Measure of Force-Momentum.

55

matter were all moving at ten feet a second, this mass would possess one hundred times the quantity of motion of the first.

A hundredweight moving at any given rate possesses double the quantity of motion of a fifty-six pound weight moving at the same

rate.

A railway train of forty railway carriages moving at the speed of thirty miles an hour, has six times the momentum of a train of twenty carriages moving at the speed of ten miles an hour.

So estimated, the motion of a sixty-two-pound cannon ball, moving a foot and a half per second, is the same (nearly) as that of an ounce bullet when it leaves the rifle.

A man's force will move a small skiff quickly, a loaded bargc very slowly, and a large ship in a degree scarcely to be perceived. Yet in each case the quantity of motion may be the same, and a true measure of the effort exerted.

By experiment it is found that if an inelastic ball of soft clay of one pound, suspended by a cord as a pendulum, be made to strike with a velocity of ten feet per second against another of nine pounds, suspended in the same way, but at rest, the two will start together at the rate of nearly one foot per second, the original quantity of motion being then diffused through ten times the original mass, and therefore exhibiting only one-tenth of the velocity.

A block of wood, floating against a man's leg with moderate velocity, would be little felt; but a loaded barge, coming at the same rate, and pressing it against the quay, might break the bones; and a large ship, moving slowly, would crush his body against any fixed obstacle.

Two huge floating icebergs meeting will crush a man-of-war as easily as we crush an egg-shell between the fingers.

127. In these instances we see the power of heavy masses in motion; but enormous momentum may be obtained from light masses, if moving with very great velocity.

Air, which is so light and gentle when slowly moved, exhibits tremendous force when blowing a hurricane; pulling down houses like so many hay-ricks, and uptearing trees as if they were but weeds.

The waves of the sea in a storm often possess a most irresistible violence, tossing about the largest iron-clad as if it were a lump of cork.

A most remarkable illustration of this fact appears in the irre

56

The Parallelogram of Forces.

sistible force of heat. It is an inconceivably rapid quivering motion of the minute molecules or atoms of coal, which, communicated to those of water vapour or steam, excites the ponderous engines that push our trains and ships along, and may indeed be said to move the world.

128. As the resulting velocity of two bodies, moving in the same line so as to meet, is the sum of their velocities, so the whole quantity of motion they possess is the sum of their momenta, and this is the measure of the shock they will produce.

If two persons running or skating strike against each other, the clash is much more violent than if one were standing still, and the result may be dangerous or even fatal.

The meeting fists of boxers not unfrequently dislocate or break bones.

When two ships in opposite courses meet at sea, the destruction may be as complete to both as if each, with a double velocity, had struck on a rock.

If a railway train dash into another moving in the opposite direction, their resultant momentum is exhibited with terrific effect; and the destruction to a passenger train is all the more disastrous if the opposing one be a long, heavy goods train.

"Composition and Resolution of Forces.

129. Force, then, being estimated by quantity of motion or mass and velocity conjointly, the rules of composition and resolution given for velocities apply equally to forces.

Two or more forces may act on a body at the same instant, as, for example, in the throwing of a stone, the force of projection and the force of gravity. Each force is equivalent to a certain velocity of the mass in its direction, and the resultant force will be represented by the mass moving with the resultant of those simultaneous velocities.

Thus-(i.) If two forces act in the same line on a body-say the force of the tide and of the screw on a steamer—the resulting force will be the sum or difference of these forces, according as they agree or oppose in direction.

(ii.) If a body be influenced by two forces in different directions, these may, just like velocities, be represented by two contiguous sides of a parallelogram, and the resultant force will be represented by the diagonal of that figure, drawn from the position of the body. A billiard ball, for instance, may be struck at the same moment

Resolution of Force.

57

with two cues, so that in virtue of the one stroke it would go in the direction A B, and in virtue of the other along A C. Now, if we take in those directions A B

and AC proportional to the forces, they will, of course, be proportional to the velocities imparted to the ball per second; and since A D represents the resultant of these velocities (see

A

Fig. 8.

Art, 115), it will be also proportional to or will represent the resultant of the two forces.

We have thus precisely the same method for finding the resultant of two forces as of two velocities: and this method is known as the Parallelogram of Forces.

(iii.) So the rule for the composition of any number of forces acting simultaneously is exactly the same as that for velocities (see Art. 117); and simply, as we see, because the consideration of forces really resolves itself into that of velocities.

(iv.) Similarly, a force acting on a body so as to move it in any one direction, may be resolved or supposed broken up into two in any two desired directions.

A stroke given to a billiard ball, which will make it move in the line A D, may be supposed resolved into two co-existing strokes in the directions A B and A C (fig. 8); and their amounts will be known from the proportion of the sides and diagonal of any representative parallelogram, such as A B D C, in the same way as for velocities (Art. 118).

130. The resolution of force is very well exemplified by the action of the wind on a sailing vessel, whereby the wind blowing in one direction may cause the ship to sail in a very different one; and, what is still more curious, the same wind may waft two vessels in nearly contrary courses. Were a ship equally ready to move towards either side, as, for instance, if it were in the shape of a round tub with a sail hoisted, it would be driven just right with the wind. But it is made so as to cut through the water easily in one direction, and can only with great force be moved broadside against it. When the wind is not blowing in the course of the ship, but obliquely towards it, this will be equivalent to two winds blowing, one in the direction of its course, and another across it. By long experience the sailor learns to turn his sail-yard so that the component force in the line of the ship's course may be as great as

58

Accelerated Forces-their Measure.

possible, and that across it comparatively insignificant. The latter is counteracted by the huge volume of water that would have to be pushed against.

Different dispositions of the sails will thus give the most effective components of force for different courses; and just as a stroke with a billiard cue in precisely the same direction will make the ball go to one side or another according to where it strikes the ball, so the same wind may be driving two ships in courses right athwart each other.

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'Uniform and variable forces; and their measure."

131. Force, expressed by a moving body, will be uniform or variable according as the velocity of the body is either constant or not; and its measure will be estimated, of course, in the same way as that of the velocity. Thus we speak of uniform, accelerated, or retarded force, according as the momentum remains the same independently of time, or increases or decreases with it. As we do not meet with absolutely isolated or free matter, so all expressed force (or moving force) is more or less variable; it comes into being gradually, never instantaneously, and it dies away also by degrees.

Variable forces alone present any difficulty in their calculation, and to these therefore—as being also the most important—we shall now turn our attention. In no case is force called into existence in a moment; whether it be produced by the earth's attraction, cohesion, chemical affinity, by magnetism, electricity, or any other

means.

We shall consider the grounds for this statement in detail.

"Examples of accelerated force."

132. Even in the most impulsive communication of motion there is more or less of continued, gradual, or accelerating action.

When we strike a billiard ball, it, being not perfectly rigid, receives the moving force gradually, the blow increasing from zero at the first instant of touching, up to a maximum, and diminishing again to zero when the action is complete and the ball is on the point of leaving the cue.

The action of gunpowder on bullets, appearing so sudden, is still not an instantaneous but a gradually, though rapidly, increasing one; for we find the power of projection to depend much on the

The Force of Gravity.

59

length of the piece along which the force pursues the ball. A small fast-sailing vessel, with a single long gun, has compelled a superior essel, whose guns were shorter, to yield.

The following are examples where the continuance or cumulation of the force is more apparent :—

Savages throw, with deadly effect, poisoned arrows, by blowing them through a long smooth tube with the mere force of the breath. The boy's pea-shooter is a more harmless and familiar illustration of the same kind.

When a powerful blow is intended, the fist, or hammer, or hatchet, or club is lifted high and carried far back, that there may be time and space for accumulating greater force.

Bulls, rams, and goats, in fighting, alternately recede and then run at each other, knowing that thus they increase the shock.

A horse kicking, from the great length of his leg, and the consequent space through which he can be adding velocity to his foot, drives it at last against the object almost like a cannon-shot.

A bow-string, propelling an arrow, follows it through a considerable space, and so gives the piercing power at last produced.

The battering-rams of the ancients accumulated in them the effects of many hands and of a considerable duration of action, so as to give one powerful, sudden shock.

A boy's catapult, in like manner, owes its power to the action of the elastic force of the stretched india-rubber continued through the space of a few inches.

If the mainspring of a watch were allowed to uncoil itself freely, we should see the hands moving round at a constantly increasing rate, till the whole force was spent ; and the sole object of the mechanism is merely to regulate this force and allow it to act uniformly.

"The accelerating force of gravity."

133. Gravity is the most obvious example of accelerating force, as it is also the most important-partly from its sharing in all mechanical concerns of a practical kind, and partly from its uniformity and readiness of calculation.

Examples to show that the force of gravity is an accelerating or constantly increasing one, meet us everywhere.

A boy letting a ball drop from his hand can catch it again in the first instant, but after a little delay his hand pursues it in vain. It he throws it up and catches it, he receives a harder blow when it