Important Definitions & Formulas

Measurement: The process of comparing an unknown physical quantity with a known standard quantity of the same nature.

Unit: The internationally accepted basic reference standard used for measurement.

Characteristics: 1) Well-defined. 2) Suitable size. 3) Easily reproducible. 4) Indestructible and unchanging with time/temperature. 5) Internationally accepted.

Fundamental: Physical quantities completely independent of each other. Ex: Mass (kg, $[M]$), Length (m, $[L]$).

Derived: Physical quantities derived from fundamental quantities using math operations. Ex: Velocity (m/s, $[LT^{-1}]$), Force (N, $[MLT^{-2}]$).

System of Units: A complete set of fundamental and derived units (e.g., SI, CGS).

7 Fundamental: Length (m), Mass (kg), Time (s), Electric Current (A), Temperature (K), Luminous Intensity (cd), Amount of Substance (mol).

2 Supplementary: Plane Angle (radian, rad), Solid Angle (steradian, sr).

Astronomical Unit (AU): Average distance between Earth and Sun. $1 \text{ AU} = 1.496 \times 10^{11} \text{ m}$.

Light Year (ly): Distance light travels in a vacuum in one year. $1 \text{ ly} = 9.46 \times 10^{15} \text{ m}$.

Parsec: Distance at which an arc of 1 AU subtends an angle of 1 arc-second. $1 \text{ Parsec} = 3.08 \times 10^{16} \text{ m} = 3.26 \text{ ly}$.

Solar Day: Time taken by Earth to complete one rotation on its axis w.r.t the Sun.

Lunar Month: Time taken by the moon to complete one revolution around Earth (~27.3 days).

Shake: Smallest practical unit of time in nuclear physics. $1 \text{ shake} = 10^{-8} \text{ s}$.

Accuracy: How close a measured value is to the true or accepted correct value.

Precision: The resolution or limit to which the quantity is measured (how close multiple measurements are to each other, regardless of being correct).

Error: Uncertainty in a measurement.

Systematic: Errors that tend to be in one direction (positive or negative) due to known causes (e.g., zero error). Can be eliminated.

Gross: Errors caused by human carelessness (e.g., misreading a scale).

Least Count: Error associated with the resolution limit of the instrument. Minimized by using higher resolution tools.

Rest & Motion: An object is at rest if it doesn't change position w.r.t surroundings. It is in motion if it does. Both are relative terms.

Frame of Reference: A coordinate system (X,Y,Z) attached to an observer with a clock, used to describe the motion of a body.

Distance: Actual total path covered. Scalar. Always positive.

Displacement: Shortest straight-line distance from initial to final position. Vector. Can be +ve, -ve, or 0.

SI Unit: m. Dim: $[L]$.

Uniform: Body covers equal displacements in equal time intervals (e.g., car cruising at constant 60 km/h).

Non-Uniform: Body covers unequal displacements in equal time intervals (e.g., train braking at a station).

Avg Speed: Total distance / total time ($v = \frac{\Delta x}{\Delta t}$). Scalar.

Avg Velocity: Total displacement / total time ($\vec{v} = \frac{\Delta \vec{x}}{\Delta t}$). Vector.

SI Unit: m/s. Dim: $[LT^{-1}]$.

Velocity at a specific, exact microscopic instant of time. It is the first derivative of displacement w.r.t time.
Formula: $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{x}}{\Delta t} = \frac{d\vec{x}}{dt}$

Acceleration: Time rate of change of velocity. Vector. $\vec{a} = \frac{d\vec{v}}{dt}$. SI: m/s². Dim: $[LT^{-2}]$.

Average: Total change in velocity / total time ($\frac{\Delta \vec{v}}{\Delta t}$).

Instantaneous: Acceleration at an exact instant ($\frac{d\vec{v}}{dt}$).

1) $v = u + at$
2) $s = ut + \frac{1}{2}at^2$
3) $v^2 - u^2 = 2as$

Condition: Valid only when acceleration ($a$) remains perfectly constant (uniform).

Free Fall: Motion of a body purely under gravity's influence ($a = g \approx 9.8 \text{ m/s}^2$).

For a body dropped from rest ($u=0$):
1) $v = gt$
2) $h = \frac{1}{2}gt^2$
3) $v^2 = 2gh$

Scalar: Quantity with magnitude only. Obeys simple algebra. Ex: Mass, Work.

Vector: Quantity with both magnitude and direction. Obeys vector addition laws. Ex: Velocity, Force.

Position Vector ($\vec{r}$): Drawn from origin to particle's location. $\vec{r} = x\hat{i} + y\hat{j}$.

Displacement Vector ($\Delta\vec{r}$): Straight line from initial to final position. $\Delta\vec{r} = \vec{r}_2 - \vec{r}_1$.

Equal: Same magnitude, same direction. Negative: Same magnitude, opposite direction.

Parallel: Angle 0°. Collinear: Act along same/parallel lines.

Coplanar: Lying in the same 2D plane. Null: Zero magnitude, arbitrary direction.

A vector with magnitude exactly 1, used solely to denote direction. Denoted by a cap ($\hat{A}$).
Formula: $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$

Triangle Law: If two vectors are represented by two sides of a triangle in order, their resultant is the third side in opposite order.

Parallelogram Law: If two co-initial vectors form adjacent sides of a parallelogram, their resultant is the diagonal passing through that common point.

Splitting a single vector into two perpendicular components. For vector $\vec{A}$ at angle $\theta$ with x-axis:
$A_x = A \cos\theta$ and $A_y = A \sin\theta$.

Product of their magnitudes and the cosine of the angle between them. Result is a scalar.
Formula: $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta$.

Yields a third vector perpendicular to both. Magnitude is product of magnitudes and sine of angle.
Formula: $\vec{A} \times \vec{B} = (|\vec{A}| |\vec{B}| \sin\theta) \hat{n}$.

Projectile: Body thrown at an angle, moving in 2D solely under gravity. Trajectory is a parabola ($y = x \tan\theta - \frac{gx^2}{2u^2 \cos^2\theta}$).

Time of Flight: $T = \frac{2u \sin\theta}{g}$

Horizontal Range: $R = \frac{u^2 \sin(2\theta)}{g}$

Max Height: $H = \frac{u^2 \sin^2\theta}{2g}$

Motion along a circular path with constant speed. Yes, it is accelerated because direction changes continuously, changing velocity.

Angular Displacement ($\theta$): Angle swept by radius. Unit: Radian.

Angular Velocity ($\omega$): $d\theta/dt$. Unit: rad/s.

Angular Acceleration ($\alpha$): $d\omega/dt$. Unit: rad/s².

Inward acceleration in circular motion, directed towards the center. Changes direction of velocity.
Formula: $a_c = \frac{v^2}{r} = r\omega^2$. Unit: m/s².

Centripetal: Real, inward force keeping body in circle ($mv^2/r$).

Centrifugal: Fictitious, outward pseudo-force experienced in the rotating (non-inertial) frame.

Inertia: Inherent property resisting change in state. Measured by mass.

1. Of Rest: Falling backward when bus starts.

2. Of Motion: Falling forward when brakes applied.

3. Of Direction: Leaning sideways during a sharp turn.

A body continues in its state of rest or uniform motion in a straight line unless compelled by an external unbalanced force. Defines force/inertia.

Total quantity of motion. Product of mass and velocity. Vector.
Formula: $\vec{p} = m\vec{v}$. Unit: kg·m/s. Dim: $[MLT^{-1}]$.

Rate of change of momentum is directly proportional to applied force, taking place in force's direction. $\vec{F} = \frac{d\vec{p}}{dt} = m\vec{a}$.

Impulse: Huge force acting for tiny time. $J = F_{avg} \times \Delta t$.

Theorem: Total impulse equals total change in linear momentum ($\vec{J} = \Delta \vec{p}$).

If net external force is zero ($\vec{F}_{ext} = 0$), total linear momentum of system remains perfectly constant ($\vec{p}_{initial} = \vec{p}_{final}$).

To every action, there is an equal and opposite reaction ($\vec{F}_{AB} = -\vec{F}_{BA}$). Action and reaction act on different bodies simultaneously.

Inertial: Non-accelerated frame (at rest or constant velocity). Newton's laws hold.

Non-Inertial: Accelerated frame (like turning car). Newton's laws fail without pseudo forces.

Electromagnetic contact force exerted by a surface, acting strictly perpendicular (90°) to the surface, preventing object from sinking in.

Friction: Opposing tangential contact force.

Static: Prevents motion before it starts. Self-adjusting.

Limiting: Maximum static friction right before sliding.

Kinetic: Constant friction during actual sliding.

Rolling: Much smaller friction when a body rolls.

Angle of Friction ($\theta$): Angle between Normal Reaction and resultant of Limiting friction/Normal. $\tan\theta = \mu_s$.

Angle of Repose ($\alpha$): Min angle of inclined plane where block just starts sliding. $\theta = \alpha$.

Raising outer edge of curved road above inner edge. Provides centripetal force via horizontal component of Normal reaction, saving tires.
$v_{max} = \sqrt{rg \frac{\mu_s + \tan\theta}{1 - \mu_s \tan\theta}}$.

Fictitious mathematical force applied to objects observed from an accelerated (non-inertial) frame, directed perfectly opposite to frame's acceleration. $F_{pseudo} = -ma_{frame}$.

Work is done only when body is physically displaced by a force. Scalar product of force and displacement. $W = \vec{F} \cdot \vec{S} = FS \cos\theta$. Unit: Joule.

Positive ($\theta < 90^\circ$): Force aids motion (Gravity pulling apple).

Negative ($\theta > 90^\circ$): Force opposes motion (Friction).

Zero ($\theta = 90^\circ$): Force perpendicular to displacement (Centripetal force).

Energy: Capacity to do work. Joule.

Kinetic (KE): Energy due to motion. $K = \frac{1}{2}mv^2$.

Potential (PE): Energy due to position/configuration. Gravity: $mgh$. Spring: $\frac{1}{2}kx^2$.

Net work done by ALL forces on a body equals change in its Kinetic Energy. $W_{net} = \Delta K$.

Conservative: Work depends ONLY on endpoints, not path. Closed loop work is 0. (Gravity, electrostatic).

Non-Conservative: Work depends on path. Energy dissipates as heat. (Friction, air drag).

Mechanical: If only conservative forces act, KE + PE = constant.

General: Energy cannot be created or destroyed, only transformed.

Rate of doing work. $P = dW/dt = \vec{F} \cdot \vec{v}$. Unit: Watt (W). 1 HP = 746 W.

Head-on (1D): Bodies move along same straight line before and after.

Oblique (2D): Bodies scatter in different directions in a plane.

Elastic: KE is fully conserved. No heat loss.

Inelastic: KE is lost as heat/sound.

Perfectly Inelastic: Bodies permanently stick together. Max KE loss.

Ratio of velocity of separation to velocity of approach. $e = \frac{v_2 - v_1}{u_1 - u_2}$.
Elastic: e=1. Perfectly Inelastic: e=0.

An ideal body with a perfectly unchanging shape. Distance between any two particles remains absolutely constant despite external forces.

Translational: Every particle has identical velocity.

Rotational: Every particle moves in circles around a common straight axis.

Centre of Mass (COM): Point where entire mass is assumed concentrated for translational motion. Depends on mass distribution.

Centre of Gravity (COG): Point where total weight acts. Depends on gravitational field.

Turning effect of a force. Axial Vector. $\vec{\tau} = \vec{r} \times \vec{F}$. Unit: Nm.

Rotational equivalent of linear momentum. $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$.
Relation: $\vec{\tau} = \frac{d\vec{L}}{dt}$.

If net external torque is zero, total angular momentum is constant ($I_1\omega_1 = I_2\omega_2$). Ex: Ice skater spinning.

Property to resist change in rotational motion. $I = \sum m_i r_i^2$. Unit: kg·m².

Perpendicular distance from axis where, if whole mass were concentrated, MOI remains same. $K = \sqrt{I/M}$.

Perpendicular: For 2D lamina, $I_z = I_x + I_y$.

Parallel: For any body, $I = I_{cm} + Md^2$.

Simultaneous combination of pure translation of COM and pure rotation about COM. Contact point velocity is zero.

Orbits: Planets move in ellipses with Sun at one focus.

Areas: Line joining planet to Sun sweeps equal areas in equal times (Areal velocity constant).

Periods: $T^2 \propto a^3$ (Square of time period proportional to cube of semi-major axis).

Force is directly proportional to product of masses, inversely proportional to square of distance. $F = G\frac{m_1 m_2}{r^2}$. It is a central, conservative, attractive force.

Uniform acceleration of a freely falling body due to Earth's pull. $g = \frac{GM}{R^2}$.

Inertial: Resistance to acceleration ($m_i = F/a$).
Gravitational: Response to gravity ($m_g = F_g/g$). They are perfectly identical.

Altitude: Decreases. $g_h = g(1 - \frac{2h}{R})$.

Depth: Decreases. $g_d = g(1 - \frac{d}{R})$. Zero at center.

Shape/Rotation: Max at poles, min at equator due to equatorial bulge and centrifugal effect.

Intensity (E): Force per unit mass. $E = F/m$. (Vector)

Potential (V): Work done per unit mass from infinity. $V = -GM/r$. (Scalar)

Potential Energy (U): Total work done. $U = -GMm/r$. (Scalar)

Min speed to permanently escape planet's gravity. $v_e = \sqrt{2GM/R} = \sqrt{2gR}$. ~11.2 km/s for Earth.

Orbital Vel: $v_o = \sqrt{GM/r}$.

Time Period: $T = 2\pi\sqrt{r^3/GM}$.

KE: $+GMm/(2r)$. PE: $-GMm/r$. TE (Total): $-GMm/(2r)$. Negative indicates a bound system.

Geostationary: Orbits equatorial plane, T=24h. Used for telecom.

Polar: Low altitude, pole-to-pole orbit. Used for weather/spy.

Astronaut and satellite fall towards Earth with same acceleration (free fall). Normal reaction is zero, causing apparent weightlessness.

Answer: Plasticity: The property of a body by virtue of which it permanently retains its deformed shape and shows no tendency to regain its original shape after the deforming force is removed.

a) Perfectly Elastic: Completely and instantly regains its original shape/size (e.g., Quartz phosphor).

b) Inelastic (Plastic): Does not regain its shape at all (e.g., Putty, wet clay).

c) Partially Elastic: Partially regains its shape, leaving some permanent deformation (most real-world materials).

Answer: Stress: The internal restoring force developed per unit cross-sectional area of a deformed body to regain its original shape.

Formula: \( \sigma = \frac{F}{A} \)

SI Unit: \( N/m^2 \) or Pascal (Pa). | Dimension: \( [ML^{-1}T^{-2}] \).

Answer:

a) Longitudinal Stress: Restoring force per unit area when the deforming force is applied normal to the cross-section, changing the length (Tensile if it stretches, Compressive if it squashes).

b) Volumetric Stress: Restoring force per unit area when force is applied uniformly all over the surface, changing the volume but not the shape.

c) Shearing Stress: Restoring force per unit area when the deforming force acts tangentially to the surface, changing the shape (angle) of the body.

Answer: Strain: The ratio of the change in dimension of a body to its original dimension. It is dimensionless because it is a ratio of two identical physical quantities.

a) Longitudinal Strain: Change in length / Original length (\( \Delta L / L \)).

b) Volumetric Strain: Change in volume / Original volume (\( \Delta V / V \)).

c) Shearing Strain: The angle (in radians) through which a face originally perpendicular to the fixed face gets turned.

Answer: Hooke's Law: It states that within the elastic limit (for small deformations), the stress developed in a body is directly proportional to the strain produced. \( \text{Stress} \propto \text{Strain} \).

Modulus of Elasticity (E): The constant of proportionality in Hooke's Law. \( E = \frac{\text{Stress}}{\text{Strain}} \).

Answer:

a) Proportional Limit: The maximum stress up to which stress is directly proportional to strain (Hooke's law is obeyed; graph is a straight line).

b) Elastic Limit (Yield Point): The maximum stress up to which the body behaves as a perfectly elastic body and completely regains its shape.

c) Permanent Set: The permanent strain remaining in the wire when the deforming force is removed after crossing the elastic limit.

d) Fracture Point: The point on the curve corresponding to the stress at which the wire physically breaks.

Answer: Elastomers: Materials that can be stretched to large values of strain (often >1000%) without breaking and still regain their original shape. Examples: Rubber, tissue of the aorta.

Difference: Their stress-strain curve is highly non-linear right from the beginning, meaning they do not obey Hooke's Law over any significant region.

Answer:

Young's Modulus (Y): Ratio of longitudinal stress to longitudinal strain. \( Y = \frac{F/A}{\Delta L/L} = \frac{FL}{A\Delta L} \). SI: \(N/m^2\).

Bulk Modulus (B): Ratio of volumetric stress to volumetric strain. \( B = \frac{-P}{\Delta V/V} = \frac{-PV}{\Delta V} \). SI: \(N/m^2\).

Compressibility (k): The exact reciprocal of Bulk Modulus. \( k = 1/B \).

Shear Modulus (G): Ratio of shearing stress to shearing strain. \( G = \frac{F/A}{\theta} \). SI: \(N/m^2\).

Answer: The work done against inter-atomic restoring forces during stretching is stored as Elastic Potential Energy (\(U\)).

Formula: \( U = \frac{1}{2} \times \text{Stretching Force} \times \text{Extension} = \frac{1}{2} F \Delta L \).

Energy Density (Energy per unit volume): \( u = \frac{1}{2} \times \text{Stress} \times \text{Strain} \).

Answer: Thrust: The total normal force exerted by a fluid at rest on a given surface in contact with it. SI Unit: Newton (N). Dim: \([MLT^{-2}]\).

Pressure: The thrust exerted strictly per unit area of a surface. \( P = \frac{F}{A} \).
SI Unit: Pascal (Pa) or \(N/m^2\). Dim: \([ML^{-1}T^{-2}]\). It is a scalar quantity.

Answer:

a) Atmospheric Pressure (\(P_a\)): The pressure exerted by the weight of the Earth's atmosphere. (1 atm = \(1.013 \times 10^5\) Pa).

b) Gauge Pressure (\(P_g\)): The pressure of a system measured relative to atmospheric pressure. \( P_g = \rho g h \).

c) Absolute Pressure (\(P\)): The actual total pressure at a point. \( P = P_a + P_g \).

Answer: Pascal's Law states that if pressure is applied to any part of an enclosed, incompressible fluid at rest, it is transmitted equally and undiminished to every other part of the fluid and to the walls of the containing vessel. (Used in hydraulic lifts).

Answer: Density (\(\rho\)): The mass per unit volume of a substance.
Formula: \( \rho = \frac{m}{V} \). | SI Unit: \( kg/m^3 \). | Dimension: \( [ML^{-3}] \).

Answer: Streamline Flow: Orderly flow where the velocity of every fluid particle crossing a particular point remains perfectly constant over time. Paths never cross.

Turbulent Flow: Chaotic, irregular flow that occurs when fluid velocity exceeds the critical velocity. Characterized by the formation of eddies and vortices.

Answer: Bernoulli's Principle: For the streamline flow of an ideal (non-viscous, incompressible) fluid, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains strictly constant along any streamline.

Equation: \( P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \).

Answer:

a) Dynamic Lift: Due to the aerofoil shape of an airplane wing, air flows faster over the top (low pressure) and slower underneath (high pressure), creating an upward lift force.

b) Magnus Effect: A spinning ball drags air around it, creating a velocity difference (and thus a pressure difference) across its sides, causing the ball's path to curve (swing).

c) Venturimeter: A device with a constricted throat used to measure the flow rate of a liquid. The velocity increases at the throat, causing a measurable pressure drop.

Answer: Torricelli's Theorem: The velocity of efflux (outward flow) of a fluid emerging from a small orifice in an open tank is equal to the velocity a body would acquire in falling freely under gravity from the free surface of the liquid to the orifice.

Formula: \( v = \sqrt{2gh} \).

Answer:

Viscosity: The internal fluid friction that opposes the relative sliding motion between adjacent layers of a fluid.

Velocity Gradient (\( \frac{dv}{dx} \)): The rate of change of fluid velocity with distance perpendicular to the direction of flow. Unit: \( s^{-1} \).

Coefficient of Viscosity (\(\eta\)): The tangential viscous force required to maintain a unit velocity gradient across a unit area. SI Unit: Poiseuille (Pa·s).

Answer: Stokes' Law: The backward viscous drag force acting on a small spherical body of radius \(r\) falling with velocity \(v\) through a viscous fluid is \( F = 6\pi\eta r v \).

Terminal Velocity: The maximum, constant velocity acquired by a body falling freely through a viscous fluid. This occurs when the net upward forces (viscous drag + buoyant force) exactly balance the downward gravitational weight, resulting in zero net acceleration.

Answer: Surface Tension (\(S\)): The property of a liquid surface at rest to act like a stretched elastic membrane, tending to contract to the minimum possible surface area. \( S = F / L \). SI: N/m.

Surface Energy: The extra potential energy possessed by molecules residing exactly on the surface compared to molecules deep inside. Numerically equal to Surface Tension.

Angle of Contact (\(\theta\)): The angle inside the liquid between the tangent to the solid surface and the tangent to the liquid surface at the point of contact.

Answer: Capillarity: The phenomenon of the spontaneous rise or fall of a liquid inside a fine capillary tube against the force of gravity.

Ascent Formula: \( h = \frac{2S \cos\theta}{r \rho g} \).

Detergent Effect: Detergents severely weaken the cohesive forces between water molecules, drastically lowering the surface tension, which helps water penetrate cloth fibers to remove dirt.

Answer: Heat: Form of kinetic energy that flows from a higher temperature body to a lower temperature body. SI Unit: Joule.

Temperature: Macroscopic property measuring the degree of hotness or coldness. It determines the direction of heat flow. SI Unit: Kelvin.

Conversion Formula: \( \frac{C}{100} = \frac{F - 32}{180} = \frac{K - 273.15}{100} \).

Answer: Boyle's Law: At constant T, \( P \propto 1/V \).
Charles's Law: At constant P, \( V \propto T \).
Avogadro's Law: Equal volumes of all gases under identical P and T contain equal numbers of molecules.

Ideal Gas Equation: \( PV = nRT \) (where \(R = 8.314 \text{ J/mol K}\) is the Universal Gas Constant).

Answer: Thermal Expansion: Increase in dimensions of a body due to heat. Types: Linear (\(\alpha\)), Superficial (\(\beta\)), Cubical (\(\gamma\)). Relation: \( \alpha = \beta/2 = \gamma/3 \).

Anomalous Behaviour: Instead of expanding, pure water contracts when heated from 0°C to 4°C, giving it maximum density exactly at 4°C.

Answer: Specific Heat (\(c\)): Heat required to raise the temperature of unit mass (1 kg) of a substance by 1°C. \( c = \frac{Q}{m\Delta T} \).

Latent Heat (\(L\)): Heat required to change the physical state of a unit mass of substance strictly without changing its temperature. \( Q = mL \).

Answer: Conduction: Heat transfer through stationary material via particle collisions (requires medium, solids).

Convection: Heat transfer via actual physical movement of heated fluid molecules (requires medium, fluids).

Radiation: Heat transfer in the absence of a medium via electromagnetic waves (vacuum).

Answer: Stefan's Law: Total heat energy emitted per second per unit area by a black body is directly proportional to the 4th power of its absolute temp. \( E = \sigma T^4 \).

Wien's Law: The wavelength corresponding to maximum spectral emissive power is inversely proportional to absolute temp. \( \lambda_m T = b \).

Answer: Zeroth Law: If systems A & B are in thermal eq. with C, they are in thermal eq. with each other.

Internal Energy (\(U\)): Sum of microscopic kinetic & potential energies. It is a State Function.

First Law: Conservation of energy. Heat supplied is used to increase internal energy and do external work. \( \Delta Q = \Delta U + \Delta W \).

Answer: Isothermal: Constant Temperature (\(\Delta T = 0\)).

Isobaric: Constant Pressure (\(\Delta P = 0\)).

Isochoric: Constant Volume (\(\Delta V = 0, W = 0\)).

Adiabatic: Zero heat exchange with surroundings (\(\Delta Q = 0\)).

Answer: Kelvin-Planck: Cannot build an engine that is 100% efficient.
Clausius: Heat cannot flow from cold to hot without external work.

Reversible: Ideal, infinitely slow process with zero dissipation, can be perfectly retraced.
Irreversible: Real-world processes, energy dissipates as heat, cannot be retraced (e.g., friction).

Answer:

• Gas consists of tiny, identical, perfectly elastic spheres.
• Actual volume of molecules is negligible compared to gas volume.
• Molecules move randomly in all directions.
• Collisions between molecules and walls are perfectly elastic.
• No intermolecular forces exist except during collisions.

Answer: Degrees of Freedom (\(f\)): Number of independent coordinates needed to describe motion. Monoatomic = 3, Diatomic = 5, Triatomic non-linear = 6.

Equipartition of Energy: In thermal equilibrium, total internal energy is equally distributed among all active degrees of freedom. Energy per degree of freedom per molecule = \( \frac{1}{2} k_B T \).

Answer: \(C_v\): Heat required to raise temp of 1 mole of gas by 1K at constant volume.
\(C_p\): Heat required at constant pressure. (\(C_p > C_v\) because work is done during expansion).

Mayer's Formula: \( C_p - C_v = R \).

Answer: Periodic: Repeats after equal time intervals (e.g., Earth's orbit).

Oscillatory: To-and-fro periodic motion about a mean position.

SHM: Special oscillatory motion where restoring force is directly proportional to displacement and directed towards mean position. \( F = -kx \).

Answer: Undamped: Frictionless, constant amplitude.
Damped: Amplitude decays due to friction/air drag.

Resonance: Occurs in forced oscillations when the external periodic force frequency exactly matches the body's natural frequency, producing maximum amplitude.

Answer: Pendulum: \( T = 2\pi\sqrt{\frac{l}{g}} \) (Independent of bob's mass).

SHM Energies:
PE: \( U = \frac{1}{2}kx^2 \)
KE: \( K = \frac{1}{2}k(A^2 - x^2) \)
TE: \( E = \frac{1}{2}kA^2 \) (Constant).

Answer: Transverse: Medium vibrates perpendicular to wave direction (crests/troughs). Ex: Light.
Longitudinal: Medium vibrates parallel to wave direction (compressions/rarefactions). Ex: Sound.

Mechanical: Needs physical medium (Sound).
Non-Mechanical: Can travel in vacuum (EM waves).

Answer: Progressive Equation: \( y(x,t) = A \sin(\omega t - kx) \). Travels carrying energy forward.

Stationary (Standing) Wave: Superposition of two identical waves traveling in opposite directions. Energy is trapped. Forms Nodes (zero vibration) and Antinodes (max vibration).

Answer: Beats: Periodic variation in loudness caused by superposition of two sound waves of slightly different frequencies.
Formula: \( f_{beat} = |\nu_1 - \nu_2| \).