MECHANICS. 248 MECHANICS. the latter, the hoop is said to 'exert a pressure' on it. In both cases the acceleration is s'/r, where s is the linear speed of the particle and r is the radius of its path, and has the direction from the moving particle toward the centre of the circle; consequently the force is imi'/r in this direction. In other words., to make a particle of mass m move at a uniform speed s in a circle of radius r requires a force acting on it directed toward the centre and with a mimcrical value tiis'/r, or, if a is the angular speed of the par- ticle, mro)'. If this force is decreased, the par- ticle will cease to move in a circle and will move farther away from the centre; if the force is re- moved at any instant — by cutting the string — the particle will continue to move with the same velocity that it has at that instant, i.e. along the tangent to the circle with a constant si)eed. This fact that, unless the force is sulTicicntly great, the particles of a rotating body will move farther away from the axis of rotation is illus- trated in many ways. The body is said to move under the action of a 'centrifugal force.' A simple pendulum is defined to be a particle of matter suspended by a long niassless string. If it swings through small angles in a vertical plane, the motion of the particle or 'bob' is prac- tically in a straight line, and is simple harmonic. Let O be the point of sus- pension of the pendulum, let OQ be its position when hanging at rest, and OP its position at any instant while it vibrates; call the angle QOP, e. There are two forces acting on the particle of mass m placed at I' : one is the tension of the string aUmg the string toward (), the other is its wi'ight ml/, vertically down. The actual motion at this instant is tangent to the circle whose radius is OP, that is, it is in the direction I'K. either up or down. The force T has no comjionent in this direction, being perpendicular to it: and that of the force mg is mg sinO (using the general for- mula for resolving a vector). Therefore the acceleration of the vibrating particle, in the di- rection Vi, the force divided by the mass, is mgainS -^— orssmfl This acceleration may be written I'Q X g Qpi or calling PQ, i, ami OP, I , g -^ If the amplitude is very small, PQ is prac- tically the path of the moving particle; and thus the motion is harmonic, in accordance with the definition of such motion; and its period, therefore, is For other illustrations of forces see Electmcitt ; M.oNETisM ; Elasticity ; Gravitation ; Cem- TKAI, FORCKS. It has been shown that, if there arc no external intluences, the centre of inertia of a system of particles or of a large body continues, if in mo- tion, to move in a straight line with a constant speed. Tliis is owing to the fact that the action and reaction of each pair of particles are eciual and opposite. If, however, there are external forces, the acceleration of the centre of inertia in any direction is the sum of the components of these forces in this direction divided by the mass of the whole system. This is equivalent to saying that the motion of the centre of inertia of a sys- tem of particles is exactly as if a single particle of the mass of the system were under the in- Uuence of the given forces. Thus if an iron beam falls from a building (without touching any- thing as it falls) the motion of its centre of inertia is like that of a falling particle — vertical — however the beam revolves. If a hammer is thrown at random into the air, its centre of inertia will describe a parabola, because that is the path of a projected particle. See Projectile. Many forces are not constant and some are abrupt, like the blow of a hammer: and in these cases it is imiiossible to measure them. Their effect is evidently to produce a sudden change in velocity; and it is measured by the total change in the linear momentum. Force itself is the rate of change of linear momentum; so if a force F acting on a particle produces a change of momentum from mv,, to mv in an interval of time (, mv — »»!'(, ^^ = 1 and thus the total change of momentum equals the product of the force and the interval of time. This product Ft is called the 'impulse' of the force, and may be measured even if both F and t are unknown. Similarly, if an impulsive force acts on a large body, the velocity of its centre of inertia will be changed from r„ to r in the direction of the force. In other words, the change of velocity of the centre of inertia, r — I'o, equals the amount of the impulse divided by the mass of the body, entirely regardless of the point of ap]>lieation of the force. The time required for a force F to change the velocity from r„ to v is The distance required for this same force F to produce this change in velocity from i-o to r in its direction is foinid by the formulse of kine- matics, which show that under a constant ac- celeration «, the distance traversed while the speed changes from .<„ to s is such that iax = S" — So'. Therefoi-e, in this case, since a = F/m, 1= p / The product For, that is, the force multiplied by a distance in its line of action, is called the 'work'; the quantity V.nix" is called the 'kinetic en<'rgy' of tr;inslation of the body whose m;iss is HI when it has the speed s. This formula is expressed in words by saying that the 'irorfc done by the force' on the body equals the in- crease in its kinetic energy of translation, pro- vided the speed is increasing, e.g. a train being set in motion. If the speed is decreasing, e.g. a train slowing up by virtue of its brakes and the resulting friction, it is said that the bo<ly loses an amount of kinetic energy of translation e(|Ual to the work it does in overcoming friction or 'iigainsl the f<jree' V. Rotation. A 'rigid body' is defined as one which is not deformed in anv way under the
