No external torque → $L = I\omega =$ constant
Sphere: $I = \frac{2}{5}MR^2$
$$I_2 = \frac{2}{5}M(2R)^2 = 4 \times \frac{2}{5}MR^2 = 4I_1$$Angular momentum $\vec{L} = I\vec{\omega}$ is conserved when net external torque is zero ($\vec{\tau}_{ext} = 0$). $L = I\omega = $ constant. If $I$ increases: $\omega$ decreases (and vice versa). Examples: ice skater spinning (arms in → $I$ decreases → $\omega$ increases), diver tucking (tuck → small $I$ → fast rotation), gymnastics. Formation of neutron stars: massive stars collapse → radius shrinks by $10^5$ → $I$ decreases by $10^{10}$ → $\omega$ increases by $10^{10}$ → pulsars (millisecond rotation periods).
$I = \sum m_i r_i^2 = \int r^2\,dm$. Depends on mass distribution AND choice of axis. Key values: Hollow ring/cylinder: $I = MR^2$. Solid disc/cylinder: $I = MR^2/2$. Solid sphere: $I = 2MR^2/5$. Hollow sphere: $I = 2MR^2/3$. Thin rod (centre): $I = ML^2/12$. Thin rod (end): $I = ML^2/3$. Parallel axis: $I = I_{cm} + Md^2$.
Torque $\tau = r \times F = I\alpha$. Rotational KE $= \frac{1}{2}I\omega^2$. Work done by torque: $W = \tau\theta$. Power $= \tau\omega$. Angular impulse $= \tau\Delta t = \Delta L$. Rotational analogues: $\tau \leftrightarrow F$, $I \leftrightarrow m$, $\alpha \leftrightarrow a$, $\omega \leftrightarrow v$, $L \leftrightarrow p$, $\frac{1}{2}I\omega^2 \leftrightarrow \frac{1}{2}mv^2$.
$v_{cm} = R\omega$. Total $KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$. For rolling: $KE_{total} = \frac{1}{2}mv^2(1 + k^2/R^2)$ where $k$ = radius of gyration. Speed at bottom of incline (height $h$): $v = \sqrt{\frac{2gh}{1+k^2/R^2}}$. Solid sphere ($k^2/R^2 = 2/5$): fastest. Hollow sphere ($2/3$): slower. Disc ($1/2$): slower. Ring ($1$): slowest.
$\vec{r}_{cm} = \sum m_i\vec{r}_i / \sum m_i$. For uniform shapes: geometric centre. For composite bodies: $x_{cm} = (m_1 x_1 + m_2 x_2)/(m_1+m_2)$. CM of uniform semicircle from diameter: $4R/(3\pi)$. CM of solid hemisphere: $3R/8$ from flat face. Velocity of CM: $\vec{v}_{cm} = \sum m_i\vec{v}_i/M$. External forces affect CM motion: $\vec{F}_{ext} = M\vec{a}_{cm}$. Internal forces do not affect CM motion.
Gyroscope: spinning object with angular momentum $L = I\omega$ (large). When external torque applied (gravity on tilted gyro): instead of falling, axis precesses (rotates around vertical). Precession angular velocity $\Omega = \tau/L = mgr/(I\omega)$. Faster spin → slower precession. Applications: gyrocompasses (aircraft, ships), stabilizers (ships, cameras), MEMS gyroscopes (phones for rotation sensing), spacecraft attitude control. Earth's axis precesses with period ~26,000 years (gravitational torque from Sun and Moon on equatorial bulge).
For static equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, $\sum \tau = 0$ (about any point). Three conditions for 2D problems. Torque can be calculated about any convenient point (usually where unknown force acts → eliminates that unknown). Ladder against wall: normal from wall (horizontal), normal from floor (vertical), weight at mid-point, friction from floor (horizontal). Condition: solve for friction and normal force. Centre of gravity: for uniform $g$, coincides with centre of mass.
Earth's spin: conservation of angular momentum in solar system formation (collapsing gas cloud → spinning planets). Tidal locking: Moon orbits Earth with same period as it rotates → always shows same face. Earth rotation slowing due to tidal friction from Moon → day getting longer (~1.4 ms/century). As Earth's rotation slows, Moon moves further away (angular momentum transferred). In 5 billion years: day will be ~47 current days and Moon will be ~1.6× current distance. Neutron stars: fastest pulsars (millisecond pulsars) spin up by accreting matter from companion star.