Q.Define the centre of mass of a system of particles. Is the centre of mass of a rigid body always located within the material of the body itself? Justify your answer with one example.

The centre of mass (c.m.) is the single hypothetical point at which the entire mass of an extended, finite-sized object (or system of objects) can be treated as concentrated, allowing Newton's laws of motion -- which strictly apply only to point masses -- to be validly applied to real, extended objects. For a system of n particles, its position is the mass-weighted average of the individual particles' positions, $\vec{r}=\frac{\sum m_i\vec{r_i}}{M}$ (or the corresponding integral, $\vec{r}=\frac{\int \vec{r}\,dm}{M}$, for a continuous mass distribution); its velocity and acceleration are similarly the mass-weighted averages of the individual particles' velocities and accelerations.

The centre of mass has many useful properties: it is a location rather than a physical quantity in itself; it can only be shifted by an external unbalanced force, never by internal forces (as in a collision or explosion); it lies on any axis or point of symmetry an object has; it need not coincide with any individual particle, or even lie physically within the body at all (a ring or horseshoe are simple examples); and only the net external force on a system determines the acceleration of its centre of mass, $\vec{F}_{ext}=M\vec{a}_{cm}$, regardless of how complicated the system's internal motions or forces are.