Physics Essentials

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1. Force, Work, Power & Energy (i) Turning Forces & Equilibrium Moment of a Force (Torque): Turning effect of a force about an axis. $M = F \times d$ (Force $\times$ perpendicular distance from pivot to line of action of force) Units: Newton-meter (N m) (SI), dyne-cm (cgs) Clockwise moments are typically negative, anti-clockwise positive. Equilibrium: Translational Equilibrium: Net force is zero ($\Sigma F = 0$). Rotational Equilibrium: Net moment (torque) is zero ($\Sigma M = 0$). Principle of Moments: For a body in rotational equilibrium, sum of clockwise moments = sum of anti-clockwise moments. Centre of Gravity (CG): Point where the entire weight of an object appears to act. (ii) Uniform Circular Motion Constant speed, but changing direction implies acceleration (centripetal acceleration). Centripetal Force ($F_c$): Force directed towards the center of the circular path, causing the object to move in a circle. $F_c = \frac{mv^2}{r}$. Centrifugal Force: Fictitious force experienced by an object in a rotating frame of reference, directed away from the center. (iii) Work, Energy & Power Work ($W$): Energy transferred when a force causes displacement. $W = F \cdot s \cos\theta$ (where $\theta$ is angle between force and displacement) If $\theta = 0^\circ$, $W = F \cdot s$ (force in direction of displacement). If $\theta = 90^\circ$, $W = 0$ (force perpendicular to displacement). Gravitational Work: $W = mgh$ (for vertical displacement). Units: Joule (J) (SI), erg (cgs). $1 \text{ J} = 10^7 \text{ erg}$. Energy: Capacity to do work. Units: Joule (J), erg, calorie ($1 \text{ cal} \approx 4.18 \text{ J}$), kilowatt-hour (kW h) ($1 \text{ kW h} = 3.6 \times 10^6 \text{ J}$), electron-volt (eV) ($1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$). Power ($P$): Rate at which work is done or energy is transferred. $P = \frac{W}{t} = F \cdot v$ Units: Watt (W) (SI). $1 \text{ W} = 1 \text{ J/s}$. Other units: kilowatt (kW), megawatt (MW), gigawatt (GW). Horsepower: $1 \text{ hp} = 746 \text{ W}$. (iv) Types of Energy Mechanical Energy: Potential Energy (PE or $U$): Stored energy due to position or state. Gravitational PE: $U = mgh$ (mass $\times$ gravity $\times$ height). Kinetic Energy (KE or $K$): Energy due to motion. Translational KE: $K = \frac{1}{2}mv^2$ (mass $\times$ velocity squared). Rotational, Vibrational KE (qualitative understanding). Other forms: Chemical, Heat, Electrical, Nuclear, Sound, Light. Energy Conversion: Energy can be converted from one form to another (e.g., chemical to electrical in a battery). (v) Machines Purpose: Force multipliers, change direction of force, or change speed/distance. Terms: Effort ($E$): Force applied to the machine. Load ($L$): Force overcome by the machine. Mechanical Advantage (MA): $MA = \frac{L}{E}$. Velocity Ratio (VR): $VR = \frac{d_E}{d_L}$ (distance moved by effort / distance moved by load). Efficiency ($\eta$): Ratio of useful work output to total work input. $\eta = \frac{\text{Work Output}}{\text{Work Input}} = \frac{L \cdot d_L}{E \cdot d_E} = \frac{MA}{VR}$. For ideal machine, $\eta = 1$ (or 100%), $MA = VR$. For practical machines, $\eta Levers: Principle: Moment of load = Moment of effort (for equilibrium). Class I: Fulcrum between effort and load (e.g., see-saw, crowbar). MA can be $>1$, $ Class II: Load between fulcrum and effort (e.g., wheelbarrow, nutcracker). MA always $>1$. Class III: Effort between fulcrum and load (e.g., fishing rod, tweezers). MA always $ Pulley Systems: Single Fixed Pulley: Changes direction of effort. $MA=1, VR=1$. Single Movable Pulley: Force multiplier. $MA \approx 2, VR=2$. Block and Tackle: Multiple pulleys, higher MA & VR. $VR = \text{number of strands supporting the movable block}$. (vi) Principle of Conservation of Energy Statement: Energy can neither be created nor destroyed, but can be transformed from one form to another. The total energy of an isolated system remains constant. Freely Falling Body: $U + K = \text{constant}$ (assuming no air resistance). At max height: $K=0, U=mgh_{max}$. Total Energy $= mgh_{max}$. At ground: $U=0, K=\frac{1}{2}mv_{max}^2$. Total Energy $=\frac{1}{2}mv_{max}^2$. Therefore, $mgh_{max} = \frac{1}{2}mv_{max}^2$. Simple Pendulum: Continuous conversion between PE and KE. 2. Light (i) Refraction of Light Definition: Bending of light as it passes from one medium to another due to change in speed. Laws of Refraction: Incident ray, refracted ray, and normal at point of incidence lie in the same plane. Snell's Law: $\frac{\sin i}{\sin r} = \text{constant} = \mu$ (refractive index). Effect on Light Properties: Speed ($V$) and Wavelength ($\lambda$): Change. $V = f\lambda$. Frequency ($f$): Remains constant. Refractive Index ($\mu$): Ratio of speed of light in vacuum ($c$) to speed of light in medium ($V$). $\mu = \frac{c}{V}$. $\mu_{12} = \frac{\mu_2}{\mu_1} = \frac{V_1}{V_2}$. Conditions for undeviated ray: $i=0^\circ$ (normal incidence) or $\mu_1 = \mu_2$. Applications: Real and Apparent Depth: $\mu = \frac{\text{Real Depth}}{\text{Apparent Depth}}$. Objects in denser medium appear shallower. Refraction through Glass Block: Produces lateral displacement. Refraction through Prism: Light bends towards the base. (ii) Total Internal Reflection (TIR) Conditions for TIR: Light travels from a denser medium to a rarer medium. Angle of incidence ($i$) in the denser medium is greater than the critical angle ($C$). Critical Angle ($C$): Angle of incidence in the denser medium for which the angle of refraction in the rarer medium is $90^\circ$. $\mu = \frac{1}{\sin C}$. Applications: Optical fibers, periscopes, binoculars, right-angle prisms (used to deviate light by $90^\circ$ or $180^\circ$). Prism vs. Plane Mirror: TIR from prism is more efficient (no absorption) than reflection from a silvered mirror. (iii) Lenses Types: Converging (Convex): Thicker in middle, converges parallel rays. Diverging (Concave): Thinner in middle, diverges parallel rays. Technical Terms: Optical centre, principal axis, principal foci ($F_1, F_2$), focal length ($f$), centre of curvature ($2F_1, 2F_2$). Ray Diagrams: Use principal rays to locate images. Ray parallel to principal axis passes through (or appears to come from) focal point after refraction. Ray passing through (or directed towards) optical centre goes undeviated. Ray passing through (or directed towards) focal point becomes parallel to principal axis after refraction. Lens Formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ (Sign convention important). Magnification ($M$): $M = \frac{\text{Image Height}}{\text{Object Height}} = \frac{v}{u}$. Power of a Lens ($P$): Reciprocal of focal length in meters. $P = \frac{1}{f}$. Unit: Dioptre (D). Convex lens: $f$ is positive, $P$ is positive. Concave lens: $f$ is negative, $P$ is negative. Magnifying Glass (Simple Microscope): Convex lens used to produce a magnified, virtual, erect image when object is placed between $F$ and $O$. (iv) Spectrum & Scattering Dispersion: Splitting of white light into its constituent colors (spectrum) when passing through a prism due to different refractive indices for different wavelengths (colors). Violet deviates most, Red least. Electromagnetic Spectrum: Arrangement of electromagnetic waves in order of increasing wavelength (decreasing frequency/energy). Radio waves, Microwaves, Infrared, Visible light (ROYGBIV), Ultraviolet, X-rays, Gamma rays. All EM waves travel at speed $c$ in vacuum, are transverse, and do not require a medium. Infrared: Heat radiation, remote controls. Ultraviolet: Sterilization, tanning, causes sunburn. Scattering of Light: Redirection of light by particles in the medium. Shorter wavelengths (blue/violet) are scattered more effectively than longer wavelengths (red). Explains blue color of sky, reddish sunrise/sunset. 3. Sound (i) Reflection of Sound (Echoes) Echo: Repetition of sound due to reflection from a surface. Condition for Echo: Minimum distance from source to reflector for distinct echo is $\approx 17 \text{ m}$ (for speed of sound $340 \text{ m/s}$ and persistence of hearing $0.1 \text{ s}$). Uses: SONAR (Sound Navigation And Ranging), medical imaging (ultrasound), bats, dolphins for navigation. (ii) Vibrations Natural Vibrations: Oscillations of a body at its own characteristic frequency when disturbed and left free. Damped Vibrations: Vibrations whose amplitude decreases over time due to energy loss (e.g., friction). Forced Vibrations: A body vibrates at the frequency of an external periodic force. Resonance: Special case of forced vibrations where the driving frequency matches the natural frequency of the body, leading to a large increase in amplitude. (iii) Characteristics of Sound Loudness: Subjective perception of intensity. Depends on amplitude of vibration. Intensity: Objective measure of sound energy passing per unit area per unit time. Unit: $\text{W/m}^2$. Sound Level: Measured in decibels (dB). Noise Pollution: Undesirable sound levels. Pitch: Subjective perception of frequency. Higher frequency = higher pitch. Quality (Timbre): Distinguishes sounds of same loudness and pitch from different sources. Depends on waveform (presence and relative intensity of overtones). 4. Electricity and Magnetism (i) Ohm's Law & Resistance Electric Current ($I$): Rate of flow of charge. $I = \frac{Q}{t}$. Unit: Ampere (A). Potential Difference (pd or $V$): Work done per unit charge in moving a charge between two points. $V = \frac{W}{Q}$. Unit: Volt (V). Electromotive Force (emf): Energy supplied by source per unit charge to move charge around a complete circuit. Ohm's Law: $V = IR$. (For ohmic resistors, $V \propto I$ at constant temperature). Resistance ($R$): Opposition to current flow. Unit: Ohm ($\Omega$). Factors affecting R: Length ($L$), cross-sectional area ($A$), material (resistivity $\rho$), temperature. $R = \rho \frac{L}{A}$. Ohmic Resistors: Obey Ohm's law (linear $V-I$ graph). Non-Ohmic Resistors: Do not obey Ohm's law (e.g., diode, filament bulb). Internal Resistance ($r$): Resistance offered by the cell/battery itself. $V = \text{emf} - Ir$. Superconductors: Materials with zero resistance below a critical temperature. Resistors in Series: $R_{eq} = R_1 + R_2 + R_3 + \dots$ (Current same, voltage divides). Resistors in Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$ (Voltage same, current divides). (ii) Electrical Power & Energy Electrical Energy ($W$): Energy consumed by an electrical appliance. $W = Q V = V I t$ Using Ohm's Law: $W = I^2 R t = \frac{V^2}{R} t$ Unit: Joule (J). Commercial unit: kilowatt-hour (kW h). Electrical Power ($P$): Rate of energy consumption. $P = \frac{W}{t} = VI$ Using Ohm's Law: $P = I^2 R = \frac{V^2}{R}$ Unit: Watt (W). Power Rating: Specifies voltage and power consumption (e.g., $220 \text{ V}, 100 \text{ W}$). (iii) Household Circuits Main Circuit: Live (red/brown), Neutral (black/blue), Earth (green/yellow). Safety Devices: Fuse/MCB (Miniature Circuit Breaker): Protects against overcurrent/short circuits. Fuse wire melts, MCB trips. Earthing: Connects metal casing of appliance to earth to prevent electric shock if live wire touches casing. Switches: Break or make circuits. Two-way switches for staircase wiring. Three-pin Plug: Live, Neutral, Earth. Earth pin is longer and thicker. (iv) Magnetic Effect of Current & EM Induction Oersted's Experiment: Electric current produces a magnetic field. Magnetic Field due to Current: Straight Wire: Concentric circles around wire. Direction by Right-Hand Thumb Rule. Loop/Solenoid: Similar to a bar magnet. Electromagnets: Temporary magnets created by current in a coil. Uses: Cranes, relays. Fleming's Left-Hand Rule (Motor Rule): For force on a current-carrying conductor in a magnetic field. (Forefinger = Field, Centre finger = Current, Thumb = Motion/Force). DC Electric Motor: Converts electrical energy to mechanical energy. Basic parts: Coil, magnets, split-ring commutator, brushes. Electromagnetic Induction (EMI): Production of induced emf/current when magnetic flux linked with a coil changes. Fleming's Right-Hand Rule (Generator Rule): For direction of induced current. (Forefinger = Field, Centre finger = Induced Current, Thumb = Motion/Force). AC Generator: Converts mechanical energy to electrical energy. Basic parts: Coil, magnets, slip rings, brushes. Produces alternating current. Transformer: Changes AC voltage levels. Works on EMI. Step-up Transformer: Increases voltage, decreases current. More turns in secondary coil. Step-down Transformer: Decreases voltage, increases current. Fewer turns in secondary coil. Power in primary $\approx$ Power in secondary (ideal transformer). $\frac{V_P}{V_S} = \frac{N_P}{N_S} = \frac{I_S}{I_P}$. 5. Heat (i) Calorimetry & Specific Heat Capacity Heat: Form of energy transferred due to temperature difference. Units: Joule (J), calorie (cal). Temperature: Measure of degree of hotness or coldness. Units: Celsius ($^\circ \text{C}$), Kelvin (K). Thermal (Heat) Capacity ($C'$): Amount of heat required to raise the temperature of an entire body by $1^\circ \text{C}$ or $1 \text{ K}$. $C' = \frac{Q}{\Delta T}$. Unit: $\text{J/K}$. Specific Heat Capacity ($C$): Amount of heat required to raise the temperature of unit mass of a substance by $1^\circ \text{C}$ or $1 \text{ K}$. $C = \frac{Q}{m \Delta T}$. Unit: $\text{J kg}^{-1} \text{K}^{-1}$. Relation: $C' = mC$. Water has high specific heat capacity ($4200 \text{ J kg}^{-1} \text{K}^{-1}$), used as coolant. Principle of Method of Mixtures: For an isolated system, Heat Lost by Hot Body = Heat Gained by Cold Body. $m_1 C_1 (T_1 - T_f) = m_2 C_2 (T_f - T_2)$. (ii) Latent Heat Change of Phase: Transition between solid, liquid, gas states. Occurs at constant temperature. Latent Heat: Heat absorbed or released during a phase change without change in temperature. Specific Latent Heat of Fusion ($L_f$): Heat required to change unit mass of a substance from solid to liquid at its melting point. $Q = mL_f$. Unit: $\text{J/kg}$. Heating Curve: Shows temperature vs. heat supplied. Plateaus indicate phase changes. Applications: Ice used for cooling (absorbs $L_f$), steam burns (releases $L_v$). 6. Modern Physics (i) Radioactivity Nucleus: Composed of protons (atomic number $Z$) and neutrons ($A-Z$). Mass number $A = Z + N$. Radioactivity: Spontaneous disintegration of unstable atomic nuclei, emitting radiation. Types of Radiations: Alpha ($\alpha$): Helium nucleus ($^4_2\text{He}$). Positively charged, low penetrating power, high ionizing power. Mass number decreases by 4, atomic number by 2. Beta ($\beta$): Fast-moving electron ($^0_{-1}e$). Negatively charged, medium penetrating power, medium ionizing power. Mass number unchanged, atomic number increases by 1. Gamma ($\gamma$): High-energy electromagnetic radiation. No charge, very high penetrating power, low ionizing power. No change in $A$ or $Z$. Uses of Radioisotopes: Medical tracers, carbon dating, industrial gauges. Harmful Effects: Ionizing radiation causes cell damage, mutations. Safety Precautions: Lead shielding, remote handling, proper disposal, minimum exposure time, maximum distance. Background Radiation: Natural radiation from cosmic rays, rocks, soil, food. (ii) Nuclear Fission & Fusion Nuclear Fission: Splitting of a heavy nucleus into two or more lighter nuclei, releasing a large amount of energy. E.g., $^1_0n + ^{235}_{92}\text{U} \to ^{141}_{56}\text{Ba} + ^{92}_{36}\text{Kr} + 3^1_0n + \text{Energy}$. Used in nuclear power plants and atomic bombs. Nuclear Fusion: Combining of two light nuclei to form a heavier nucleus, releasing an even larger amount of energy. E.g., $^2_1\text{H} + ^3_1\text{H} \to ^4_2\text{He} + ^1_0n + \text{Energy}$. Powers the sun and hydrogen bombs. SI Units Fundamental Units Quantity Unit Name Symbol Mass kilogram kg Length metre m Time second s Electric current ampere A Temperature kelvin K Luminous intensity candela cd Amount of substance mole mol Derived Units (Examples) Quantity Unit Name Symbol Derived From Volume cubic metre $\text{m}^3$ Length cubed Density kilogram per cubic metre $\text{kg m}^{-3}$ Mass/Volume Velocity metre per second $\text{m s}^{-1}$ Length/Time Acceleration metre per second squared $\text{m s}^{-2}$ Velocity/Time Momentum kilogram metre per second $\text{kg m s}^{-1}$ Mass $\times$ Velocity Force newton N $\text{kg m s}^{-2}$ Pressure pascal Pa $\text{N m}^{-2}$ Energy, Work joule J $\text{N m}$ Power watt W $\text{J s}^{-1}$ Frequency hertz Hz $\text{s}^{-1}$ Electric charge coulomb C $\text{A s}$ Electric resistance ohm $\Omega$ $\text{V A}^{-1}$ Electromotive force volt V $\text{J C}^{-1}$ Standard Prefixes Multiple Prefix Symbol $10^9$ giga G $10^6$ mega M $10^3$ kilo k $10^{-1}$ deci d $10^{-2}$ centi c $10^{-3}$ milli m $10^{-6}$ micro $\mu$ $10^{-9}$ nano n $10^{-12}$ pico p $10^{-15}$ femto f