Newton's 2nd Law for Rotation: $\vec{\tau}_{net} = I\vec{\alpha}$

Work in Rotation: $W = \int_{\theta_i}^{\theta_f} \tau d\theta$ (constant $\tau$: $W = \tau \Delta \theta$)

Power in Rotation: $P = \tau \omega$

Angular Momentum:

Newton's 2nd Law (Angular Momentum Form): $\vec{\tau}_{net} = \frac{d\vec{L}}{dt}$

Conservation of Angular Momentum: $\vec{L}_{net} = \text{constant}$ if $\vec{\tau}_{net,ext} = 0$.

Static Equilibrium Conditions:
- Translational: $\sum \vec{F}_{net} = 0 \Rightarrow \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$
- Rotational: $\sum \vec{\tau}_{net} = 0$ (about any axis)
Stress: Force per unit area ($P = F/A$)
Strain: Fractional change in dimension ($\Delta L/L$, $\Delta V/V$)
Hooke's Law (Elastic Moduli): Stress = Modulus $\times$ Strain
- Young's Modulus (tension/compression): $E = \frac{F/A}{\Delta L/L}$
- Shear Modulus (shearing): $G = \frac{F/A}{\Delta x/h}$
- Bulk Modulus (volume compression): $B = -\frac{\Delta P}{\Delta V/V}$

Newton's Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$ ($G = 6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$)
Gravitational Acceleration: $g = G \frac{M_E}{R_E^2}$ (at surface), $g(r) = G \frac{M_E}{r^2}$ (at distance $r$ from center)
Gravitational Potential Energy: $U = -G \frac{m_1 m_2}{r}$ (zero at infinity)
Escape Speed: $v_{esc} = \sqrt{\frac{2GM}{R}}$
Kepler's Laws:
- 1st: Orbits are ellipses with Sun at one focus.
- 2nd: A line joining a planet and the Sun sweeps out equal areas in equal times.
- 3rd: $T^2 \propto a^3$ (for circular orbit $T^2 = \left(\frac{4\pi^2}{GM}\right)r^3$)

Density: $\rho = \frac{m}{V}$
Pressure: $P = \frac{F}{A}$
Pressure in a Fluid at Depth: $P = P_0 + \rho gh$
Pascal's Principle: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel. ($\frac{F_1}{A_1} = \frac{F_2}{A_2}$)
Archimedes' Principle (Buoyancy): $F_B = \rho_{fluid} g V_{disp}$ (Buoyant force equals weight of displaced fluid)
Equation of Continuity: $A_1 v_1 = A_2 v_2$ (for incompressible fluid)
Bernoulli's Equation: $P_1 + \frac{1}{2}\rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g y_2$

Simple Harmonic Motion (SHM):
- Displacement: $x(t) = x_m \cos(\omega t + \phi)$
- Velocity: $v(t) = -x_m \omega \sin(\omega t + \phi)$
- Acceleration: $a(t) = -x_m \omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$
- Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (mass-spring), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum)
- Period: $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$ (mass-spring), $T = 2\pi \sqrt{\frac{L}{g}}$ (simple pendulum)
- Frequency: $f = \frac{1}{T}$
- Energy: $E = \frac{1}{2}kx_m^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$
Damped SHM: $x(t) = x_m e^{-bt/2m} \cos(\omega' t + \phi)$