Motion in a Straight Line & Plane

Newtonian Mechanics & Force Equilibrium: \Sigma F = dp/dt = ma; an isolated system conserves total linear momentum (\Sigma p_i = \Sigma p_f).

Work-Energy Theorem: Total work done by all forces (conservative + non-conservative + external) equals change in kinetic energy (W_{net} = \Delta K).

Conservative Fields & Potential Energy: Conservative force is the negative gradient of potential energy F = - abla U = -dU/dx; Mechanical energy E = K + U is conserved in conservative systems.

Rotational Analogues of Translation: Torque au = r imes F = I\alpha, Angular momentum L = r imes p = I\omega, Rotational Kinetic Energy K_{rot} = 0.5 I\omega^2.

Kinematic Cartesian Decomposition: One-dimensional and projectile motion under constant acceleration follow ( v = u + at ), ( s = ut + \frac{1}{2}at^2 ), and ( v^2 = u^2 + 2as ) applied independently along orthogonal Cartesian axes without cross-coupling.

Newton's Laws & Free-Body Isolation: Resolves all real external forces (gravitational, normal, friction, tension) acting on an isolated mass; net vector resultant equals mass times acceleration (( \sum \vec{F} = m\vec{a} )).

Work-Energy Theorem & Conservation of Energy: Net work performed by all acting forces strictly equals the change in kinetic energy (( W_{\text{net}} = \Delta K )). In conservative force fields, mechanical energy (( K + U )) is strictly conserved.

Linear & Angular Momentum Conservation: In the absence of an external net force, linear momentum (( \vec{p} = m\vec{v} )) is conserved; in the absence of net external torque, angular momentum (( \vec{L} = I\vec{\omega} )) is invariant.

Simple Harmonic Motion (SHM) Mechanics: Defined by a restorative force proportional to displacement (( F = -kx )); oscillatory acceleration is ( a = -\omega^2 x ), and time period for a simple pendulum is ( T = 2\pi \sqrt{\frac{L}{g}} ).

Thermodynamics & Heat Engine Limits: Heat and work are mutually convertible (( \Delta Q = \Delta U + \Delta W )). The maximum achievable thermodynamic efficiency between hot and cold reservoirs is bounded by the Carnot limit (( \eta = 1 - \frac{T_C}{T_H} )).

Electrostatics & Gauss's Flux Theorem: Electric force follows inverse-square Coulomb's law. Net electric flux through any closed Gaussian surface equals enclosed charge divided by vacuum permittivity (( \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0} )).

Kinematic Cartesian Decomposition: One-dimensional and projectile motion under constant acceleration follow ( v = u + at ), ( s = ut + \frac{1}{2}at^2 ), and ( v^2 = u^2 + 2as ) applied independently along orthogonal Cartesian axes without cross-coupling.

Newton's Laws & Free-Body Isolation: Resolves all real external forces (gravitational, normal, friction, tension) acting on an isolated mass; net vector resultant equals mass times acceleration (( \sum \vec{F} = m\vec{a} )).

Work-Energy Theorem & Conservation of Energy: Net work performed by all acting forces strictly equals the change in kinetic energy (( W_{\text{net}} = \Delta K )). In conservative force fields, mechanical energy (( K + U )) is strictly conserved.

Linear & Angular Momentum Conservation: In the absence of an external net force, linear momentum (( \vec{p} = m\vec{v} )) is conserved; in the absence of net external torque, angular momentum (( \vec{L} = I\vec{\omega} )) is invariant.

Simple Harmonic Motion (SHM) Mechanics: Defined by a restorative force proportional to displacement (( F = -kx )); oscillatory acceleration is ( a = -\omega^2 x ), and time period for a simple pendulum is ( T = 2\pi \sqrt{\frac{L}{g}} ).

Thermodynamics & Heat Engine Limits: Heat and work are mutually convertible (( \Delta Q = \Delta U + \Delta W )). The maximum achievable thermodynamic efficiency between hot and cold reservoirs is bounded by the Carnot limit (( \eta = 1 - \frac{T_C}{T_H} )).

Electrostatics & Gauss's Flux Theorem: Electric force follows inverse-square Coulomb's law. Net electric flux through any closed Gaussian surface equals enclosed charge divided by vacuum permittivity (( \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0} )).

Kinematic Cartesian Decomposition: One-dimensional and projectile motion under constant acceleration follow ( v = u + at ), ( s = ut + \frac{1}{2}at^2 ), and ( v^2 = u^2 + 2as ) applied independently along orthogonal Cartesian axes without cross-coupling.

Newton's Laws & Free-Body Isolation: Resolves all real external forces (gravitational, normal, friction, tension) acting on an isolated mass; net vector resultant equals mass times acceleration (( \sum \vec{F} = m\vec{a} )).

Work-Energy Theorem & Conservation of Energy: Net work performed by all acting forces strictly equals the change in kinetic energy (( W_{\text{net}} = \Delta K )). In conservative force fields, mechanical energy (( K + U )) is strictly conserved.

Linear & Angular Momentum Conservation: In the absence of an external net force, linear momentum (( \vec{p} = m\vec{v} )) is conserved; in the absence of net external torque, angular momentum (( \vec{L} = I\vec{\omega} )) is invariant.

Simple Harmonic Motion (SHM) Mechanics: Defined by a restorative force proportional to displacement (( F = -kx )); oscillatory acceleration is ( a = -\omega^2 x ), and time period for a simple pendulum is ( T = 2\pi \sqrt{\frac{L}{g}} ).

Thermodynamics & Heat Engine Limits: Heat and work are mutually convertible (( \Delta Q = \Delta U + \Delta W )). The maximum achievable thermodynamic efficiency between hot and cold reservoirs is bounded by the Carnot limit (( \eta = 1 - \frac{T_C}{T_H} )).

Electrostatics & Gauss's Flux Theorem: Electric force follows inverse-square Coulomb's law. Net electric flux through any closed Gaussian surface equals enclosed charge divided by vacuum permittivity (( \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0} )).

Fundamental governing laws and conservation principles.

Asymptotic bounds and mathematical derivations.

Limiting behavior and boundary constraints.

Fundamental governing laws and conservation principles.

Asymptotic bounds and mathematical derivations.

Limiting behavior and boundary constraints.

Fundamental governing laws and conservation principles.

Asymptotic bounds and mathematical derivations.

Limiting behavior and boundary constraints.