to use a flat rod, the weight of which is also = m. Let the bottom of the box A be placed on a level with O on the scale, the whole mass being as described above = 63m, perfectly balanced in equilibrio. Now let the rod, the weight of which =m, be placed on the upper surface of A; this body will descend along the scale precisely in the same manner as when the moving force was applied in the form of a circular weight. Suppose the mass A, fig. 6, to have descended by constant acceleration of force of m, for any given time, or through a given space: let a circular frame be so affixed to the scale, contiguous to which the weight descends, that A may pass centrally through it, and that this circular frame may intercept the rod m, by which the body A has been accelerated from quiescence. After the moving force m has been intercepted at the end of the given space or time, there will be no force operating on any part of the system which can accelerate or retard its motion: this being the case, the weight A, the instant after m has been removed, must proceed uniformly with the velocity which it had acquired that instant in the subsequent part of its descent, the velocity, being uniform, will be measured by the space described in any convenient number of seconds.

Other uses of the instrument it is needless to describe particularly, but it may not be improper to mention some of them; such as the experimental estimation of the velocities communicated by the impact of bodies elastic and non-elastic; the quantity of resistance opposed by fluids, as well as for various other purposes. These uses we shall not insist on; but the properties of retarded motion being a part of the present subject, it may be necessary to shew in what manner the motion of bodies resisted by constant forces are reduced to experiment by means of the instrument above described, with as great ease and precision as the properties of bodies uniformly accelerated. A single instance will be sufficient: Thus, suppose the mass contained in the weights A and B, fig. 6, and the wheels to be 61m, when perfectly in equilibrio; let a circular weight m be applied to B, and let two long weights or rods, each m, be applied to a, then will A descend by the action of the moving force m, the mass moved being 64m: suppose that when it has described any given space by constant acceleration, the two rods m are intercepted by the circular frame above described, while a is descending through it, the velocity acquired by that descent is known; and when the two rods are intercepted, the weight A will begin to move on with the velocity acquired, being now retarded by the constant force m; and since the mass moved is 62m, it follows that the force of retardation will be part of

T

that force whereby gravity retards bodies thrown perpendicularly upwards. The weight A will therefore proceed along the graduated scale in its descent with an uniformly retarded motion, and the space described, times of motion, and velocities destroyed by the resisting force, will be subject to the same measures as in the examples of accelerated motion above described.

In the foregoing descriptions, two suppositions have been assumed, neither of which is mathematically true: but it may be easily shewn that they are so in a physical sense: the errors occasioned by them in practice being insensible.

1. The force which communicates motion to the system has been assumed constant; which will be true only on a supposition that the line, at the extremities of which the weights A and B, fig. 1. are affixed, is without weight. In order to make it evident that the line's weight and inertia are of no sensible effect, let a case be referred to, wherein the body a descends through 48 inches from rest by the action of the moving force m, when the mass moved is 64m; the time wherein a describes 48 inches is increased by the effects of the line's weight by no more thanth parts of a second; the time of descent being S-9896 seconds, when the string's weight is not considered, and the time when the string's weight is taken into account = 4'0208 seconds; the difference between which is wholly insensible by observation.

2. The bodies have also been supposed to move in vacuo, whereas the air's resistance will have some effect in retarding their motion: but as the greatest velocity communicated in these experiments cannot much exceed that of about 26 inches in a second (suppose the limit 26-2845), and the cylindrical boxes being about 14 inches in diameter, the air's resistance can never increase the time of descent in so great proportion as that of 240:241; its effects therefore will be insensible in experiment.

The effects of friction are almost wholly removed by the friction-wheels; for when the surfaces are well polished and free from dust, &c. if the weights A and B be balanced in perfect equilibrio, and the whole mass consists of 63m, according to the example already described, a weight of 14 grain, or at most 2 grains, being added either to A or B, will communicate motion to the whole; which shews that the effects of friction will not be so great as a weight of 14 or 2 grains. In some cases, however, especially in experiments relating to retarded motion, the effects of friction become sensible; but may be very readily and exactly removed by adding a small weight of 1.5 or 2 grains to the descending body, taking care that the weight added is such as is in the least degree smaller than that which is just

sufficient to set the whole in motion, when A and B are equal, and balance each other before the moving force is applied. (Atwood on Motion, p. 316.)

BALANCE, as distinguished from the Steelyard, is a lever with equal arms, whose fulcrum or centre of motion is situated immediately above the centre of gravity of the beam, when horizontal: it is used chiefly in determining the equality or difference in the weights of different bodies.

Some remarks on the nature of the balance were made when we treated of the lever in the first volume; where also we shewed how to correct the deception occasioned by a false balance: in addition to what was there stated, we shall now present a few such observations as may be most serviceable in directing the accurate construction of this instrument.

1. The axis of motion of the balance should be above the centre of gravity of the beam.

2. A slender index, or tongue (as it is called), passes through the centres of gravity and motion of the beam, perpendicular to its axis: by this index the horizontal position of the beam, when loaded, in the comparison of weights, is determined.

3. When the balance unloaded is quiescent, and therefore horizontal, if the index which passes through the fulcrum be directed to any fixed point; and again when the balance is reversed, it be directed to the same fixed point; it is in the right line which joins the centre of gravity and the fulcrum.

By this means the position of the index is adjusted.

4. The perpendicular distances of the points of application of the weights to be compared, from the right line which joins the centres of gravity and of motion, should be equal, that is, the arms of the balance ought to be equal.

5. The points of application from which the weights are suspended, should be in the same right line perpendicular to the line joining the centres of gravity and of motion.

6. The nearer the right line joining the points of application is to the centre of motion, the larger vibrations of the balance, and a more sensible effect, will be produced.

7. If the centre of motion be situated below the line joining the points of application, the beam, when loaded with equal weights, will overset, rest in any position, or equilibrate according to the weight.

8. When two given weights, suspended from the arms of a balance, are in equilibrio, if these weights when transferred to the opposite scales be still in equilibrio, the arms of the balance are equal.

9. The various adjustments of the balance are these: 1st. Equal weights are readily found, whatever be the state of the

balance; for, if they reduce the beam to the same position, when successively applied to the same arm, they must be equal: then if these equal weights transposed do not disturb the position of the beam, the arins are equal. 2dly. If unequal weights transposed produce equal deflections of the beam, the points of suspension are in the same right lines, perpendicular to that which joins the centre of gravity and motion; and therefore the line joining these points will be horizontal when the beam hangs freely. 3dly. Let the index be directed to any fixed point, then the beam being reversed, if it still pass through the same point, the index is perpendicular to the axis of the beam.

10. The equilibrium of the balance will be effected by the tongue, unless it be continued below the centre of motion, so that the momenta on both sides may be equal and opposite.

11. Minute differences of weight are rendered more discernible by diminishing the friction upon the axis, as by suspending it in a fork with springs, &c.

Indeed when balances are required for accurate philosophical purposes, much caution is requisite in the various parts of the construction, and many peculiar contrivances have been adopted: some of the best of these are given in the following article.

Hydrostatic BALANCE, is an instrument contrived to determine accurately the specific gravity of both solid and fluid bodies. One of the most ingenious forms of this balance is exhibited in fig. 5. pl. VI. where vCG is the stand or pillar, which is to be fixed in a table. From the top A hangs by two silk strings the horizontal bar BB, from which is suspended by a ring i the fine beam of a balance b: which is prevented from descending too low on either side by the gently springing piece txyz, fixed on the support M. The harness is annulated at o, to shew distinctly the perpendicular position of the examen, by the small pointed index fixed above it.

The strings by which the balance is suspended, passing over two pulleys, one on each side the piece at A, go down to the bottom on the other side, and are hung over the hook at v; which hook, by means of a screw P, is moveable to about the distance of an inch and a quarter backward and forward, and therefore the balance may be raised or depressed so much. But if a greater elevation or depression be required, the sliding piece which carries the screw P, is readily removed to any part of the square brass rod VK, and fixed by means of a screw.

The notion of the balance being thus adjusted, the rest of the apparatus is as follows: HH is a small board fixed upon the piece D, under the scales d and e, and is moveable up and down in a low slit in the pillar above c, and fastened at any part by a screw behind. From the point in the middle of the bottom of

each scale bangs, by a fine hook, a brass wire ad, and ac: these pass through holes m, m, in the table. To the wire ad is suspended a curious cylindric wire rs, perforated at each end for that purpose: this wire rs is covered with paper graduated by equal divisions, and is about five inches long.

In the corner of the board at E is fixed a brass tube, on which a round wire hl is so adapted as to move neither too tight nor too freely, by its flat head 1. Upon the lower part of this moves another tube g, which has sufficient friction to make it remain in any position required: to this is fixed an index T, › moving horizontally when the wire hl is turned about, and may therefore be easily set to the graduated wire rs. From the

lower end of the wire rs hangs a weight L; and from that a wire pn, with a small brass ball g about one-fourth of an inch diameter. On the other side from the wire ac hangs a large glass bubble R, by a horse-hair.

Now, let us suppose the weight L taken away, and the wire pn suspended from s: and on the other side let the bubble R be taken away, and a weight, as F, suspended at c in its room. This weight F we suppose to be sufficient to keep the several parts hanging from the other scale in equilibrio; at the same time that the middle point of the wire pn is at the surface of the water in the vessel o. The wire pn is to be of such a size that the length of one inch shall weigh four grains.

Now it is evident, since brass is about eight times heavier than water, that for every inch the wire sinks in the water, it will become half a grain lighter; and half a grain heavier for every inch it rises out of the water: consequently, by sinking two inches below the middle point, or rising two inches above it, the wire will become in effect one grain lighter or heavier. If, therefore, when the middle point is at the surface of the water in equilibrio, the index T be set to the middle point of the graduated wire rs, and the distance of r and of s from the index be each reckoned to contain 100 equal parts; then, if in weighing bodies the weight is required to the hundredth_part of a grain, it may be easily obtained by proceeding thus:-Let the body to be weighed be placed in the scale e; and let this be so determined that one grain more shall be too much, and one grain less too little. Then the balance being moved gently up or down by the screw p till the equilibrium be nicely shewn at o, if the index T be at the middle point of the wire rs, it shews. that the weights put into the scale e are just equal to the weight of the body.

But if the index T stand nearer to r than to s, as suppose 36 of the 100 parts, it shews the number of grains in the scale e were less than equal to the weight of the body in scale d, by 36

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