body may be made to roll by the uncoiling of a thread or rib band wound about it.

If w denote the weight of a body, s the space described by a body falling freely, or sliding freely down an inclined plane, then the spaces described by rotation in the same time by the following bodies, will be in these proportions.

1. A hollow cylinder, or cylindrical surface, s = s tension of the cord in the first case = =w.

2. In a solid cylinder, s = s, tens. = 1 W.

3. In a spheric surface, or thin spherical shell, s =

tens. =

} w.

4. In a solid sphere, s = s, tens. =

2 w.

B

If two cylinders be taken of equal size and weight, and with equal protuberances upon which to roll, as in the marginal figures then, if lead be coiled uniformly over the curve surface of B, and an equal quantity of lead be placed uniformly from one end to the other near the axis in the cylinder A, that cylinder will roll down any inclined plane quicker than the other cylinder B. The reason is that each particle of matter in a rolling body, resists motion in proportion to the SQUARE of its distance from the axis of motion: and the particles of lead which most resist motion are placed at a greater distance from the axis in the cylinder в than in A.

7. The friction between the surface of any body and a plane, may be easily ascertained by gradually elevating the plane until the body upon it just begins to slide. The friction of the body is to its weight as the height of the plane to its base, or as the tangent of the inclination of the plane to the radius. Thus, if a piece of stone in weight 8 pounds, just begins to slide when the height of the plane is 2 feet, and its base 24 then the friction will be the weight, or of 8 lbs. = 6 lbs.

:

8. After motion has commenced upon an inclined plane, the friction is usually much diminished. It may easily be ascertained experimentally, by comparing the time occupied by a body in sliding down a plane of given height and length, or given inclination, with that which the simple theorem for t, would give. For, iff be the value of the friction in terms of the pressure, the theorem for t will be

Example.-Suppose that a body slides down a plane in length 30 feet, height 10, in 23 seconds, what is the value of the friction.

Here t =

2 s g sin

=

sin i ƒ

Hence (26) (2.366)::: 27603

Consequently, 33333276030573 value of the fric-. tion, the weight being unity.

9. When a weight is to be moved either up an inclined plane, or along an horizontal plane, the angle of traction P w B, that the weight may be drawn with least effort, will vary with the value of f. The magnitude of that angle P w B for several values off are exhibited below.

B

10. If, instead of seeking the line of traction so that the moving force should be a minimum, we required the position such that the suspending force to keep a load from descending should be a minimum, or a given force should oppose motion with the greatest energy; then the angles in the preceding table will be still applicable, only the angle in any assigned case must be taken below, as в w p. This will serve in the building and fastening walls, banks of earth, fortifications, &c. and in arranging the position of land-lies, &c.

SECTION III.-Motions about a Centre or Axis.

Pendulum, simple and compound; Centres of Oscillation, Percussion, and Gyration.

DEF. 1. 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.

Or, since the force of gravity upon the whole body may be considered as a single force (equivalent to the weight of the body) applied at its centre of gravity, the centre of oscillation is that point in a vibrating body into which, if the whole were concentrated and attached to the same axis of motion, it would then vibrate in the same time the body does in its natural state. COR. From the first definition it follows that the centre of oscillation is situated in a right line passing through the centre of gravity, and perpendicular to the axis of motion. It is always farther from the point of suspension than the centre of gravity.

DEF. 2. The centre of gyration is that point in which, if all the the matter contained in a revolving system were collected, same angular velocity will be generated in the same time by a given force acting at any place as would be generated by the same force acting similarly in the body or system itself.

When the axis of motion passes through the centre of gravity, then is the centre called the principal centre of gyration.

The distance of the centre of gyration from the point of suspension, or the axis of motion, is a mean proportional between the distances of the centres of oscillation and gravity from the same point or axis.

If s represent the point of suspension, & the place of the centre of gravity, o that of the centre of oscillation, and R that of the centre of gyration. Then

SRSO.SG

(1)

and so . s G = a constant quantity for the same body and the same plane of vibration.

DEF. 3. The Centre of Percussion is that point in a body revolving about an axis, at which, if it struck an immoveable obstacle, all its motion would be destroyed, or it would not incline either way.

When an oscillating body vibrates with a given angular velocity, and strikes an obstacle, the effect of the impact will be the greatest if it be made at the centre of percussion.

any

For, in this case the obstacle receives the whole revolving motion of the body; whereas, if the blow be struck in other point, a part of the motion of the pendulum will be employed in endeavouring to continue the rotation.

If a body revolving on an axis, strike an immovable obstacle at the centre of percussion, the point of suspension will not be affected by the stroke.

We can ascertain this property of the point o when we give a smart blow with a stick. If we give it a motion

round the joint of the wrist only, and, holding it at one extre-. mity, strike smartly with a point considerably nearer or more remote than of its length, we feel a painful wrench in the hand but if we strike with that point which is precisely at of the length, no such disagreeable strain will be felt. If we strike the blow with one end of the stick, we must make its centre of motion at of its length from the other end; and then the wrench will be avoided.

PROP. The distance of the centre of percussion from the axis of motion is equal to the distance of the centre of oscillation from the same supposing that the centre of percussion is required in a plane passing through the axis of motion and centre of gravity.

DEF. 4. A Simple Pendulum, theoretically considered, is a single weight, regarded as a point, or as a very small globe, hanging at the lower extremity of an inflexible right line, void of weight, and suspended from a fixed point or centre, about which it oscillates.

DEF. 5. A Compound Pendulum is one that consists of several weights moveable about one common centre of motion, but so connected together as to retain the same distance both from one another and from the centre about which they vibrate.

Or any body, as a cone, a cylinder, or of any shape, regular or irregular, so suspended as to be capable of vibrating, may be regarded as a compound pendulum; and the distance of its centre of oscillation from any assumed point of suspension, is considered as the length of an equivalent simple pendulum.

Any such vibrating body will have as many centres of oscillation as you give it points of suspension; but when any one of those centres of oscillation is determined, either by theory or experiment, the rest are easily found by means of the property that so. s G is a constant product, or of the same value for the same body.

DEF. 6. When a body either revolves about an axis, or oscillates, the sum of the products of each of the material elements, or particles of that body, into the squares of their respective distances from the axis of rotation, is called the momentum of inertia of that body. (See art. 6, p. 241).

A point, or very small body, on descending along the successive sides of a polygon in a vertical plane, loses at each angle a part of its actual velocity equal to the product of that velocity into the versed sine of the angle made by the side which it has just quitted, and the prolongation of the side upon which it is just entering.

A

D

B

7. A heavy body which descends along a curve posited in a vertical plane, by the force of gravity, has, in any point whatever, the same velocity as it would have if it had fallen through a vertical line equal to that between the top and the bottom of the arc run over and when it has arrived at the bottom of any such curve, if there be another branch either similar or dissimilar, rising on the opposite side, the body will rise along that branch (apart from the consideration of friction) until it has reached the horizontal plane from which it set out. Thus, after having descended from A to v, it will have the same velocity as that acquired by falling through D v, and it will ascend up the opposite branch until it arrives at B.

V

8. If the body describe a curve by a pendulous motion, the same property will be shown, free from the effects of friction. Thus, let a ball hang

by a flexible cord s D from a pin at s: then, after it has descended through the arc D E, it will pass through an equal and similar. arc E A, going up to A in the same horizontal line with D, and ascending from E to A in an interval of time equal to that which it descended from D to E. But, if a pin projecting from P or p stop the cord in its course, the ball will still rise to в or to c, in the same horizontal line with A and D; but will describe the ascending portions of the curve in shorter intervals of time than the descending branch.

S

9. When a pendulum is drawn from its vertical position, it will be accelerated in the direction of the tangent of the curve it would describe, by a force which is as the sine of its angular distance from the vertical position. Thus, the accelerating force at A, would be to the accelerating force at B, as AF to B E. (See art. 5, on the Centre of Gravity).

This admits of an easy experimental proof.

E

10. If the same pendulous body descend through different arcs, its velocity at the lowest point will be proportional to the chords of the whole arcs described. Thus, the velocity at D,