CENTRE OF PARALLEL FORCES.

61

entire mass at once, requiring, however, ten times the number of minutes in which he raised the 200 pounds.

86. Thus it is, in limine, obvious that we exchange time for power in using simple machines; and this is true with all the apparatus to which that term has been applied. The simple machines may be divided into three species:

1, The lever,

2, The pulley,

3, The inclined plane,

the theoretical properties and peculiarities of which, with their chief modifications, we will now briefly describe.

1. THE LEVER.

87. The lever, theoretically considered, is an inflexible rod, destitute of weight, perfectly straight, and moving without friction on a fulcrum or support, corresponding to the centre of motion.

Referring to what has been already stated with regard to the centre of parallel forces (31), we find that whenever a series of forces, perfectly parallel in direction, act upon a mass, they may be replaced by one force, which may be considered as their centre, or resultant. The following are the chief properties of this resultant. A. It is equal to the sum of the forces if they are all exerted in one direction, and to their difference if exerted in opposite directions. B. It is parallel to the forces of which it is the resultant. c. It is placed at a certain point & in such a manner that the distances GC, GA

XA

are in the inverse ratio of the forces CD and AB; and it is

this point which is termed the centre of parallel forces. D. This point remains the same when the forces change their absolute directions, providing they remain parallel; for if the above forces act in the direction CN, AM, instead of CD, AB, the centre will still be G, because they have not changed their intensity, and their power is in the inverse ratio of

GC, GA.

W

F

B

88. Let the bar AB be balanced on a fulcrum in its centre, it will, of course, remain in equilibrium: suspend from the end A a weight; immediately this mass is added, the centre of gravity will no longer be over the fulcrum but near the end a, and this power, being unsupported, will be drawn down by the attraction of the earth. Then suspend from the end B a similar weight P; the centre of gravity of the whole, as the bar and the weight may be considered as forming one mass, will be once more over the fulcrum, and the whole will be supported in equilibrio. If, instead of p being equal it be one fourth less, then the centre of gravity will be no longer over the fulcrum, but nearer a, as at c, and the earth drawing this down will cause w to preponderate; nor can the state of equilibrium be obtained, unless weights be added to P, until it becomes equal to w. This form of lever is evidently nothing more than the ordinary balance; and we find that when the weights Pw are equal, the length of the lever on both sides of F must be equal; these portions FA, FB are termed the arms of the lever.

to w,

EQUILIBRIUM OF LEVERS.

63

89. If the weights pw remain equal, but the position of the fulcrum be changed to F, then equilibrium will no longer occur, for the arm FB will preponderate; this necessarily occurs, for the centre of gravity, c, being no longer supported,

the force of gravitation draws it towards the earth. But let AB be graduated into four equal portions AF, FC, CE, EB, and let the point r be supported: it is obvious that things remaining as they were, a state of equilibrium can only be obtained by throwing the centre of gravity over the point of support, or fulcrum. This necessarily occurs by diminishing the weight P; and if this be done gradually, the centre of gravity will be found to approach F in proportion as we diminish P; and when this is equal to one third of w, the centre of gravity will be exactly over the fulcrum, and equilibrium obtained. Now, as P is equal to 1, and w equal to 3, whilst the arm to which P is attached is thrice as long as that to which w is suspended, we deduce the general law that the power è and weight w are always in equilibrio, when they are to each other in the inverse ratio of the arms of the lever, to which they are attached. Consequently, any weights will keep each other in equilibrio, on the arms of a straight lever, when the products arising from multiplying each weight by its distance from the fulcrum are equal on each side of the fulcrum; and, as in the above example, P=1 and w = 3;

=

whilst AF1 and FB 3, it follows that (w X AF)=(PX FB) and both being equal to 3, equilibrium necessarily results. Of the lever with unequal arms, the common steelyard, used for weighing heavy weights, is a good example.

90. As a smaller weight is made to counterbalance a greater, by lengthening one of the arms of the lever when arranged as a balance, it frequently tempts the dishonest vender to thus alter his scales, to cheat the unsuspicious buyer; of course this is readily detected, by weighing the substance to be purchased first in one scale-pan, and then in the other. If the balance be correct it will weigh the same in both; but if incorrect, its apparent weight will be different in each scale-pan. To determine the true weight of a substance with such a balance, weigh it first in one scale-pan, then in the other; multiply these two weights together, and take the square root of the product. Thus, if a substance weighed 253 pounds in one scale and 251 in the other, √251 x 253 252 pounds, the true weight.

91. In the lever, with unequal arms, we see that the velocity with which its extremities move is very different. Let the line AFC represent a lever, turning on the fulcrum F as on a centre, and suppose weights to be attached to the end c, and a force applied to a sufficient to move the weight,

then, whilst the latter describes the arc DCB, the force applied will pass through the arc EG, the length of each arc being in the inverse ratio of the force applied to each, and in the direct ratio of the arms of the lever. We see also that a small weight attached to A and passing through the space

LEVERS IN ACTION.

65

EAG, will, by its velocity, generate a degree of momentum sufficient to counterbalance a much heavier weight attached to c, moving, as the latter necessarily must, with less velocity; for, as from the conditions of this lever, the arcs DB, EG must be described in equal times, and as EG is much larger than DB, it is obvious that the end must move with as much greater velocity than c as the arc EG is larger than DB. As the momentum of a body is equal to its quantity of matter, multiplied by its velocity or quantity of motion (66); we learn that equilibrium must occur in a lever, when the weights at either end, multiplied by the velocities with which they move, are equal to each other. From this reasoning, we also become convinced of the truth of the statement we set out with, that the application of the mechanical powers is an exchange of time for power.

92. These levers have been termed levers of the first class, and are characterized by having the fulcrum at some point between the power applied and the resistance to be overcome. Those levers in which the fulcrum is applied at one end, and the resistance at an intermediate point, have been termed levers of the second class; whilst those in which the power is applied between the fulcrum and resistance are placed in a third class. The only real distinction that it is necessary to make, is between levers in which the fulcrum is between the force applied and the resistance, and those in which the fulcrum is at one end. The proportion between the forces to produce equilibrium is expressed in the same terms in each case, the great difference between them being that when the fulcrum is central, as in the lever already adverted to, the pressure upon it is equal to the sum of the forces applied, and to their difference, when the fulcrum is terminal.

93. That modification of the lever in which the force is applied between the fulcrum and resistance, is not very frequently met with; indeed, on account of the mechanical disadvantage in which the force is necessarily exerted, it is