pulley or wheel of radius r upon the same axis, and let s be the space through which the weight P descends in the time t, the proposed body whose weight is w turning upon the same axis with the same angular velocity: then

CR = C=

J

2s w

2 s prs

Example.-A body which weighs 100 lbs. turns upon a norizontal axis, motion being communicated to it by a weight of 10 lbs. hanging from a very light wheel of 1 foot diameter. The weight descends 2 feet in 3 seconds. Required the distance of the centre or circle of gyration from the axis of motion.

Here, I take g = 32, instead of 321, and obtain an approximative result. Whence

5. When the impulse communicated to a body is in a line passing through its centre of gravity, all the points of the body move forward with the same velocity, and in lines parallel to the direction of the impulse communicated. But when the direction of that impulse does not pass through the centre of gravity, the body acquires a rotation on an axis, and also a progressive motion, by which its centre of gravity is carried forward in the same straight line, and with the same velocity, as if the direction of the impulse had passed through the centre of gravity.

The progressive and rotatory motion are independent of one another, each being the same as if the other had no existence.

6. When a body revolves on an axis, and a force is impressed, tending to make it revolve on another, it will revolve on neither, but on a line in the same plane with them, dividing the angle which they contain, so that the sines of the parts are in the inverse ratio of the angular velocities with which the body would have revolved about the said axis separately.

7. A body may begin to revolve on any line as an axis that passes through its centre of gravity, but it will not continue to revolve permanently about that axis, unless the opposite rotatory forces exactly balance one another.

This admits of a simple experimental illustration. Suspend a thin circular plate of wood or metal by a cord tied to its edge, from a hook to which a rapid rotation can be given.

The plate will at first turn upon an axis which is in the continuation of the cord of rotation. As the velocity augments, the plane will soon quit that axis, and revolve permanently upon a vertical axis passing through its centre of gravity, itself having assumed a horizontal position.

The same will happen if a ring be suspended, and receive rotation in like manner.

And if a flexible chain of small links be united at its two ends, tied to a cord and receive rotation, it will soon adjust itself so as to form a ring, and spin round in a horizontal plane.

Also, if a flattened spheroid be suspended from any point, however remote from its minor axis, and have a rapid rotation given it, it will ultimately turn upon its shorter axis posited vertically.

This evidently serves to confirm the motion of the earth upon its shorter axis.

8. In every body, however irregular, there are three axes of permanent rotation, at right angles to one another. These are called the principal axes of rotation: they have this remarkable property, that the momentum of inertia with regard to any of them is either a maximum or a minimum.

Central Forces.

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.

4. When a body describes a circle by means of a force directed to its centre, its actual velocity is everywhere equal to that which it would acquire in falling by the same uniform force through half the radius.

5. This velocity is the same as that which a second body would acquire by falling through half the radius, whilst the first describes a portion of the circumference equal to the whole radius.

6. In equal circles the forces are as the squares of the times inversely.

7. If the times are equal, the velocities are as the radii, and the forces are also as the radii.

8. In general, the forces are as the distances or radii of the circles directly, and the squares of the times inversely.

9. The squares of the times are as the distances directly, and the forces inversely.

10. Hence, if the forces are inversely as the squares of the distances, the squares of the times are as the cubes of the disThat is,

tances.

if Fƒ:: d2 : D', then T2: :: D3 : d3.

11. The right line that joins a revolving body and its centre of attraction, called the radius vector, always describes equal areas in equal times, and the velocity of the body is inversely as the perpendicular drawn from the centre of attraction to the tangent of the curve at the place of the revolving body.

12. If a body revolve in an elliptic orbit by a force directed to one of the foci, the force is inversely as the square of the distance: and the mean distances and the periodic times have the same relation as in art. 10. This comprehends the case of the planetary motions.

13. If the force which retains a body in a curve increase in the simple ratio as the distance increases, the body will still describe an ellipse; but the force will in this case be directed to the centre of the ellipse; and the body in each revolution will twice approach towards it, and again twice recede from that point.

14. On the principles of central forces depend the operation of a conical pendulum applied as a governor or regulator to steam engines, water mills, &c.

This contrivance will be readily comprehended from the marginal figure, where A a is a vertical shaft capable of turning freely upon the sole a. C D, C F, are two bars which move freely upon the centre c, and carry

с

P

H

at their lower extremities two equal weights P, Q; the bars c D, C F, 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 ▲ a. When this shaft and connected apparatus are made to revolve, in virtue of the centrifugal force, the balls PQ fly out more and more from a a, as the rotatory velocity increases; if, on the contrary, the rotatory velocity slackens, the balls descend and approach A a. The ring 1 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 cylinHer; 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.

Here, if the vertical distance of P or q below c, be denoted by d, the time of one rotation of the regulator by t, and 3.141593 by x, the theory of central forces gives

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. Also, when I c Q=1 C P 45°, the centrifugal force of each ball is equal to its weight.

Inquiries connected with Rotation and Central Forces.

1. Suppose the diameter of a grindstone to be 44 inches, and its weight half a ton; suppose also that it makes 326 revolutions in a minute. What will be the centrifugal force, or its tendency to burst?

2x2w×3.141593a w

Here F

the measure of the required tendency.

= 47.22 w= 23.6 tons,

2. If a fly wheel 12 feet diameter, and 3 tons in weight, revolve in 8 seconds: and another of the same weight revolves in 6 seconds: what must be the diameter of the last, when their centrifugal force is the same?

By art. 8, Central Forces, F: f::

D

d

:

Therefore, since F

3. If a fly of 12 feet diameter revolve in 8 seconds, and another of the same diameter in 6 seconds: what is the ratio of their weights when their central forces are equal?

By art. 6, Central Forces, the forces are as the squares of the times inversely when the weights are equal: therefore when the weights are unequal, they must be directly as the squares of the times, that the central forces may be equal.

Hence w: w:: 36: 64:: 1:17

That is, the weight of the more rapidly to that of the more slowly revolving fly, must be as 1 to 17, in the case proposed.

other of these kinds, the usual theories are but of little service in practical mechanics, except as they may suggest an extension to the actual circumstances of nature and art.

2. The general principle for determining the motions of bodies from percussion, and which belongs equally to both elastic and non-elastic bodies, is this: viz. that there exists in the bodies the same momentum, or quantity of motion, estimated in any one and the same direction, both before the stroke and after it. And this principle is the immediate result of the law of nature or motion; that reaction is equal to action, and in a contrary direction; from whence it happens, that whatever motion is communicated to one body by the action of another, exactly the same motion does this latter lose in the same direction, or exactly the same does the former communicate to the latter in the contrary direction.

From this general principle too it results, that no alteration. takes place in the common centre of gravity of bodies by their actions upon one another; but that the said common centre of gravity perseveres in the same state, whether of rest or of uniform motion, both before and after the impact.

3. If the impact of two perfectly hard bodies be direct, they will, after impact, either remain at rest, or move on uniformly together with different velocities, according to the circumstances under which they met.

Let в and b represent two perfectly hard bodies, and let the velocity of в be represented by v, and that of b by v, which may be taken either positive or negative, according as b moves in the same direction as B, or contrary to that direction, and it will be zero when b is at rest. This notation being understood, al! the circumstances of the motions of the two bodies, after collision, will be expressed by the formula:

which being accommodated to the three circumstances under which v may enter become

These formulæ arise from the supposition of the bodies being perfectly hard, and consequently that the two after impact move on uniformly together as one mass. In cases