tities, with their respective weights, will be equal to | A famillar example of a lever of this kind is shown each other; that is, the effect of 12 .cs. moving through under PARING-KNIFE, Fig. 1591, the hook at the end 1 inch, or of 6 lbs. moving through 2 inches of space, being the fulcrum; the wood to be cut is placed is the same. And although we out the consideration under it, and is the load or resistance to be moved or of motions, and deal only with pressures, the same overcome; and the power is the hand of the workman principle applies to them. Any 2 pressures, however at the extremity of the blade. A wheel-barrow is unequal (a pressure of 1b. and cae of 1,000 lbs., also a lever of the second kind, the wheel being the for instance), will balance each other, if they are so fulcrum, the contents of the barrow, the weight, and applied that the motion of the first through 1,000 the man wheeling it, the power. inches would be necessarily accompanied by a motion of the second through one inch, and rice ters. Any means by which this connexion between the 2 pressures is effected, is called a machine.

A

Fig. 2029.

or rope being the efficient agent; the real mechanical | plane: it is regarded in mechanical science as a peradvantage is founded on the fact, that the cord must fectly hard, smooth, inflexible surface, inclined obundergo the same tension in every part of its length. liquely to the weight or resistance. The line a C, Pulleys are fixed or mocable, according as their blocks Fig. 2029, is called the length of the inclined plane or frames are fixed or not. In Fig. 2027 the power BC its height, and AB its base. A heavy body & and the load are equal, whether the rope pass over placed upon it will act in the vertical direction G v of a fixed pulley or a fixed ring. In such a case, as a line passing through its centre of gravity G. This already observed, there is no mechanical advantage, line G v may be made the diagonal of a parallelogram but only a convenience in being able to apply the GwVX, so that if power in any required direction. In Fig. 2028, the GV represent the weight w is equal to 4 times the magnitude and direcpower P. Now, the rope must have tion of the weight, the same tension everywhere through- it may be resolved out its length, or the system would into the two forces not be in equilibrium, and in order to represented in direcbe in equilibrium the tension must be tion and magnitude equal to the power; the power P is by G w and GX, one of which is parallel, and the supported by the tension of that part other perpendicular to the plane: hence the pressure of the rope situated between c and GV is equivalent to two other pressures, Gw and P, and as the tension is everywhere GX, the former of which, aw, is destroyed by the equal, it follows that the 4 portions resistance of the plane; and the latter, G x, only acts of the rope which connect the 2 so as to cause the descent of the body down the pulleys, are each adequate to the plane. Now GX is to VG as BC is to AB; that is support of the power, and calling this to say, a weight placed upon an inclined plane is pro1 lb. then w will be 4 lbs. In systems pelled down the plane by a force bearing such proporof pulleys with one rope and one tion to the weight as the height of any section of the movable block, the load is as many plane bears to its length. If, therefore, it were times the power as there are different required to draw the heavy body G up the plane, any Fig. 2028. parts of the rope engaged in support-pressure in the direction x G exceeding GX, and the ing the movable block; and in general when the friction, would be sufficient to do so; and any pressure power acts downwards, the number of pulleys required in the same direction which with the friction equals is equal to the number of times that the power is to GX, would hold the weight in equilibrium. be concentrated; but when the power acts upwards, one pulley may be dispensed with, since, in Fig. 2028, the power P may be applied to pull up the cord a, without the intervention of the fixed pulley c, which adds nothing to the mechanical effect.

W

In applying the dynamic principle of virtual velocitics to the pulley, it will be found that whatever is gained in force is lost in velocity: the ascent of the weight is as many times less than the descent of the power, as the weight itself is greater than the power. Thus, in Fig. 2028, if the power be 1lb. and the weight 4 lbs., and it be required to raise the weight 1 foot, the power must descend through 4 feet; for, in order to raise the movable block 1 foot, each of the 4 portions of cord by which it hangs must be shortened 1 foot; but as they all form parts of one continued cord, this must altogether be shortened 4 feet, or in other words, 4 feet of cord must pass out from the system between the blocks. Thus, no power is gained by the pulley: its sole advantage is to enable us to economize power, and expend it gradually. In raising a weight of 50 lbs. 1 foot high, the expenditure of power is obviously the same, whether we accomplish the task by raising 1lb. through 50 feet, or 50 separate pounds through 1 foot; and in the pulley or any other machine, a weight of 50 lbs. cannot be raised to a given height with a less expenditure of power than is required to raise 100 lbs. half that height, or 1 lb. 50 times that height.

The fourth so-called mechanical power is the inclined

Now to test this case of the inclined plane by the principle of virtual velocities, let the weight & be at the foot of the plane, and the power P at the top; then let P descend in the direction C B, until G arrive at the top of the plane. P will have descended through a depth equal to the length of the plane, while G will have ascended through a depth equal to its height; hence the perpendicular spaces through which the weight and the power move in the same time are in the proportion of their velocities. The proportion of the weight to the power is that of the length to the height: hence the power and the weight are reciprocally as their virtual velocities. P x the space through which it moves=wx the space through which it moves. Hence, if the height of the plane be 2 feet, and its length 50 feet, P will descend 50 feet, while w is raised 2 feet in vertical height : and accordingly P must beth of the weight of w, in order to effect this; or rather greater than this, or account of friction.

When the inclined plane is movable, it is termed the wedge, and has been called the fifth mechanical power. In its simplest form, as used for raising weights, by thrusting an inclined plane under the load, instead of lifting a load by moving it along an inclined plane, the theory is this:-the moving power must bear to the resistance moved the ratio which the height of the plane bears to its base, and not as in the fixed plane, Fig. 2029, the ratio of the height to the length. In the fixed plane, therefore, the power

always balances a load greater than itself, however | omitted; and this, in the case of the screw, is very steep the slope may be; but in the wedge the power great, and is generally sufficient by itself, as in the and the load will be equal, if the slope be 45°; and wedge, to balance the longitudinal force, without any if steeper than this, the power must be greater than assistance at P. Indeed, it is seldom that any amount the load. The wedge, however, is commonly used of longitudinal force is sufficient to turn the screw, for separating two surfaces that are pressed together for the threads would be destroyed rather than turn. by some force which constitutes the resistance; and The condition of equilibrium in the screw is, that in such case it must be regarded as a double wedge, the power, multiplied by the circumference which it or two inclined planes joined base to base. Such a describes, is equal to the weight or resistance, multiwedge is commonly used for cleaving timber, in which plied by the distance through which the screw or nut case the power acts by percussion. But when re- can move longitudinally during one turn; that is, the garded as a simple machine, the same rule is applied distance between the centres, or other corresponding to it as to other simple machines, the force acting on parts of two contiguous threads, or rather turns of dB the wedge being considered to move the same thread, which distance (which is twice a B, through its length dc, Fig. 2030, Fig. 2031, in double-threaded screws) is called the while the resistance yields to the pitch of the screw; or the power: the weight :: the extent of its breadth A B. The force pitch the circumference described by the power; of percussion differs so entirely from which agrees with the principle of virtual velocities. continued forces, that it admits of no numerical comparison with them; that is, it is not possible to define the proportion between a blow and a Fig. 2030. pressure. Hence the theory of the wedge is of very little practical value; and indeed its most valuable property, the friction between its surfaces and the substance which they divide, as in the case of nails used for binding substances together, is omitted in theory.

V

The mechanical efficacy of the screw may be increased, either by causing the power to move through a greater space by increasing the length of the lever, or secondly, by increasing the number of turns of the thread. For if, in the above example, the pitch were

instead of an inch, the other conditions remaining the same, the efficacy of the machine would be doubled, and the power of 1 lb. would sustain 240 lbs. instead of 120 lbs.

Such is a very brief notice of the mechanical powers. For a more extended notice of them and their applications, we must refer to other works, such as those mentioned below,' the first of which will also introduce the student to the study of dynamics, which can scarcely be entered upon in this place; nor, indeed, will such details be expected of us, in a work devoted to the useful arts and manufactures, rather than to the principles of science. But knowing how greatly the success in practice, and the appreciation in study, of manufacturing processes depend on the thorough knowledge and skilful application of scientific principles, we have been induced to insert a few short articles on those principles, in order that the reader may be induced to study them more fully in other works, and adopt them as a basis in working his own peculiar pursuit, or in studying manufacturing details.

The sixth mechanical power, the screw, is another variety of the movable inclined plane. We have seen, under SCREW, that the thread may be formed by wrapping an inclined plane round the surface of a cylinder; and in its application as a simple machine the power is usually transmitted by causing the screw to move through an internal screw or nut N, Fig. 2031, and the power may be applied either to turn the nut while the screw is prevented from moving, or to turn the screw while the nut remains fixed. Neither effect takes place without producing a Fig. 2031. longitudinal motion of one or the other, whichever meets with least longitudinal resistance; but this resistance may exceed the turning power in the proportion that the revolving motion exceeds the longitudinal motion. Thus power is gained at the expense of motion, as in all other cases where power appears to be increased. But in the application of the screw, shown in Fig. 2031, a compound machine, consisting of the lever and the screw, is produced. The power is applied to the end of the for the subject of statics and dynamics, the Editor begs to refer lever at P, while the weight or pressure w is sustained by the screw, as in the common screw-press. Supposing the distance, A B, between any two threads of the screw to be half an inch, and the circumference of the circle described by turning round the end of the lever p to be 5 feet, or 120 half-inches, then a force or pressure of 11b. at P would sustain 120 lbs. at w. In such an example all consideration of friction is

In the science of statics, force is regarded simply as that which is necessary to oppose or balance force. In dynamics, force is regarded as the cause of change of motion. Mechanical forces are considered as motions actually produced, or tending to be produced, without any reference to the nature of

(1) For a more extended notice of the mechanical powers, and

to "Rudimentary Mechanics." In conjunction with this work, he would recommend the study of Mr. Baker's "Principles and Practice of Statics and Dynamics," with examples wrought out by common arithmetic; and also "Elements of Mechanism," by the same writer. In this work are elucidated the scientific prin

ciples of the practical construction of machines; while in Mr. Law's "Rudiments of Civil Engineering," with a continuation by Mr. Burnell, the most important applications of statics to the equilibrium of fixed structures are treated of. All the abovenamed works are in Mr. Weale's cheap and useful Rudimentary Series.

the force, or its generating cause. Hence two forces, | on arriving there, its velocity is proportional to the which impart to the same body the same degree square root of the vertical height which it has deof speed in the same direction, are regarded as scended; 4th, that if it start from 0 with this same identical, whether they originate from animal power, a weight descending by its gravity, the impact of a heavy body, or the elasticity of steam, &c. Force is not required for the maintenance of motion, but only for its change, that is, for producing, first, a change of state from rest to motion, or from motion to rest; secondly, a change in the velocity of motion either by accelerating or retarding it; or thirdly, a change in its direction, by deflecting it upwards, downwards, to the right or to the left. And since matter is inert, that is, has no tendency either to rest or motion, a body impressed with that motion must persist in that motion, in a straight line, and with uniform velocity, for ever, unless some new force, (such as friction, resistance of the air, &c.) act upon it, either to change its state, its direction, or its velocity; for it cannot of itself change either its state of rest or its state of motion, its velocity or its direction.

We may thus regard as being in equilibrium, not only such bodies as are at rest, but such also as are performing uniform rectilinear motion; for it is only while their velocity or direction is changing, that is, while they are being accelerated, retarded, or are moving in a curve, that the forces acting on them can be unbalanced, or can produce a resultant pressure; and as long as this pressure remains unbalanced, the motion will continue changing in velocity, or direction, or both; whenever it becomes straight and uniform, the resultant of all the forces acting on the body=0, or it is not subject to any unbalanced force. The dynamical effect of force being then a change in motion, a continued force must produce a continuous change, whether in velocity or direction. The simpler effect of a sudden change of velocity, or an angular deflection, can only be produced by an instantaneous exertion of force, or an impact, as it is called. The force of any moving body, or its momentum, or quantity of motion, is the product of its mass by its velocity. Now it is established, 1. that when equal masses are in motion, their forces are proportional to their velocities; 2. that when the velocities are equal, the forces are proportional to the masses or quantities of matter; 3. that when neither the masses nor the velocities are equal, the forces are in the proportion of both taken jointly, that is, the proportion of their products.

These theorems may be illustrated by two balls of clay, AB, Fig. 2032, or of some other comparatively non-elastic substance, suspended from c by strings, so as to hang in contact in the middle of a graduated arc DE. The arc should be cycloid, and divided not into equal parts, but as shown in the figure, so that the numbers 1, 2, 3, &c. may be proportional to the perpendicular heights above the level of the point 0. Now it has been proved in the case of gravity, 1st, that when a ball thus suspended is let fall from any point of the arc, its velocity will be the same whatever be its mass; 2d, that this velocity will continually increase until it reaches the point 0; 3d, that

B

Fig. 2032.

velocity, it will ascend to the same height from which it must have fallen to have acquired that velocity and no higher, because its velocity is by the action of gravity constantly diminished, until at this precise. height it is destroyed. The velocities of the balls, therefore, at the moment of their arrival at, or departing from the point 0, may be exactly measured by noting the divisions on the scale from which they have descended, or to which they ascend, provided the 4th division be reckoned 2, the 9th division 3, and so on. Now, suppose these balls to be equal in mass, and to be moved in opposite directions, a towards D, and B towards E, and then allowed to fall at the same moment; if the balls fall through equal arcs, they will of course impinge upon each other with equal velocities, and each will destroy the force of the other and remain at rest, for equal masses having equal velocities, must have equal forces. If, however, the ball a be double the weight or mass of B, and a be raised towards D, as far as the first division, and towards E as far as the fourth division; when allowed to descend at such an interval of time as to bring them both at once to the point 0, their velocities will be as 1 /4, or as 1: 2; but as their masses are as 2 to 1, their forces will be as 2 x 1 to 1 x 2, or equal. Accordingly, after impact, these two bodies will remain at rest, because the equal and opposite forces have destroyed each other. So also if the masses of the balls be inversely as their velocities, their forces will be equal, and they will remain at rest after impact. If we now suppose A and B, Fig. 2032, to be unequal in mass, but equal in velocity; that a is twice the size of B, and that each is allowed to descend from the same height at D and E; if the velocity of each be called 6, the quantity of motion in a may be expressed by 2 × 6=12, while that in B will be 1 x 66. After impact the 6 parts of motion in B will destroy 6 parts of the 12 in A, leaving only 6 parts in both bodies. Now, the combined mass of both being = 3, and their momentum

=

:

6, their velocity must be 6 divided by 3, or = 2; so that both will move on together with a velocity of 2, that is, d of their velocity before impact, and this will carry them th the height from which they descended. Similar results will be obtained when the