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These three equations contain the whole theory of projectiles in vacuo; the second equation shows that the horizontal motion is uniform, being proportional to the time; the third expresses that the motion in the perpendicular is uniformly accelerated, being as the square of the time.
Theory of Falling Bodies.
99. If the particle begins to move from a state of rest, b=0, and the equations of motion are
The first shows that the velocity increases as the time; the second shows that the space increases as the square of the time, and that the particle moving uniformly with the velocity it has acquired in the time t, would describe the space 2z, that is, double the space it has moved through. Since gt expresses the velocity v, the last of the preceding equations gives
2gz = got2 = vo,
where z is the height through which the particle must have descended from rest, in order to acquire the velocity v. In fact, were the particle projected perpendicularly upwards, the parabola would then coincide with the vertical: thus the laws of parabolic motion include those of falling bodies; for the force of gravitation overcomes the force of projection, so that the initial velocity is at length destroyed, and the body then begins to fall from the highest point of its ascent by the force of gravitation, as from a state of rest. By experience it is found to acquire a velocity of nearly 32.19 feet in the first second of its descent at London, and in two seconds it acquires a velocity of 64.38, having fallen through 16.095 feet in the first second, and in the next 32.19 + 16.095 48.285 feet, &c. The spaces described are as the odd numbers 1, 3, 5, 7, &c.
These laws, on which the whole theory of motion depends, were discovered by Galileo.
Comparison of the Centrifugal Force with Gravity.
100. The centrifugal force may now be compared with gravity, for if v be the velocity of a particle moving in the circumference of a circle of which r is the radius, its centrifugal force is f = Let
h be the space or height through which a body must fall in order to acquire a velocity equal to v; then by what was shown in article 99, v2hg, for the accelerating force in the present case If we suppose h=r,
is gravity; hence
the centrifugal force becomes equal to gravity.
101. Thus, if a heavy body be attached to one extremity of a thread, and if it be made to revolve in a horizontal plane round the other extremity of the thread fixed to a point in the plane; if the velocity of revolution be equal to what the body would acquire by falling through a space equal to half the length of the thread, the body will stretch the thread with the same force as if it hung vertically.
102. Suppose the body to employ the time T to describe the circumference whose radius is r; then being the ratio of the circumference to the diameter, v =
Thus the centrifugal force is directly proportional to the radius, and in the inverse ratio of the square of the time employed to describe the circumference. Therefore, with regard to the earth, the centrifugal force increases from the poles to the equator, and gradually diminishes the force of gravity. The equatorial radius, computed from the mensuration of degrees of the meridian, is 20920600 feet, T365.2564, and as it appears, by experiments with the pendulum, that bodies fall at the equator 16.0436 feet in a second, the preceding formulæ give the ratio of the centrifugal force to gravity at the equator equal to Therefore if the rotation of the earth were 17 times more rapid, the centrifugal force would be equal to gravity, and at the equator bodies would be in equilibrio from the action of these two forces.
103. A particle of matter suspended at the extremity of a thread, supposed to be without weight, and fixed at its other extremity, forms the simple pendulum.
104. Let m, fig. 27, be the particle of matter, Sm the thread, and S the point of suspension. If an impulse be given to the particle, it will move in a curve mADC, as if it were on the surface of the sphere of which S is the centre; and the greatest deviation from the vertical Sz would be measured by the sine of the angle CSm. This motion. arises from the combined action of
105. The impulse may be such as to make the particle describe a curve of double curvature; or if it be given in the plane xSz, the particle will describe the arc of a circle DCm, fig. 28; but it is evident that the extent of the arc will be in proportion to the fig. 28. intensity of the impulse, and it may be so great as to cause the particle to describe an indefinite number of circumferences. But if the impulse be small, or if the particle be drawn from the vertical to a point B B and then left to itself, it will be urged in the vertical by gravitation, which will cause it to describe the arc mC with an accelerated velocity; when at C it will have acquired so much velocity that it will overcome the force of gravitation, and having passed that point, it will proceed to D; but in this half of the arc its motion will be as much retarded by gravitation as it was accelerated in the other half; so that on arriving at D it will have lost all its velocity, and it will descend through DC with an accelerated motion which will carry it to B again. this manner it would continue to move for ever, were it not for the resistance of the air. This kind of motion is called oscillation.
The time of an oscillation is the time the particle employs to move through the arc BCD.
106. Demonstration.-Whatever may be the nature of the curve, it has already been shown in article 99, that at any point m, v2 = 2gz, g being the force of gravitation, and z = Hp, the height through which the particle must have descended in order to acquire the velocity v. If the particle has been impelled instead of falling from rest, and if I be the velocity generated by the impulse, the equation becomes v=I+2gz. The velocity at m is directly as the element of the space, and inversely as the element of the time; hence
The sign is made negative, because z diminishes as t augments. If the equation of the trajectory or curve mCD be given, the value of ds = Am may be obtained from it in terms of z = Hp, and then the finite value of the preceding equation will give the time of an oscillation in that curve.
107. The case of greatest importance is that in which the trajectory is a circle of which Sm is the radius; then if an impulse be given to the pendulum at the point B perpendicular to SB, and in the plane roz, it will oscillate in that plane. Let h be the height through which the particle must fall in order to acquire the velocity given by the impulse, the initial velocity I will then be 2gh; and if BSC a be the greatest amplitude, or greatest deviation of the pendulum from the vertical, it will be a constant quantity. Let the variable angle mSC = 0, and if the radius be r, then Sp=r cos 0; SH = r cos a; Hp = Sp – SH = r (cos 0 cos α), and the elementary arc mArde; hence the expression for the time becomes
This expression will take a more convenient form, if x = Cp = - cos 0) be the versed sine of mSC, and B = (1 - cos α)
the versed sine of BSC; then do =
√2x - x2
Since the versed sine never can surpass 2, if h + rß > 2r, the velocity will never be zero, and the pendulum will describe an indefinite number of circumferences; but if h + rß <2r, the velocity v
will be zero at that point of the trajectory where x =
h + rß
the pendulum will oscillate on each side of the vertical. If the origin of motion be at the commencement of an oscillation,
But the integral must be taken between the limits x = ẞ and x = 0, that is, from the greatest amplitude to the point C. Hence
being the ratio of the circumference to the diameter. From the same author it will be found that
between the same limits. Hence, if T be the time of half an oscil