System of Particles and Rotational Motion

For a system of particles, the centre of mass is the single point that behaves as though the system's entire mass were concentrated there and every external force acted directly on it. For two particles of masses $m_1$ and $m_2$ at positions $x_1$ and $x_2$ on a line, it is defined as $x_{cm}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}$; for $n$ particles this generalises to the vector form $\vec{R} =

Torque (or moment of force) about a point is the rotational analogue of force, defined as $\vec{\tau }=\vec{r}\times \vec{F}$, where $\vec{r}$ is the position vector from that point to where $\vec{F}$ acts. Its magnitude is $\tau =rF\sin \theta$, with $\theta$ the angle between $\vec{r}$ and $\vec{F}$, so a force directed along $\vec{r}$ itself produces no torque — only its component perpend

The moment of inertia of a rigid body about a given axis is the rotational analogue of mass: it measures the body's resistance to angular acceleration about that axis. For a system of particles it is $I=\sum m_i{r}_i^2$, and for a continuous body $I=\int r^{2}dm$, where $r_i$ (or $r$) is the perpendicular distance of each mass element from the axis. Unlike mass, moment of inertia is not fix

Rolling without slipping is the combined motion of translation of a body's centre of mass together with rotation about an axis through that centre. For a round body of radius $R$, the two motions are locked together by the condition $v_{cm}=R\omega$. This follows because the point of the body instantaneously touching the ground has zero velocity: its forward velocity $v_{cm}$ from translatio