector ($\hat{A}$):** A vector with magnitude 1, used to specify direction. $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$. - **Null Vector (Zero Vector):** A vector with zero magnitude and arbitrary direction. - **Equal Vectors:** Two vectors are equal if they have the same magnitude and direction. - **Negative Vector:** A vector with the same magnitude but opposite direction. - **Collinear Vectors:** Vectors acting along the same line (parallel or anti-parallel). - **Coplanar Vectors:** Vectors lying in the same plane. - **Position Vector ($\vec{r}$):** A vector from the origin to a point. - **Vector Addition:** - **Triangle Law:** If two vectors are represented by two sides of a triangle taken in order, their resultant is represented by the third side taken in opposite order. - **Parallelogram Law:** If two vectors acting simultaneously at a point are represented by the two adjacent sides of a parallelogram, their resultant is represented by the diagonal passing through that point. - Magnitude of Resultant: $R = \sqrt{A^2 + B^2 + 2AB\cos\alpha}$ (where $\alpha$ is the angle between $\vec{A}$ and $\vec{B}$) - Direction: $\tan\theta = \frac{B\sin\alpha}{A+B\cos\alpha}$ (angle of resultant with $\vec{A}$) - **Polygon Law:** For more than two vectors. - **Vector Subtraction:** $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ - **Resolution of Vectors:** Breaking a vector into components (e.g., $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$). - **Vector Multiplication:** - **Scalar (Dot) Product:** $\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$ - Properties: Commutative, distributive. If $\vec{A} \cdot \vec{B} = 0$, then $\vec{A} \perp \vec{B}$ (if non-zero vectors). - **Vector (Cross) Product:** $\vec{A} \times \vec{B} = (AB\sin\theta)\hat{n}$ (where $\hat{n}$ is a unit vector perpendicular to both $\vec{A}$ and $\vec{B}$ following right-hand rule). - In components: $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$ - Properties: Not commutative ($\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$), distributive. If $\vec{A} \times \vec{B} = 0$, then $\vec{A} \parallel \vec{B}$ (if non-zero vectors). - Magnitude of $\vec{A} \times \vec{B}$ represents the area of the parallelogram formed by $\vec{A}$ and $\vec{B}$. - **Relative Velocity:** $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ (velocity of A relative to B). ### Dynamics - **Kinematics (Study of Motion without considering its causes):** - **Distance:** Total path length covered (scalar). - **Displacement ($\vec{s}$):** Shortest distance between initial and final position (vector). - **Speed:** Rate of change of distance (scalar). - **Velocity ($\vec{v}$):** Rate of change of displacement (vector). - Average Velocity: $\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t}$ - Instantaneous Velocity: $\vec{v} = \frac{d\vec{r}}{dt}$ - **Acceleration ($\vec{a}$):** Rate of change of velocity (vector). - Average Acceleration: $\vec{a}_{avg} = \frac{\Delta\vec{v}}{\Delta t}$ - Instantaneous Acceleration: $\vec{a} = \frac{d\vec{v}}{dt}$ - **Equations of Motion (for constant acceleration):** - $v = u + at$ - $s = ut + \frac{1}{2}at^2$ - $v^2 = u^2 + 2as$ - $s_n = u + \frac{a}{2}(2n-1)$ (distance covered in the n-th second) - **Free Fall:** Motion under gravity only. $a = g$ (downwards). - **Projectile Motion:** Motion of an object thrown into space and subjected to gravity only. - **Horizontal motion:** Constant velocity ($v_x = u\cos\theta$). - **Vertical motion:** Constant acceleration ($a_y = -g$). - **Trajectory Equation:** $y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}$ - **Maximum Height ($H$):** $H = \frac{u^2\sin^2\theta}{2g}$ - **Horizontal Range ($R$):** $R = \frac{u^2\sin(2\theta)}{g}$. Maximum range at $\theta = 45^\circ$. - **Time of Flight ($T$):** $T = \frac{2u\sin\theta}{g}$ - **Circular Motion:** - **Angular Displacement ($\Delta\theta$):** Angle swept by the radius vector. - **Angular Velocity ($\omega$):** $\omega = \frac{d\theta}{dt}$. Relation $v = r\omega$. - **Angular Acceleration ($\alpha$):** $\alpha = \frac{d\omega}{dt}$. Relation $a_t = r\alpha$ (tangential acceleration). - **Centripetal Acceleration ($a_c$):** $a_c = \frac{v^2}{r} = r\omega^2$. Always directed towards the center. - **Total Acceleration:** $\vec{a} = \vec{a}_t + \vec{a}_c$. Magnitude $a = \sqrt{a_t^2 + a_c^2}$. ### Newtonian Mechanics - **Force:** An external agent that changes or tends to change the state of rest or uniform motion of a body. - **Inertia:** The tendency of a body to resist changes in its state of motion. - **Momentum ($\vec{p}$):** Product of mass and velocity. $\vec{p} = m\vec{v}$. - **Newton's Laws of Motion:** - **First Law (Law of Inertia):** A body remains in its state of rest or uniform motion in a straight line unless acted upon by an external unbalanced force. - **Second Law:** The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force. $\vec{F} = \frac{d\vec{p}}{dt}$. If mass is constant, $\vec{F} = m\vec{a}$. - **Third Law:** To every action, there is always an equal and opposite reaction. (Forces occur in pairs, act on different bodies). - **Impulse ($\vec{J}$):** Change in momentum. $\vec{J} = \int \vec{F} dt = \Delta\vec{p}$. - **Conservation of Linear Momentum:** In the absence of an external force, the total linear momentum of an isolated system remains constant. $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$. - **Friction:** A force that opposes relative motion between surfaces in contact. - **Static Friction ($F_s$):** Force that opposes the initiation of motion. $F_s \le \mu_s N$ (where $\mu_s$ is coefficient of static friction, $N$ is normal force). - **Kinetic (Dynamic) Friction ($F_k$):** Force that opposes motion once it has started. $F_k = \mu_k N$ (where $\mu_k$ is coefficient of kinetic friction). - Rolling friction is generally less than sliding friction. - **Centripetal Force ($F_c$):** The force required to keep an object moving in a circular path. It is always directed towards the center of the circle. - $F_c = \frac{mv^2}{r} = m\omega^2 r$ - **Banking of Roads:** Tilting of a road to provide the necessary centripetal force for vehicles to take turns safely. - $\tan\theta = \frac{v^2}{rg}$ - **Centrifugal Force:** An apparent force experienced by an observer in a rotating frame of reference, directed away from the center of rotation. It's a pseudo force. ### Work, Power & Energy - **Work ($W$):** Done when a force causes a displacement. - $W = \vec{F} \cdot \vec{s} = Fs\cos\theta$ - Units: Joule (J). - Work done by a variable force: $W = \int \vec{F} \cdot d\vec{s}$ - Work done by a spring: $W = \frac{1}{2}kx^2$ (where k is spring constant, x is displacement). - **Energy ($E$):** The capacity to do work. - **Kinetic Energy ($E_k$):** Energy due to motion. $E_k = \frac{1}{2}mv^2$. - **Potential Energy ($E_p$):** Energy due to position or configuration. - Gravitational Potential Energy: $E_p = mgh$ - Elastic Potential Energy (Spring): $E_p = \frac{1}{2}kx^2$ - **Work-Energy Theorem:** The net work done on an object is equal to the change in its kinetic energy. $W_{net} = \Delta E_k$. - **Power ($P$):** The rate at which work is done or energy is transferred. - $P = \frac{W}{t} = \frac{dE}{dt}$ - Also, $P = \vec{F} \cdot \vec{v} = Fv\cos\theta$. - Units: Watt (W). - **Conservation of Energy:** Energy can neither be created nor destroyed, but it can be transformed from one form to another. For an isolated system, total mechanical energy ($E_k + E_p$) remains constant in the absence of non-conservative forces. - **Conservative and Non-Conservative Forces:** - **Conservative Forces:** Work done by these forces depends only on initial and final positions, not on the path taken (e.g., gravitational, elastic). - **Non-Conservative Forces:** Work done by these forces depends on the path taken (e.g., friction, air resistance). - **Efficiency ($\eta$):** The ratio of useful output energy (or power) to total input energy (or power). - $\eta = \frac{\text{Energy Output}}{\text{Energy Input}} \times 100\% = \frac{\text{Power Output}}{\text{Power Input}} \times 100\%$ ### Gravitation & Gravity - **Newton's Law of Universal Gravitation:** Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. - $F = G\frac{m_1m_2}{r^2}$ - **Gravitational Constant ($G$):** $G = 6.673 \times 10^{-11} \text{ N m}^2/\text{kg}^2$. - **Acceleration due to Gravity ($g$):** The acceleration experienced by an object due to Earth's gravitational pull. - $g = G\frac{M_e}{R_e^2}$ (where $M_e$ is Earth's mass, $R_e$ is Earth's radius). - **Variation of $g$:** - **With Altitude ($h$):** $g_h = g\left(\frac{R_e}{R_e+h}\right)^2 \approx g\left(1 - \frac{2h}{R_e}\right)$ (for $h \ll R_e$) - **With Depth ($d$):** $g_d = g\left(1 - \frac{d}{R_e}\right)$ - **With Latitude:** $g$ increases from equator to poles due to Earth's rotation and shape. - **Gravitational Potential Energy:** $U = -\frac{GMm}{r}$ (for two masses $M$ and $m$ separated by $r$). - **Gravitational Potential:** $V = -\frac{GM}{r}$ (potential energy per unit mass). - **Escape Velocity ($v_e$):** The minimum velocity required for an object to escape the gravitational field of a planet. - $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$ - **Satellites:** Objects orbiting a planet. - **Orbital Velocity ($v_o$):** Velocity required to maintain a stable orbit. $v_o = \sqrt{\frac{GM}{r}}$ (where $r$ is orbital radius). - **Time Period ($T$):** Time taken for one complete orbit. $T = \frac{2\pi r}{v_o} = 2\pi\sqrt{\frac{r^3}{GM}}$ - **Geostationary Satellites:** Orbit Earth with a period of 24 hours, appearing stationary relative to a point on Earth. - **Kepler's Laws of Planetary Motion:** - **First Law (Law of Orbits):** All planets move in elliptical orbits with the Sun at one focus. - **Second Law (Law of Areas):** The radius vector drawn from the Sun to a planet sweeps out equal areas in equal intervals of time. (Implies conservation of angular momentum). - **Third Law (Law of Periods):** The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. $T^2 \propto r^3$. ### Elasticity - **Deforming Force:** A force that changes the size or shape of a body. - **Restoring Force:** An internal force that tends to bring the body back to its original size and shape. - **Elasticity:** The property of a body by virtue of which it tends to regain its original size and shape after the removal of the deforming force. - **Plasticity:** The property of a body by virtue of which it does not regain its original size and shape after the removal of the deforming force. - **Stress ($P$ or $\sigma$):** Restoring force per unit area. - $P = \frac{F}{A}$ - Units: N/m$^2$ or Pascal (Pa). - **Types of Stress:** Longitudinal (Tensile/Compressive), Volume (Hydraulic), Shearing (Tangential). - **Strain ($\epsilon$):** The fractional change in dimension produced by the deforming force. - $\epsilon = \frac{\Delta L}{L}$ (Longitudinal strain) - $\epsilon_v = \frac{\Delta V}{V}$ (Volume strain) - $\epsilon_s = \tan\phi \approx \phi$ (Shearing strain, angle of shear) - Unitless and dimensionless. - **Hooke's Law:** Within the elastic limit, stress is directly proportional to strain. - Stress = Modulus of Elasticity $\times$ Strain - **Moduli of Elasticity:** - **Young's Modulus ($Y$):** Ratio of longitudinal stress to longitudinal strain. - $Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{F/A}{\Delta L/L}$ - **Bulk Modulus ($K$):** Ratio of volume stress to volume strain. - $K = \frac{\text{Volume Stress}}{\text{Volume Strain}} = \frac{F/A}{\Delta V/V}$ - **Shear Modulus (Modulus of Rigidity, $\eta$):** Ratio of shearing stress to shearing strain. - $\eta = \frac{\text{Shearing Stress}}{\text{Shearing Strain}} = \frac{F/A}{\phi}$ - **Poisson's Ratio ($\sigma$):** For a homogeneous isotropic material, the ratio of lateral strain to longitudinal strain. - $\sigma = -\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = -\frac{\Delta D/D}{\Delta L/L}$ (The negative sign indicates that if length increases, diameter decreases). - Theoretical limits: $-1 ### Fluid Mechanics - **Fluids:** Substances that can flow (liquids and gases). - **Density ($\rho$):** Mass per unit volume. $\rho = \frac{m}{V}$. - **Relative Density (Specific Gravity):** Ratio of the density of a substance to the density of water at $4^\circ C$. - **Pressure ($P$):** Force per unit area. $P = \frac{F}{A}$. - Units: Pascal (Pa) = N/m$^2$. Other units: atm, bar, torr. - **Pressure in a Fluid at Rest:** $P = P_0 + h\rho g$ (where $P_0$ is atmospheric pressure, $h$ is depth). - **Pascal's Principle:** Pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel. (Basis of hydraulic systems). - **Buoyancy (Upthrust):** The upward force exerted by a fluid on an immersed object. - **Archimedes' Principle:** When a body is wholly or partially immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by it. - Buoyant Force ($F_B$) = $V_{disp}\rho_{fluid}g$. - **Fluid Dynamics (Fluids in Motion):** - **Ideal Fluid:** Incompressible (constant density) and non-viscous (no internal friction). - **Types of Flow:** - **Streamline (Laminar) Flow:** Fluid particles follow smooth paths, and the velocity at any point remains constant over time. - **Turbulent Flow:** Irregular and chaotic fluid motion with eddies and vortices. - **Equation of Continuity:** For an incompressible fluid in streamline flow, the product of the cross-sectional area and the fluid speed is constant along a streamline. $A_1v_1 = A_2v_2 = \text{constant}$. (Implies conservation of mass). - **Bernoulli's Principle (or Equation):** For an ideal fluid in streamline flow, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline. - $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ - **Viscosity:** The internal friction within a fluid that opposes its flow. - **Coefficient of Viscosity ($\eta$):** Defined from Newton's Law of Viscosity ($F = -\eta A \frac{dv}{dy}$). Units: Poise (P) or Pa.s. - **Stokes' Law:** The viscous drag force experienced by a small spherical body falling through a viscous fluid. $F_v = 6\pi\eta rv$. - **Terminal Velocity:** Constant velocity attained by a body falling through a viscous fluid when the gravitational force is balanced by the buoyant force and viscous drag. - **Surface Tension ($T$):** The property of a liquid surface at rest to behave like a stretched elastic membrane. It is the force per unit length acting perpendicular to a line drawn on the liquid surface. - $T = \frac{F}{L}$ - Units: N/m. - **Surface Energy:** Energy per unit area of the liquid surface. Numerically equal to surface tension. - **Angle of Contact:** The angle between the tangent to the liquid surface and the solid surface inside the liquid. Determines whether a liquid wets a surface or not. - **Capillarity (Capillary Action):** The phenomenon of rise or fall of a liquid in a narrow tube (capillary). - Capillary Rise/Fall: $h = \frac{2T\cos\theta}{r\rho g}$ (where $\theta$ is angle of contact, $r$ is capillary radius). ### Oscillations - **Periodic Motion:** Motion that repeats itself after a regular interval of time. - **Oscillatory Motion:** A type of periodic motion where a body moves back and forth about a mean position. - **Simple Harmonic Motion (SHM):** A special type of oscillatory motion where the restoring force is directly proportional to the displacement from the mean position and acts opposite to the displacement. - **Condition:** $F = -kx$ or $a = -\omega^2 x$. - **Displacement Equation:** $x(t) = A\sin(\omega t + \phi)$ or $x(t) = A\cos(\omega t + \phi)$ - $A$: Amplitude (maximum displacement). - $\omega$: Angular frequency ($\omega = \sqrt{k/m}$). - $\phi$: Phase constant (initial phase). - **Velocity in SHM:** $v(t) = \frac{dx}{dt} = A\omega\cos(\omega t + \phi) = \pm \omega\sqrt{A^2 - x^2}$ - Maximum velocity: $v_{max} = A\omega$ (at mean position $x=0$). - **Acceleration in SHM:** $a(t) = \frac{dv}{dt} = -A\omega^2\sin(\omega t + \phi) = -\omega^2 x$ - Maximum acceleration: $a_{max} = A\omega^2$ (at extreme positions $x=\pm A$). - **Time Period ($T$):** Time taken for one complete oscillation. $T = \frac{2\pi}{\omega}$. - **Frequency ($f$):** Number of oscillations per unit time. $f = \frac{1}{T} = \frac{\omega}{2\pi}$. - **Energy in SHM:** - **Kinetic Energy ($E_k$):** $E_k = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2(A^2 - x^2)$. - **Potential Energy ($E_p$):** $E_p = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 x^2$. - **Total Energy ($E$):** $E = E_k + E_p = \frac{1}{2}m\omega^2 A^2 = \frac{1}{2}kA^2$. (Constant). - **Simple Pendulum:** An ideal simple pendulum consists of a point mass suspended by a massless, inextensible string in a frictionless environment. - **Time Period:** $T = 2\pi\sqrt{\frac{L}{g}}$ (for small angles of oscillation). - **Compound Pendulum:** A rigid body capable of oscillating freely about a horizontal axis passing through it. - **Time Period:** $T = 2\pi\sqrt{\frac{I}{mgL}}$ (where $I$ is moment of inertia about pivot, $L$ is distance from pivot to center of mass). - **Damped Oscillations:** Oscillations where the amplitude decreases over time due to dissipative forces (e.g., air resistance, friction). - **Forced Oscillations and Resonance:** - **Forced Oscillations:** An oscillator driven by an external periodic force. - **Resonance:** Occurs when the frequency of the external driving force matches the natural frequency of the oscillating system, leading to a large amplitude of oscillation. ### Waves - **Wave:** A disturbance that travels through a medium, transferring energy without transferring matter. - **Types of Waves:** - **Mechanical Waves:** Require a medium for propagation (e.g., sound waves, water waves). - **Electromagnetic Waves:** Do not require a medium (e.g., light, radio waves, X-rays). - **Transverse Waves:** Particles of the medium oscillate perpendicular to the direction of wave propagation (e.g., light waves, waves on a string). - **Longitudinal Waves:** Particles of the medium oscillate parallel to the direction of wave propagation (e.g., sound waves). - **Wave Characteristics:** - **Amplitude ($A$):** Maximum displacement of particles from their mean position. - **Wavelength ($\lambda$):** Distance between two consecutive crests or troughs (or compressions/rarefactions). - **Frequency ($f$):** Number of waves passing a point per second. - **Time Period ($T$):** Time taken for one complete wave to pass a point. $T = 1/f$. - **Wave Velocity ($v$):** Speed at which the wave propagates. $v = f\lambda = \frac{\omega}{k}$. - **Wave Number ($k$):** $k = \frac{2\pi}{\lambda}$. - **Angular Frequency ($\omega$):** $\omega = 2\pi f = \frac{2\pi}{T}$. - **Wave Equation (General form for a progressive wave moving in +x direction):** - $y(x,t) = A\sin(kx - \omega t + \phi)$ or $y(x,t) = A\cos(kx - \omega t + \phi)$ - **Speed of Sound:** - **In Solids:** $v = \sqrt{\frac{Y}{\rho}}$ (for longitudinal waves, where Y is Young's Modulus). - **In Liquids:** $v = \sqrt{\frac{K}{\rho}}$ (where K is Bulk Modulus). - **In Gases (Newton's Formula):** $v = \sqrt{\frac{P}{\rho}}$ (isothermal process, incorrect). - **In Gases (Laplace's Correction):** $v = \sqrt{\frac{\gamma P}{\rho}}$ (adiabatic process, where $\gamma = C_p/C_v$). - Speed of sound increases with temperature. - **Intensity of Sound:** Power per unit area. $I = \frac{P}{A}$. Proportional to $A^2f^2$. - **Reflection of Waves:** A wave bounces back after hitting a boundary. - From a denser medium (fixed end): Phase change of $\pi$ (180 degrees). - From a rarer medium (free end): No phase change. - **Refraction of Waves:** A wave changes direction as it passes from one medium to another. - **Superposition Principle:** When two or more waves overlap, the resultant displacement at any point is the vector sum of the displacements due to individual waves. - **Interference:** Phenomenon of two or more waves superposing to form a resultant wave of greater, lower, or the same amplitude. - **Constructive Interference:** Waves are in phase, amplitudes add up. - **Destructive Interference:** Waves are out of phase, amplitudes subtract. - **Beats:** Periodic variations in the intensity of sound heard when two sound sources of slightly different frequencies are sounded together. - Beat frequency = $|f_1 - f_2|$. - **Standing Waves (Stationary Waves):** Formed when two identical progressive waves traveling in opposite directions superpose. - **Nodes:** Points of zero displacement. - **Antinodes:** Points of maximum displacement. - **In Strings Fixed at Both Ends:** Fundamental frequency $f_1 = \frac{v}{2L}$. Harmonics $f_n = n f_1$. - **In Open Organ Pipes:** Fundamental frequency $f_1 = \frac{v}{2L}$. Harmonics $f_n = n f_1$. - **In Closed Organ Pipes:** Fundamental frequency $f_1 = \frac{v}{4L}$. Only odd harmonics $f_n = (2n-1)f_1$. - **Doppler Effect:** The apparent change in the frequency of a wave due to the relative motion between the source and the observer. - **General Formula:** $f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right)$ - $f'$: Apparent frequency. - $f$: Original frequency. - $v$: Speed of sound in the medium. - $v_o$: Speed of observer. - $v_s$: Speed of source. - **Sign Convention:** - Numerator ($v \pm v_o$): Use '+' if observer moves towards source, '-' if away. - Denominator ($v \mp v_s$): Use '-' if source moves towards observer, '+' if away.