Summary

Centre of mass of a two-particle system, $\vec R = (m_1\vec r_1+m_2\vec r_2)/(m_1+m_2)$, extends to any system of particles or rigid body, $\vec R = \sum m_i\vec r_i / M$, and need not lie inside the body's own material (e.g. a ring). Motion of the centre of mass: $M\vec a_{cm} = \vec F_{ext}$, so internal forces (an explosion, a collision) never move the centre of mass off its original path; when $\vec F_{ext}=0$, total linear momentum is conserved. Torque, $\vec\tau = \vec r\times\vec F$, is the rotational analogue of force. Angular momentum, $\vec l = \vec r\times\vec p$ for a particle, obeys $d\vec L/dt = \vec\tau_{ext}$; when $\vec\tau_{ext}=0$, $\vec L$ is conserved (a spinning skater pulling in the arms). A rigid body is in equilibrium only when both net force and net torque are zero; a lever balances under the principle of moments, $F_1d_1=F_2d_2$; a couple has zero net force but a nonzero moment $Fd$. Centre of gravity coincides with centre of mass in a uniform gravitational field. Moment of inertia, $I=\sum m_ir_i^2$, is the rotational analogue of mass, depending on the axis; standard values are quoted for a rod, ring, disc, sphere and cylinder. Radius of gyration, $K=\sqrt{I/M}$. The theorem of parallel axes, $I=I_{cm}+Md^2$, and the theorem of perpendicular axes (planar bodies only), $I_z=I_x+I_y$, exte …