Putting a for the altitude due to the velocity v, since (by art. 158) v2 = 2ag, 2ag we have =

whence there results

φ:9::α : gr.

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 ø and r remain constant, the velocity v will be constant.

211. If both members of the equation 1 be multiplied by the mass M of the body, and we put F to represent the centrifugal force of that mass, we shall have Mv2 F = In like manner, if F' 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

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

214. And, if we assume T2 T2: p3: r3, as in the planetary 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.

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

This contrivance will be readily comprehended from the marginal figure,

where Aa 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 CD, cF, are united, by a proper articulation, to the bars G, H, which latter are attached to a ring, I, 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 P, 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 I ascends in the former case, descends in the latter and a lever connected with I may be made to correct appropriately, the energy of the moving power. Thus, in the steam-engine, 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 versá; but the general principle is the If FQ: = FI = DP = DI, then I, P, Q, are always in some one hori zontal plane but that is not essential to the construction.

same.

Now, let t denote the time of one revolution of the shaft, x the variable horizontal distance of each ball from that shaft, m as usual = 3141593 : then will

the velocity of each ball be

270

2

x=

4π2x
t2

.

The balls being operated upon simultaneously 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. ICQ to sin. ICQ, or that of the vertical distance of Q below C to its horizontal distance from Aa. Call the former d, the latter being a :

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.

FARTHER EXAMPLES.

1. A ball whose weight is 10lbs. is whirled round in a circle whose radius is 10 feet, with a velocity of 30 feet per second : what is the measure of p the centrifugal force?

2. If the same ball be made to move uniformly through the same circle, in 2 seconds, what will the centrifugal force then be?

3. Given the diameter of the orbit, 10 feet, and the centrifugal force equal to

the weight of the revolving body; required the time of a revolution, and velocity per second.

4. Given the diameter, 14.59 feet, and the velocity, 15:279, to find the periodic time and central force.

5. Given the diameter, 14·59 feet, the central force equal to twice the weight of the body; what is the velocity and time of a revolution?

6. Let the diameter be 29.18 feet, the time of a revolution 3 seconds; required the velocity and central force.

7. If a fly, 12 feet diameter, and 3 tons weight, revolves in 8 seconds, and another of the same weight revolves in 3 seconds; what must be the diameter of the last, when they have the same centrifugal force?

8. If a fly, 12 feet diameter, revolves in 8 seconds, and another of the same diameter in 3 seconds; what is the ratio of their weights, when the central forces are equal?

9. If a fly, 2 tons weight, and 16 feet diameter, is sufficient to regulate an engine, when it revolves in 4 seconds; what must be the weight of one 12 feet diameter, when it revolves in 2 seconds, so that it may have the same power upon the engine ?

ON ROTATORY MOTION.

216. PROP. If a body revolve about an axis, the particles of which that body is composed resist, by their inertia, the communication of motion to any given point, with forces which are as the particles themselves, and the squares of their distances from the axis of motion jointly.

C

P

P

Let an axis of rotation pass through C perpendicular to the plane of the figure, and let a body fixed firmly to this axis be acted upon by several accelerating forces; we are to enquire into the circumstances of the motion produced. Suppose, at first, that a particle p, situated at P in the figure, is urged about the fixed axis by a force , applied in the direction PD: that force tends to impress upon the particle p a certain velocity in the direction PD, yet, in consequence of the mutual cohesion between the different molecule of the body, and the connection of the whole with the fixed axis, the velocity can only be produced actually in the initial direction Pd perpendicular to CP. Drawing, therefore, Pd perpendicular to CP, the force must be decomposed into two others, in the directions Pd, PC, of which the one Pe will be extinguished by the resistance of the axis, and the other Pd has place; where, of consequence, we have Pd o cos. DPd. If, therefore, we denote the distance CP of p from the axis of rotation by r, and the perpendicular distance DC of the force from the axis by d, we have, in the triangle CPD, sin. DPC = cos.

r

DPd= ; and, consequently, Pd=47· This expression would represent the effective accelerating force of the particle p, if that particle were alone; but the connection of this particle with the others, and the operation of the forces acting on the latter, change this effect: if, then, the arc a be run over at the

end of the time t by a particle at a unit of distance from the fixed axis, ar will be the arc described in the same time by the particle p; so that the velocity, and

a

the effective accelerating force of this latter will be r and r

Now comparing this force with the former, and observing that they are both directed in the same right line, we may reason thus. The accelerating force

δ

which is impressed in the direction of the initial motion, must be decomposed into

r

r

...

r

accel. force destroyed by the action of the other powers in the system.

In like manner we may proceed to investigate the effects of other forces p', ", &c. acting upon the moleculæ p', p'. &c. (whose distances are r', r", &c.) in the directions P'D', P'D", &c., and at the distances d', d", &c. the expressions being the same with the letters accented similarly: the whole, therefore, will be in equilibrio when impressed by the moving forces

Now as the system is attached to a fixed axis, only one equation is necessary to express the state of equilibrium. And, if we suppose that the forces 4, ', q", &c. are parallel to the plane of the figure, or perpendicular to the plane of rotation (and they may all be resolved to such planes by an obvious process), it will be merely requisite to make the sum of the moments of the powers with respect to the fixed axis equal to nothing. Here, that of the first force will be δορ

r2p, and the moments of the other forces will be expressed in the same manner, adding the accents. Thus, then, we shall have

døp + dø'p' + &c. = (r2p + r'2p' + &c.)

t2

Or taking the character as before, and denoting the angular velocity by U,

we shall have ƒ pop = fr2p: whence,

Ü a

=

r2p

Hence, if the quantity frp, which is the sum of the products of the several moleculæ into the squares of their respective distances from the axis, be called the momentum of inertia, and if be called the angular accelerating force, the

Ü
t

equation just given may be thus stated in words at length:

The angular accelerating force is the quotient of the sum of the momenta of the moving forces, or of their resultant, divided by the momentum of inertia.

Corol. 1. The force which accelerates the point A of any body revolving on an axis, to which point that force is applied, is equal to the product of the force into the square of the distance AC, divided by the sums of the products of all the moleculæ into the squares of their respective distances from C, the centre of motion.

CP2

For the mass moved is, and the moving force is : but the accele

rating force is equivalent to the quotient of the moving force by the mass, and is therefore represented by

AC

p. PC2

Corol. 2. The angular velocity of a system, generated in a given time, by any force at A, perpendicular to AC, is proportional to the rectangle of the force into the distance at which it acts, divided by the sums of the products of all the moleculæ into the squares of their respective distances.

For the absolute velocity of the point A is as the accelerating force, and the angular velocity is as the absolute velocity directly and the distance reciprocally; . AC2 1 Р PC2

therefore the angular velocity is as f

AC or as . AC÷

Sp. PC2. Corol. 3. The angular motion of any system, generated by a uniform force, will be a motion uniformly accelerated.

This is evident, because the accelerating force is in a constant ratio to the uniform force p.

Corol. 4. What has been here shewn with respect to moleculæ situated on a right line passing through a centre of motion will hold equally with regard to a body or system moving upon an axis: for all the particles of such body may be conceived to be transferred to the plane in which the axis of suspension CP performs its motion, by an orthographical projection, the lines of transference being all parallel to the axis of motion; this supposition will, it is obvious, neither affect the place of the centre of gravity (with regard to the axis of motion) nor the angular motion of the body.

Corol. 5. From the above final equation we may readily obtain an expression

for the angular velocity.

ข

For

=

&c. give ƒ ø&p==&p=

бр +

vòp + v'dp' +&c.

r2p+r'2p' + &c.

; and, taking the fluents,

but may

be

From this it appears that cor. 2 is not confined to a single force; extended to as many forces as we please, and applied to the body in any directions whatever.

Corol 6 As to the part Pe of the force o, which operates as a pressure upon the fixed axis, and is entirely destroyed by its re-action, it may be easily determined. For Pe cos. DPC, and DP =√(x2 82); therefore, cos. DPC

And the same may be shown with regard to the effect of any other forces, p', p', &c. upon the axis of motion.

217. DEF. The centre of oscillation is that point in the axis of suspension of a vibrating body in which, if all the matter of the system were collected, any force applied there would generate the same angular velocity in a given time as the same force at the centre of gravity, the parts of the system revolving in their respective places.