< R$): $E_g = \frac{GMr}{R^3}$, $V = -\frac{GM}{2R^3}(3R^2 - r^2)$

Uniform Spherical Shell (Mass M, Radius R):

Uniform Thin Circular Ring (Mass M, Radius R):

At axial point (distance $x$ from center): $E_g = \frac{GMx}{(R^2+x^2)^{3/2}}$, $V = -\frac{GM}{\sqrt{R^2+x^2}}$
At center ($x=0$): $E_g=0$, $V = -\frac{GM}{R}$

Differential Equation: $\frac{d^2x}{dt^2} + \omega^2 x = 0$
Displacement: $x(t) = A \sin(\omega t + \phi)$ or $A \cos(\omega t + \phi)$
Velocity: $v(t) = A\omega \cos(\omega t + \phi) = \pm \omega \sqrt{A^2 - x^2}$
Acceleration: $a(t) = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$
Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (spring-mass), $\omega = \sqrt{\frac{g}{L_{eff}}}$ (pendulum)
Time Period: $T = \frac{2\pi}{\omega}$
Energy in SHM: $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$
Damped Oscillations: $x(t) = A_0 e^{-bt/2m} \cos(\omega' t + \phi)$ where $\omega' = \sqrt{\omega_0^2 - (b/2m)^2}$
Forced Oscillations & Resonance: Amplitude maximum when driving frequency equals natural frequency.

Pressure: $P = \frac{F}{A}$
Pressure in Fluid: $P = P_0 + \rho gh$
Pascal's Law: Pressure applied to an enclosed fluid is transmitted undiminished.
Archimedes' Principle: Buoyant force $F_B = \rho_{fluid} V_{submerged} g$
Equation of Continuity: $A_1 v_1 = A_2 v_2$ (for incompressible fluid)
Bernoulli's Equation: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
Viscosity: $F = -\eta A \frac{dv}{dy}$ (Newton's Law of Viscosity)
Stokes' Law: $F_v = 6\pi\eta rv$ (for spherical body)
Terminal Velocity: $v_t = \frac{2r^2(\rho_p - \rho_f)g}{9\eta}$
Surface Tension: $S = \frac{F}{L}$
Excess Pressure in Bubble/Drop:
- Liquid drop: $\Delta P = \frac{2S}{R}$
- Soap bubble: $\Delta P = \frac{4S}{R}$
Capillary Rise: $h = \frac{2S \cos\theta}{\rho rg}$

Stress: $\sigma = F/A$
Strain: Longitudinal ($\Delta L/L$), Volumetric ($\Delta V/V$), Shear ($\phi$)
Young's Modulus: $Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}} = \frac{F/A}{\Delta L/L}$
Bulk Modulus: $B = \frac{\text{Volumetric Stress}}{\text{Volumetric Strain}} = \frac{-\Delta P}{\Delta V/V}$
Shear Modulus: $G = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{\phi}$
Poisson's Ratio: $\sigma = -\frac{\text{lateral strain}}{\text{longitudinal strain}}$

Heat required: $Q = mc\Delta T$ (for temp change), $Q = mL$ (for phase change)
Conduction: $\frac{dQ}{dt} = -KA\frac{dT}{d