Page images
PDF
EPUB

At any intermediate latitude the tangents to the meridians at two points differing in longitude by 15°, such as A and B (Fig. 72), will meet the axis at O, and the angle AOB measures the change in direction of the meridian per hour. Consequently a Foucault pendulum in that latitude will shift in one hour through an angle equal to AOB. This interesting experiment was carried out by Foucault in 1851. He used as pendulum a massive ball of copper, hung by a wire more than 50 meters long, from the dome of the Pantheon in Paris.

FIG. 72.

B

144. Conservation of Angular Momentum. In any body or system of bodies the total angular momentum of the system cannot be changed by any internal forces: for suppose A and B (Fig. 73) are two parts of the system which act on each other, since action and reaction are equal and opposite the force on A is equal and opposite to the force on B; and since the distance from the axis to the line of action of the forces is the same for both, the moments of the forces about the axis will be equal and opposite, so that in the same time they will give equal and opposite angular momenta to the system, and consequently the total angular momentum will not be changed.

For example, in the solar system the planets have not only

O Axis

angular momenta about their own axes, but also angular momenta about the common center of gravity of the system. These angular momenta may be represented as vectors and their resultant found from the vector diagram, and neither the direction nor amount of this resultant is changed by any internal forces, such as the attraction of one planet for another or any possible collisions between them.

FIG. 73.

145. Angular Momentum of Projectiles.-A body having angular momentum tends to keep the direction of its axis of revolution constant, and the greater the angular momentum the harder it is to disturb the direction of the rotation; that is,

the slower its axis of revolution will change in direction under any given torque.

So the spin of the rifle bullet or shell from a rifled gun causes it to keep pointing in a nearly constant direction as it flies through the air in spite of the tendency of a long bullet to turn sidewise.

K

146. Motion of a Top.-When a rotating body is acted on by forces which tend to turn it about an axis perpendicular to its axis of rotation the effect is to change the direction of the axis of rotation without producing any change in the amount of the angular momentum about that axis; precisely as when a force acts on a body at right angles to the direction of its linear motion ($113) it changes the direction, but not the speed of the motion.

H

D

A

CB

W'

FIG. 74.—Top with point fixed.

The motion of a top affords an excellent illustration of this principle. The top in figure 74 is represented as spinning in the direction indicated by the arrow, but in the inclined position shown it is subject to a downward force W due to its own weight acting through its center of gravity G, and the upward pressure of the floor against the point of the top at A. These two forces are equal and constitute a couple which tends to turn the top about an axis DA perpendicular to its axis of revolution. The effect of the couple is to cause a steady change in the direction of the axis of revolution, the upper end of the top moving around in the circle EFL H. This change in the direction of the axis of the top may be called its precessional motion.

The precession of the top may be explained as follows: let the vector AB represent in amount and direction the angular momentum of the top about its axis, the vector being drawn so that the top is seen to revolve clockwise by an observer looking along the vector AB in the direction in which it points. Similarly the vector AD may represent the angular momentum which would be given to the top in a very small interval of time t by the couple consisting of the forces W and W'. The

resultant of the two vectors AD and AB is the vector AC, showing that the resultant angular momentum will have AC as its axis, and the axis of the top will accordingly move through the angle BAC in the time t. And as the vector DA is always at right angles to the plane EKA, the top will move at right angles to this plane, and therefore its upper end E will describe a circle about the vertical axis AK.

In the case just discussed the friction of the floor is supposed to be sufficient to keep the point of the top fixed at A. But when the top spins on a frictionless level surface it remains at a constant inclination and its precessional motion is about a vertical axis through its center of gravity, as shown in figure 75.

B

FIG. 75.

How it is possible for a top to rise to a vertical position as it spins was first explained by Lord Kelvin. It depends on the fact that the peg of the top is rounded and the friction between it and the floor causes it to roll around in a circle; and when this rolling of the peg on the floor urges the top around faster than the regular precessional motion, it causes the inclination of the top to gradually diminish until it stands vertical, and "goes to sleep." On a perfectly frictionless surface a top could not rise in

[graphic]

this way.

147. Gyroscope.-In the gyroscope shown in figure 76 a wheel with heavy rim is mounted in two pivoted rings so that the axis of rotation of the wheel may be inclined at any angle and the whole may also turn freely about a vertical axis. When the wheel is in rapid rotation a sharp blow given with the hand to one of the rings as if to change the direction. of the axis of rotation, will cause the wheel to vibrate as though it were held in its position by stiff springs.

FIG. 76.

When a small weight is hung on near one end of the axis of rotation, the wheel, instead of tipping down, rotates slowly around the vertical axis as indicated by the arrow; if the weight

is hung from the other end of the axis this precessional motion is reversed.

A bicycle wheel serves admirably as a gyroscope.

For an interesting and simple discussion of the motion of the top and gyroscope see Spinning Tops, by John Perry (published by the S. P. C. K., London).

N

E

148. Precession of the Equinoxes.-The earth itself illustrates the precessional motion of the gyroscope. It is a rotating body with enormous angular momentum. But as it is not a sphere and its axis is not perpendicular to the plane of its orbit, the attraction of the sun on the bulging equatorial belt tends to turn it over and make its axis perpendicular to the ecliptic. The effect of this rotational force is a slow precessional motion of the axis of the earth, just as in the gyroscope. The axis remains inclined 234° to the pole of the ecliptic, but describes a circle about that pole in a period of about 25,800 years.

FI

S

FIG. 77

ECLIPTIC

If we take the pole of the ecliptic as a center and describe a circle of 234° radius it will pass through the present pole star and will mark the path which is being described by the polar axis of the earth. In about 13,000 years the bright star Vega in the constellation of the Lyre will be very nearly at the pole.

V. UNIVERSAL GRAVITATION

149. Kepler's Laws.-The German astronomer Kepler in the year 1609, having made a careful study of the observations. made by Tycho Brahé, came to the conclusion that the orbits of the planets were not circular as had been supposed, but elliptical, and announced his discovery in the following laws:

1. The orbits of the planets are ellipses having the sun at one focus.

2. The area swept over per hour by the radius joining sun and planet is the same in all parts of the planet's orbit. Hence the planet moves faster in its orbit when near the sun than when farther away.

After nine years more of persistent search for some relation between the periodic times of the planets and their distances from the sun, he discovered and announced his third law:

3. The squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun.

150. Newton's Principia.-In 1687 Sir Isaac Newton published his great work the Principia in which he clearly enunciated the fundamental principles of mechanics and applied them to a great variety of important problems. In this work he showed from the laws of mechanics that if the planets moved about the sun in ellipses in the manner described in the first two laws of Kepler, then each planet as it moves in its orbit must be subject to a force which is directed toward the sun, and varies inversely as the square of the distance between them.

151. Universal Gravitation.-From the above result Newton concluded that probably all masses, great and small, attract each other with a force proportional to their masses and inversely proportional to the square of the distance between them.

According to this law, the attractive force between any two masses m and M is expressed by the formula

[blocks in formation]

where r is the distance between the centers of the masses if they are spherical. The quantity C is an absolute constant for all kinds of matter and depends only on the units in which force, mass, and distance are measured. It is called the gravitation constant and is equal to the force with which two unit masses attract each other when placed unit distance apart.

152. Moon's Motions Connected with Fall of Apple.-Newton conceived that the weight of a body near the surface of the earth is due to this gravitation attraction between the earth and the body, and that an apple drops toward the earth in accordance with the same gravitation law which determines the motion of the moon in its orbit.

To test this point let us, following Newton, find the acceleration which the apple would have if it were dropped toward the earth when as far off as the moon, and compare this acceleration with that which the moon is known to have.

According to the law of gravitation ($151), the earth attracts

« PreviousContinue »