Page images
PDF
EPUB

would have acquired by falling freely through a height equal to that due to the initial velocity, increased by that of the plane. Hence, if a body start from

D

B

Fig. 102.

E

a state of rest at A, and, after having passed over one inclined plane AB, enters upon a second plane BC, without loss of velocity, it will reach the bottom of the second plane with the same velocity that it would have acquired by falling freely through DC, the sum of the heights of the two planes. Were there a succession of inclined planes, so arranged that there would be no loss of velocity in passing from one to another, it might be shown, by a similar course of reasoning, that the terminal velocity would be equal to that due to the vertical distance of the terminal point below the point of starting.

By a course of reasoning entirely analagous to that employed in discussing the laws of motion of bodies projected vertically upwards, it might be shown that, if a body were projected upwards, in the direction of the lower plane, with the terminal velocity, it would ascend along the several planes to the top of the highest one, where the velocity would be reduced to 0. The body would then, under the action of its own weight, retrace its path in such a manner that the velocity at every point in descending would be the same as in ascending, but in a contrary direction. The time occupied by the body in passing over any part of its path in descending, would be exactly equal to that occupied in passing over the same portion in ascending.

In the preceding discussion, we have supposed that there is no loss of velocity in passing from one plane to another. To ascertain under what circumstances this condition will be fulfilled, let us take the two planes AB and BC. Prolong BC upwards, and denote the angle ABE, by q. Denote. the velocity of the body on reaching B, by v'. Let v′be resolved into two components, one in the direction of BC, and the other at right angles to it. The effect of the latter

will be destroyed by the resistance of the plane, and the former will be the effective velocity in the direction of the plane BC. From the rule for decomposition of velocities, we have, for the effective component of ', the value v' cosp. Hence, the loss of velocity due to change of direction, is v'v' cosp; or, v'(1- cosp), which is equal to ' ver-simp. But when is infinitely small, its versed-sine is 0, and there will be no loss of velocity. Hence, the loss of velocity due to change of direction will always be 0, when the path of the body is a curved line. This principle is general, and may be enunciated as follows: When a body is constrained to describe a curvilinear path, there will be no loss of velocity in consequence of the change in direction of the body's

motion.

Periodic Motion.

121. Periodic motion is a kind of variable motion, in which the spaces described in certain equal periods of time are equal. This kind of motion is exemplified in the phenoinena of vibration, of which there are two cases.

1st. Rectilinear vibration. Theory indicates, and experiment confirms the fact, that if a particle of an elastic fluid be slightly disturbed from its place of rest, and then abandoned, it will be urged back by a force, varying directly as its distance from the position of equilibrium; on reaching this position, the particle will, by virtue of its inertia, pass to the other side, again to be urged back, and so on. Το determine the time required for the particle to pass from one extreme position to the opposite one and back, let us denote the displacement at any time t by s, and the acceleration due to the restoring force by ; then, from the law of the force, we shall have = n's, in which n is constant for the same fluid at the same temperature. Substituting for Ф its value, Equation (61), and recollecting that acts in a direction contrary to that in which s is estimated, we have,

[ocr errors][merged small][merged small]
[blocks in formation]

The velocity v will be 0 when s is greatest possible; denoting this value of s by a, we shall have,.

n'a2 + C = 0; whence, C n'a2.

=

Substituting this value of C in the preceding equation, it

[merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Taking the integral between the limits s = + a and sa, and denoting the corresponding time by 1, being the time of a double vibration, we have,

Int =T; whence, T=

T

27

n

The value of is independent of the extent of the excursion, and dependent only upon n. Hence, in the same medium, and at the same temperature, the time of vibration is constant.

These principles are of utility in discussing the subjects of sound, light, &c.

2ndly. Curvilinear vibration. Let ABC be a vertical

plane curve, symmetrical with respect to DB. Let AC be a horizontal line, and denote the distance EB by h. If a body were placed at A and abandoned to the action of its own weight, being constrained to remain on the curve, it would, in accordance with the principles of the

D

E

P

K

H

B

Fig. 103.

The body would then be at A, and would, conseascend to A, whence it Were there no retarding

last article, move towards B with an accelerated motion, and, on arriving at B, would possess a velocity due to the height h. By virtue of its inertia, it would ascend the branch BC with a retarded motion, and would finally reach C, where its velocity would be 0. in the same condition that it was quently, descend to B and again would again descend, and so on. causes, the motion would continue for ever. From what has preceded, it follows that the time occupied by the body in passing from A to B is equal to that in passing from B to C, and also the time in passing from C to B is equal to that in passing from B to A. Further, the velocities of the body when at G and H, any two points lying on the same horizontal, are equal, either being that due to the height EK. These principles are of utility in discussing the pendulum.

Angular Velocity.

122. When a body revolves about an axis, its points being at different distances from the axis, will have different velocities. The angular velocity is the velocity of a point whose distance from the axis is equal to 1. To obtain the velocity of any other point, we multiply its distance from the axis by the angular velocity. To find a general expression for the velocity of any point of a revolving body, let us denote the angular velocity by w, the space passed over by a point at the unit's distance from the axis in the time dt,

by de. The quantity de is an infinitely small arc, having a radius equal to 1; and, as in Art. 113, it is plain that we may regard the angular motion as uniform, during the infinitely small time dt. Hence, as in Article 113, we have,

[merged small][merged small][ocr errors][ocr errors][merged small]

If we denote the distance of any point from the axis by l, and its velocity by v, we shall have,

[ocr errors][merged small][merged small][merged small][merged small]

123. A PENDULUM is a heavy body suspended from a horizontal axis, about which it is free to vibrate. In order to investigate the circumstances of vibration, let us first consider the hypothetical case of a single material point vibrating about an axis, to which it is attached by a rod destitute of weight. Such a pendulum is called a SIMPLE PENDULUM. The laws of vibration, in this case, will be identical with those explained in Art. 121, the are ABC being the arc of a circle. The motion is, therefore, periodic.

Let ABC be the arc through

which the vibration takes place, and denote its radius by . The angle CDA is called the amplitude of vibration; half of this angle ADB,

denoted by a, is called the angle of deviation; and 7 is called the length of the pendulum. If the point starts from rest, at A, it will, on reaching

A

K

H

B

Fig. 104.

any point H, of its path, have a velocity v, due to the height EK, denoted by h. Hence,

[blocks in formation]

If we denote the variable angle HDB by 8, we shall

« PreviousContinue »