• Angular displacement ($\theta$): Radian measure of rotation angle
• Angular velocity ($\omega$): $\omega = \frac{d\theta}{dt}$ (rad/s)
• Angular acceleration ($\alpha$): $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ (rad/s2))))

$\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha\Delta\theta$

• Analogs of linear kinematic equations
• Apply separately from linear motion

• $v = r\omega$ (rolling without slipping condition)
• $a = r\alpha$ (rolling without slipping)
• $x = r\theta$, $v = r\omega$, $a = r\alpha$
• Connects linear and angular quantities
• Essential for rolling problems

• Point of contact is instantaneously at rest relative to surface
• $v_{cm} = R\omega$ (no slipping)
• $a_{cm} = R\alpha$
• Friction provides necessary force (static friction)
• $f_s \leq \mu_s N$ must hold for no slip

$\tau = rF\sin\theta = F_\perp d$

• Rotational analog of force
• Causes angular acceleration
• Units: N - m
• Depends on force magnitude, distance from axis, angle

• Perpendicular distance from axis to line of action of force
• $d_\perp = r\sin\theta$
• Maximizes torque when force applied perpendicular to lever arm
• Longer lever arm = more torque for same force

• For given force, maximize distance from axis (perpendicular)
• Apply force perpendicular to lever arm
• $\tau_{max} = rF$ (when $\theta = 90 degrees$)
• $\tau = rF\sin\theta$

$I = \sum m_i r_i^2$

• Rotational analog of mass
• Measure of resistance to angular acceleration
• Depends on mass distribution and axis of rotation
• Units: kg - m2
• Point mass: $I = mr^2$

• Mass farther from axis increases I more
• Hollow objects have larger I than solid objects of same mass
• Shape matters: ring > disk > solid sphere (same mass, different I)
• Can modify rotational inertia by redistributing mass

Parallel Axis Theorem:

Moment of inertia about any axis.

$I = I_{cm} + Md^2$

Where:

• $I_{cm}$ = moment about center of mass
• M = total mass
• d = distance between axes

Common shapes (about center of mass):

• Point mass: $mr^2$
• Thin rod (length L, ⟂): $\frac{1}{12}mL^2$
• Solid cylinder/disk (radius R): $\frac{1}{2}mR^2$
• Thin cylindrical shell: $mR^2$
• Solid sphere (radius R): $\frac{2}{5}mR^2$
• Thin spherical shell: $\frac{2}{3}mR^2$

• Translational equilibrium: $\sum \vec{F} = 0$
• Rotational equilibrium: $\sum \tau = 0$ about any axis
• Both conditions required for static equilibrium
• No net force, no net torque

• Can place pivot at any convenient location
• Unknown forces at pivot eliminated from torque equation
• Strategic choice simplifies problem-solving
• Always check translational equilibrium too

$\tau_{net} = I\alpha$

• Rotational analog of $F = ma$
• Causes angular acceleration
• Sum of all torques about axis
• Applies to rigid bodies

• Combination of translation and rotation
• $F_{net} = Ma_{cm}$ (translational)
• $\tau_{net} = I_{cm}\alpha$ (rotational about center of mass)
• Need to solve both simultaneously
• Examples: rolling objects, swinging objects