207. For military men it is a good practice to have a portable pendulum, made of painted tape with a brass bob at the end, so that the whole, except the bob, may be rolled up within a box, and the whole enclosed in a shagreen case. The tape is marked 200, 190, 180, 170, 160, &c. 80, 75, 70, 65, 60, at points, which being assumed respectively as points of suspension, the pendulum will make 200, 190, &c. down to 60 vibrations in a minute. Such a portable pendulum is highly useful in experiments relative to falling bodies, the velocity of sound, &c.

For the comparison of the times of oscillation in indefinitely small arcs of circles, in finite arcs of circles, and in cy cloidal arcs, the student may turn to probs. 13 and 14, in Practical Exercises on Forces, and prob. 42, in Promiscuous Exercises near the end of this volume.

CENTRAL FORCES.

208. Def. 1. Centripetal force is a force which tends constantly to solicit or to impel a body towards a certain fixed point or centre,

2. Centrifugal force is that by which it would recede from such a centre, were it not prevented by the centripetal force.

3. These two forces are, jointly, called central forces.

209. PROP. If a body, м, drawn continually towards a fixed point, c, by a constant force, o, and projected in a direction, MB, perpendicular to cм, describe the circumference of a circle about the centre o, the central force p, is to the weight of the body, as the altitude due to the velocity of projection, is to half the radius CM.

Letv be the velocity of projection in the tangent MB, and r the radius Cм. Independently of the action of the central force, the body would describe, along MB, during the very small time 1, a space мN = tv, and would recede from the point c by the quantity IN, which may, without

GAM

error, be regarded as equal to GM, when the arc mi is exceedingly small. If, therefore, the body, instead of moving in the tangent, were kept in the circumference by the central

force o, its operation in the time t, would (art. 130) be equal top, and at the same time = MG. But by the nature of the

Putting a for the altitude due to the velocity v, since (by

art. 154) v3 = 2ag, we have

sults.

2ag

=

: gar.

; whence there re.

Thus far, we have, in reality, considered only the unit of mass; but, if we multiply the first two terms of the above proportion by the mass of the body, the whole will still remain a correct proportion, and the general result may be thus enunciated: viz.

The centripetal force of any body, if it be free, or its centrifugal force, if it be retained to the centre c, by a thread (or otherwise), is to the weight of that body, as the height due to the velocity v, is to the half of the radius CM.

210. Hence, it appears that, so long as and r remain constant, the velocity v will be constant.

211. If both members of the equation 1 be multiplied by the mass м of the body, and we put F to represent the cen

Mv2 trifugal force of that mass, we shall have F = In like

g

manner, if r' is the centrifugal force of another body which revolves with the velocity v' in a circle whose radius is r', we shall have

212. If T and T' denote the times of revolution of the two

bodies, because v =

we have

T

T'

213. If the times of revolution are equal, we shall have

214. And, if we assume r2: T:3:73, as in the plane. tary motions, the proportion (3) will become

215. The subject of central forces is too extensive and momentous to be adequately pursued here. The student may consult the treatises of mechanics by Gregory and Poisson, and those on fluxions by Simpson, Dealtry, &c. We shall simply present in this place, one example connected with practical mechanics.

EXAM. Investigate the characteristic property of a conical pendulum applied as a regulator or governor to steam-engines, &c.

G

This contrivance will be readily comprehended from the marginal figure, where sa is a vertical shaft capable of turning freely upon the sole a. CD, CF, are two bars which move freely upon the centre c, and carry at their lower extremities two equal weights, P, q: the bars cn, cr, are united, by a proper articulation, to the bars G, H, which latter are attached to a ring, 1, capable of sliding up and down the vertical shaft, aa. When this shaft and connected apparatus are made to revolve, in virtue of the centrifugal force, the balls r, q, fly out more and more from Aa, as the rotatory velocity increases: if, on the contrary, the rotatory velocity slackens, the balls descend and approach Aa. The ring 1 ascends in the former case, descends in the latter and a lever connected with 1 may be made to correct appropriately, the energy of the moving power. Thus, in the steamengine, the ring may be made to act on the valve by which the steam is admitted into the cylinder; to augment its opening when the motion is slackening, and reciprocally diminish it when the motion is accelerated.

The construction is, often, so modified that the flying out of the balls causes the ring I to be depressed, and vice versa ; but the general principle is the same. If FQ=FI= DP =DI, then I, P, Q, are always in some one horizontal plane : but that is not essential to the construction.

Now, let t denote the time of one revolution of the shaft, x the variable horizontal distance of each ball from that shaft, as usual = 3.141593: then will the velocity of each

2xx

ball be= and (art. 209.) its centrifugal force

+x= = 4727. The balls being operated upon si

12

multaneously by the centrifugal force and the force of gravity, of which one operates horizontally, the other vertically, the resultant of the two forces is, evidently, always in the actual position of the handle CD, CF. It follows, therefore, that the ratio of the gravity to the centrifugal force, is that of cos. ica to sin. icq, or that of the vertical distance of a below c to its horizontal distance from sa. Call the former d, the latter being x :

Hence, the periodic time varies as the square root of the altitude of the conic pendulum, let the radius of the base be what it may.

Hence, also, when ICQ ICP 45°, the centrifugal force = = of each ball is equal to its weight.

ON THE CENTRES OF PERCUSSION,
OSCILLATION, AND GYRATION.

216. The Centre of Percussion of a body, or a system of bodies, revolving about a point, or axis, is that point, which striking an immoveable object, the whole mass shall not incline to either side, but rest, as it were, in equilibrio, without acting on the centre of suspension.

217. The Centre of Oscillation is that point, in a body vibrating by its gravity, in which if any body be placed, or if the whole mass be collected, it will perform its vibrations in the same time, and with the same angular velocity, as the whole body, about the same point or axis of suspension.

218. The Centre of Gyration is that point, in which if the whole mass be collected, the same angular velocity will

be generated in the same time, by a given force acting at any place, as in the body or system itself.

219. The angular motion of a body, or system of bodies, is the motion of a line connecting any point and the centre or axis of motion; and is the same in all parts of the same revolving body. And in different unconnected bodies, each revolving about a centre, the angular velocity is as the absolute velocity directly, and as the distance from the centre inversely; so that, if their absolute velocities be as their radii or distances, the angular velocities will be equal.

220. PROP. To find the centre of percussion of a body or system of bodies.

Let the body revolve about an axis passing through any point s in the line SGO, passing through the centres of gra vity and percussion, & and o. Let MN be the section of the body, or the plane in which the axis soo moves. And conceive all the particles of the body to be reduced to this plane, by perpendiculars let fall from them to the plane : position which will not affect the centres G, o, nor the angular motion of the body.

sup.

M

B

S

Let A be the place of one of the particles, so reduced; join sa, and draw AP perpendicular to as, and a perpendicular to sco: then AP will be the direction of a's motion as it revolves about s; and the whole mass being stopped at o, the body A will urge the point r, forward, with a force proportional to its quantity of matter and velocity, or to its matter and distance from the point of suspension s; that is, as A. SA; and the efficacy of this force in a direction perpendicular to so, at the point P, is as A. sa, by similar triangles ; also, the effect of this force on the lever, to turn it about o, being as the length of the lever, is as ▲ . sa. Po A. sa. (so — SP)=A. sɑ. so — A . sɑ. sp = A. sɑ . 80 — a In like manner, the forces of в and c, to turn the system about o, are as

B

- в . SB2, and

c. sc. so — c. sc2, &c.

SA?.

But, since the forces on the contrary sides of o destroy one another, by the definition of this force, the sum of the positive parts of these quantities must be equal to the sum of the negative parts,

that is, A. sa. so + B. sb .so + c. sc. so, &c. =

A.

SA2+ B SB2+c. sc2, &c; and

hence so =

A SA2 + B . SB2 + c . sc2, &c.