Page:Elementary Text-book of Physics (Anthony, 1897).djvu/46

32 The density is then defined as the ratio of the mass of the body to its volume, or as the mass contained in a unit of volume. By supposing the mass of the body uniformly distributed throughout its volume, so that the ratio of mass to volume has the same value no matter how small the volume is, we may represent the mass contained in any infinitesimal volume by the product of the density and the volume. The concept of density used in this way is an artificial one, and the validity of the results obtained by it is due to the fact that the particles constituting a body are so small that their distribution is practically uniform in a homogeneous body in any volume which can be examined by experiment.

The formula for density is $$D = \frac{M}{V}$$, and the dimensions are $$[D] = ML^{-3}$$. The unit of density is the density of a homogeneous body so constituted that unit of mass is contained in unit of volume.

By using the hypothesis of a continuous distribution of matter in a body, we may define the density at a point in a body which is not homogeneous as the ratio of the mass contained in a sphere described about that point as centre to the volume of the sphere, when that volume is diminished indefinitely. 30. Centre of Mass.— The centre of mass of two particles is defined as the point which divides the straight line joining the particles into two segments, the lengths of which are inversely proportional to the masses of the particles at their extremities.

Thus if $$A$$ and $$B$$ be the positions of the two particles of which the masses are $$m_{a}$$ and $$m_{b}$$ respectively, then the point $$C$$, lying on the line joining $$A$$ and $$B$$, is the centre of mass if it divide $$AB$$ so that $$m_{a} \cdot AC = m_{b} \cdot BC.$$

The centre of mass of more than two particles is found by finding the centre of mass of two of them, supposing a mass equal to their sum placed at that centre, finding the centre of mass of this ideal particle and a third particle, and proceeding in a similar way until all the particles of the system have been brought into combination. The final centre thus found is the centre of mass of the