Motion on inclined Planes.

1. When bodies move down inclined planes, the accelerating force (independently of the modification occasioned by the poh sition of the centre of gyration) is expressed by the quo

יד

tient of the height of the plane divided by its length, or by what is equivalent, the sine of the inclination of the plane,. that is to say, sin i. In this case, therefore, the formulæ be

g

sin i

υ

t

Farther, if v be the velocity with which a body is projected up or down a plane, then

Making v = 0, in equa. 4, and the latter member of equation 5, the first will give the time at which the body will cease to rise, the latter the space.

Example.-Suppose a body be projected up a smooth inclined plane whose height is 12 and length 193 feet, with a velocity of 20 feet per second, how high will it rise up the plane before its motion is extinguished?

2. With regard to the velocities acquired by bodies in falling down planes of the same height, this proposition holds; viz. that they are all equal, estimated in their respective directions. Thus, if

A D, BE, C F, be grooves of different inclinations, and A c, DF, horizontal lines, the balls A, B, C, after

D

E

descending through those planes will have equal velocities when they arrive at D, E, F, respectively.

3. Also, all the chords, such as A D, B D, C D, that terminate either in the upper or the lower extremity of the vertical diameter of a circle, will be described in the same time by heavy bodies A, B, C, running down them; and that time will be equal to the time of vertical descent through the diame

ter.

three

4. If weights, as A, B, c, be drawn up three planes of different inclina

tions, by three equal weights hanging

from

C

cords over pulleys at P, then if the length of the middle plane be twice its height, the body в will be drawn up that plane, quicker than either of the other weights A or c.

Or, generally, to ensure an ascent up a plane in the least time, the length of the plane must be to its height, as twice the weight to the power employed.

5. If it be proposed to construct a roof over a building of a given width, so that the rain shall' run quickest off it, then each side of the roof must be inclined 45° to the horizon, or the angle at the ridge must be a right angle.

6. The force by which spheres, cylinders, &c. are caused to revolve as they move down an inclined plane (instead of sliding) is the adhesion of their surfaces occasioned by the pressure against the plane this pressure is part of the body's weight; for the weight being resolved into its components, one in the direction of the plane, the other perpendicular to it, the latter is the force of the pressure; and, while the same body rolls down the plane, will be expressed by the cosine of the plane's elevation. Hence, since the cosine decreases while the arc or angle increases, after the angle of elevation arrives at a certain magnitude, the adhesion may become less than what is necessary to make the circumference of the body revolve fast enough, in this case the body descends partly by sliding and partly by rolling. And the same may happen in smaller elevations, if the body and plane are very smooth. But at all elevations the

Y

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. =

} w.

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

tens. =

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

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

H

B

resist motion are placed at a greater distance from the axis in the cylinder B 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 2: 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, if ƒ 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.

g sin i

32 X
§

Hence (2-6): (2-366) :: 3 : 27603 = sin i f Consequently, 33333

tion, the weight being unity.

276030573 value of the fric

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 of ƒ are exhibited below

B

fPWB f PWB f PWB f PWBPW Bf PW 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 matter contained in a revolving system were collected, the 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

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.

For, in this case the obstacle receives the whole revolving motion of the body; whereas, if the blow be struck in any 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