CENTRE 



67 



remote. The former are therefore retarded by the 

 hitter, while the latter are accelerated by the 

 former. There is thus one particle which will he 

 accelerated and retarded to ;ui equal amount, and 

 which will therefore move as if it were a simple 

 pendulum unconnected with the rest of the body. 

 The jioint in the Ixxly occupied by this particle 



i* c.'llled the i-fiifn nf ofii'illdtinii. 



\-all the particles of the body are rigidly con- 



i ted, they all vibrate in the same time. Hence 



it follows that the time of vibration of the rigid 

 Uidy will be the same as that of a simple pendulum, 

 called the equivalent or isochronous simple pendu- 

 lum, whose length is equal to the distance between 

 the centres of suspension and oscillation. 



The determination of the centre of oscillation of a 

 Ixxly requires the aid of the calculus. It may be 

 stated, however, that it is always farther from the 

 ;i\i> of suspension than the centre of inertia, and is 

 always in the line joining the centres of suspension 

 and oscillation. Let A be the centre of sus- 

 pension, B the centre of inertia, and C the 

 centre of oscillation, and let AB be equal to h, 

 and k be the radius of gyration of the body 

 about an axis through B parallel to the fixed 

 B axis, then it is easily shown that 



AC = T 



Fig. 3. From this there follows the important pro- 

 position that the centres of oscillation and 

 suspension are convertible, a proposition which was 

 taken advantage of by Kater for the practical 

 determination of the force of gravity at any 

 station. 



CENTRE OF PERCUSSION. If a body receive a 

 blow which makes it begin to rotate about a fixed 

 axis without causing any pressure on the axis, the 

 point in which the direction of the blow intersects 

 the plane in which the fixed axis and the centre of 

 inertia lie is called the centre of percussion. That 

 such a point must exist is easily snown by suspend- 

 ing a straight rod by a long string attached to one 

 end, and striking it with a hammer in different 

 points. If the rod is stnick near the top the foot 

 will move in one direction, and if the blow be 

 applied near the foot the top will move in the 

 opposite direction. It is thus evident that there 

 must be some point which does not move at all at 

 the instant of the blow. If a line through this 

 point be regarded as an axis of rotation, the point 

 at which the body was struck is the centre of per- 

 cussion, since no pressure is ^-reduced on the axis. 

 It is easily proved by means of higher mathematics 

 that the centre of percussion with respect to any 

 axis is the same point as the centre of oscillation. 



From what has \yeen said it is obvious that in 

 order that no jar may be felt on the hand a cricket 

 ball must be hit in the centre of percussion of the 

 bat with respect to an axis through the hand. 



There are, it may be mentioned, many positions 

 which the axis may have in which there will be no 

 centre of percussion. For example, there is no 

 centre of percussion when the axis is a principal 

 axis through the centre of inertia. 



CENTRE OF PRESSURE. When a plane surface 

 is immersed in a fluid at rest, and held in any 

 position, the pressures at different points of the 

 surface are perpendicular to the surface. These 

 pressures may therefore be looked upon as con- 

 stituting a system of parallel forces whose resultant 

 is the whole pressure. The point at which this 

 n->. ul tan t acts is called the centre of pressure, and 

 may be defined as the point at which the direction of 

 the single force which is equivalent to the fluid 

 pressures on the plane surface meets the surface. 

 The resultant action of fluids on a curved surface is 

 not always reducible to a single force. The defini- 



tion given above its therefore, limited to plane 

 surfaces. In the case of a heavy fluid it i clear 

 that the centre of prewture of a horizontal area 

 corresponds with the centre of gravity. When, 

 however, the plane is inclined at any angle to the 

 surface of the fluid, the pressure i* not the same at 

 all points, being greater as the depth increases; 

 since in the same liquid the pressure varies with 

 the depth. In general, the centre of pressure will 

 be below the centre of gravity. The determination 

 of the centre of pressure requires the use of the 

 integral calculus, but special cases may be treated 

 by ordinary algebra. In the case of a parallelogram, 

 one edge of which is in the surface of the fluid, the 

 centre of pressure is at a distance of one-third up 

 the middle line from the base. In the case of a 

 triangle, having one side in the surface of the fluid, 

 the centre of pressure is at the middle point of the 

 median corresponding to the vertex immersed ; 

 while in the case of a triangle, with its apex in the 

 surface, and the base horizontal, the centre of 

 pressure is on the median corresponding to the 

 vertex and at a distance of three-fourths of the 

 median from the vertex. 



CENTRE OF BUOYANCY. The pressures which 

 act on every point of a surface immersed in a fluid 

 can be resolved into horizontal and vertical com- 

 ponents. The former balance one another. The 

 resultant pressure must therefore be vertical ; and, 

 as the pressure increases with the depth, it is clear 

 that the upward pressures must be greater than the 

 downward. Hence the resultant pressure on an 

 immersed body must be a force acting vertically 

 upwards. Now it is easily shown that the magni- 

 tude of this pressure is equal to the weight of the 

 fluid displaced. The point in the displaced fluid 

 at which the resultant vertical pressure may be 

 supposed to act is called the centre of buoyancy, or 

 centre of displacement. Hence, we see that when a 

 body floats in a fluid, it is kept at rest by two 

 forces, the weight of the body acting downwards 

 through its centre of gravity, and the weight of the 

 fluid acting vertically upwards through its centre 

 of gravity, or centre of buoyancy. The relative 

 positions of the centre of gravity and the centre of 

 buoyancy have an important bearing on the safety 

 of ships at sea. If the centre of buoyancy be 

 above the centre of gravity, the equilibrium is 

 stable ; in other words, if the ship is displaced, it 

 will tend to return to its original position. If, on 

 the other hand, the centre of buoyancy be below 

 the centre of gravity, the equilibrium will generally 

 be unstable, although a body may float in stable 

 equilibrium even if the centre of buoyancy be 

 below the centre of gravity, as is explained in the 

 article HYDROSTATICS. 



CENTRAL FORCES. Central forces are forces 

 whose action is to cause a moving body to tend 

 towards a fixed point called the centre of force. 

 By Newton's first law of motion we know that 

 'every body continues in its state of rest or of 

 uniform motion in a straight line, except in so far 

 as it is compelled by forces to change that state.' 

 From this we learn that, if the speed of a body 

 changes, or if the line of motion be not straight, 

 whether the speed be unaltered or not, some force 

 must l>e acting. In the latter case the forces acting 

 are called central forces. The doctrine of central 

 forces considers the paths which bodies will describe 

 round centres of force, and the varying velocity 

 with which they will pass along these paths. It 

 investigates the law or the force in order that a 

 given curve may be described, and many other 

 problems which can only be solved by mathematical 

 methods. Gravity affords the simplest illustration 

 of a central force. If a stone be slung from a string, 

 gravity deflects it from the rectilinear path which 

 it would otherwise pursue, and makes it move in a 



