SUMMARY

This unit's key results, gathered together:

• A rigid body is one in which the distances between its different particles remain fixed, however large a force is applied to it.
• For a regular-shaped body with uniform mass distribution, the center of mass always coincides with the geometric center; more generally it is located via $\vec r_{CM}=\dfrac{\sum m_i\vec r_i}{M}$ (discrete) or $\vec r_{cm}=\dfrac{\int \vec r\,dm}{\int dm}$ (continuous), and with no external force it moves at constant velocity regardless of internal motion.
• Torque, $\vec\tau=\vec r\times\vec F$, is what produces turning (rotational) motion in a rigid body; a couple (equal, opposite, offset forces) produces pure rotation with zero net force.
• A rigid body is in translational equilibrium if the total external force on it is zero, and in rotational equilibrium if the total external torque on it is zero; stability (stable/unstable/neutral) further classifies how it responds to a small disturbance.
• The center of gravity of an extended body is the point through which the total gravitational torque on the body is zero.
• Moment of inertia, $I=\sum m_ir_i^2=MK^2$, is the rotational analogue of mass; the parallel and perpendicular axis theorems convert it between different axes.
• If the external torque on a body is zero, its angular momentum, $L=I\omega$, along the axis of rotation stays constant (conservation of angular momentum).
• Every translational quantity has a rotational equivalent — force/torque, mass/moment of inertia, momentum/angular momentum, work/kinetic energy/power all carry across directly. …