66 



CENTRAL PROVINCES 



CENTRE 



weaving and the smelting and working of iron ores. 

 Iron is abundant, especially in the south, and there 

 are also large coalfields, but the coal is of a very 

 inferior quality. There is considerable trade, but 

 its progress is retarded by the want of means of 

 communication ; this drawback, however, is being 

 removed, roads are being made, and the railway 

 system steadily pushed forward. Of the popula- 

 tion, three-fourths are Hindus, and one-seventh 

 belong to the so-called aboriginal or non-Aryan 

 tribes, Avho have found a refuge in the Satpura 

 plateau, and still adhere to their primitive faiths 

 (see GONDS). From these hill-tribes the Hindus 

 throughout the province have contracted beliefs 

 and habits which they have grafted upon the 

 usual worship of their sect ; adoration of the dead, 

 worship of the goddess of smallpox, and belief in 

 witchcraft are universal. The population is almost 

 entirely rural, only 6 per cent, residing in the 52 

 towns of above 5000 inhabitants, of which three 

 Nagpur, Jubbulpore, and Kampti have over 

 50,000 inhabitants. Central India (q.v.) is a term 

 of quite distinct meaning. 

 Centre and Central Forces. CENTRE OF 



INERTIA (MASS). 

 Aj If m l and m 2 be the 



masses of two par- 

 ^ tides placed at the 

 ' points A! and A 2 , 



and if the right line 



AjAjj be divided in 

 Fig. 1. Bj, so that 



B 2 



the point B x is called the centre of inertia, or 

 centre of mass, of the two particles. If m 3 be a 

 third mass at A 3 , and if BjAj be divided in B 2 , so 

 that 



B 2 is called the centre of inertia of the three 

 particles. In general, if there be any number of 

 particles, a continuation of the above process will 

 enable us to find their centre of inertia. Every 

 body may be supposed to be made up of a multitude 

 of particles connected by cohesion. From this it 

 is obvious that the centre of inertia is a definite 

 point for every piece of matter. 



In general, the determination of the centre of 

 inertia requires the use of the integral calculus. 

 In the case of some bodies, such as those 

 which have a simple geometrical form and are 

 of uniform density, elementary mathematical 

 methods will generally be sufficient. Any straight 

 line or plane that divides a homogeneous body 

 symmetrically must contain its centre of in- 

 ertia. For the particles of the body may be 

 arranged in pairs of equal mass and at equal dis- 

 tances from the straight line or plane ; and, since 

 the centre of inertia of each pair lies in the line or 

 plane, the centre of inertia of the whole must also 

 lie in the same line or plane. For example, the 

 centre of inertia of a uniform thin straight rod is 

 its middle point ; that of a uniform thin rod bent in 

 the form of a parallelogram, the point of intersec- 

 tion of its diagonals ; that of a lamina, uniform in 

 thickness and density and in form a circle, ellipse, 

 or parallelogram, its centre of figure; that of a uni- 

 form spherical shell, its centre ; that of a homogene- 

 ous sphere, its centre ; that of a parallelepiped, the 

 intersection of its diagonals ; that of a circular 

 cylinder with parallel ends, the middle point of its 

 axis. 



An important case is that of a uniformly thin 

 triangular plate. Let ABC be the plate. Bisect 

 AB in P and join CP. Let the triangle be divided 

 by right lines parallel to AB into an indefinitely 

 great number of indefinitely narrow strips. The 

 centre of inertia of each strip is its middle point. 



Fig. 2. 



But all the middle points lie on CP. The centre of: 

 inertia of the whole plate must therefore lie on CP. 

 Again, if BC be 

 bisected in Q, and 

 AQ be joined, the 

 centre of inertia of 

 the whole plate must 

 lie in AQ. The cen- 

 tre of inertia must 

 therefore be O, the 

 point of intersection 

 of CP and AQ. It 

 is easily proved by 

 elementary geo- 

 metry that OP = one-third of CP. Hence, the 

 centre of inertia of a triangular plate is obtained 

 by joining a vertex to the middle point of the 

 opposite side and taking the point two-thirds of 

 this line measured from the vertex. By a similar 

 method the centre of inertia of other plane figures 

 may be obtained. 



CENTRE OF GRAVITY. If a body be sufficiently 

 small, relatively to the earth, the weights of its par- 

 ticles may be considered as constituting a system of 

 parallel forces acting on the body. Now, the mag- 

 nitude of the weight of a particle is proportional to 

 its mass. Hence, the line of action or the result- 

 ant of the parallel forces will approximately pass 

 through the centre of inertia. For this reason 

 such bodies are said to have a centre of gravity. 

 Strictly speaking, there is no such point of 

 necessity for every body, since the directions of the 

 forces acting on the body are not accurately 

 parallel. Hence, it is only approximately that we 

 can say of a body that it has a centre of gravity. 

 On the other hand, every piece of matter has, as is 

 shown above, a centre 01 inertia. For all heavy 

 bodies of moderate dimensions it is, however, suf- 

 ficiently accurate to assume that the centre of 

 inertia and gravity coincide. For example, the 

 centre of gravity of a uniform homogeneous cylinder 

 with parallel ends is the middle point of its axis, 

 that of a uniformly thin circular lamina its centre, 

 and so on. 



The centre of gravity of a body of moderate 

 dimensions may be approximately determined by 

 suspending it by a single cord in two different 

 positions, and finding the single point in the body 

 which, in both positions, is intersected by the axis 

 of the cord. 



The term centre of gravity is also used in 

 a stricter sense than the one just explained. 

 Thus, if a body attracts and is attracted by all 

 other gravitating matter as if its whole mass 

 were concentrated in one point, it is said to have a 

 true centre of gravity at that point, and the body 

 itself is called a centrobaric body. A spherical 

 shell of uniform gravitating matter attracts an ex- 

 ternal particle as if its whole mass were condensed 

 at its centre. Such a body has a true centre of 

 gravity. When such a point exists, it necessarily 

 coincides with the centre of inertia. 



CENTRE OF OSCILLATION. A heavy particle 

 suspended from a point by a light inextensible 

 stnng constitutes what is called a simple or mathe- 

 matical pendulum. For such a pendulum it is 

 easily proved that the time of an oscillation from 

 side to side of the vertical is proportional to the 

 square root of its length for any small arc of vibra- 

 tion. A simple pendulum is, however, a thing of 

 theory, as in all physical problems we have to deal 

 with a rigid mass, and not a particle, oscillating 

 about a horizontal axis. In a pendulum of this 

 kind the time of oscillation will not vary as the 

 square root of the length of the string, for it is 

 obvious that those particles of the body which are 

 nearest the point of suspension will have a tend- 

 ency to vibrate more rapidly than those more- 



