Page:The American Cyclopædia (1879) Volume XI.djvu/336

324 MECHANICS equal bodies traverse with equal velocity une- qual circles, their centrifugal forces will be in the inverse ratio of the diameters. It follows from the first proposition that the centrifugal forces of any two bodies revolving around their common centre of gravity are equal. These propositions can be verified experimentally by employing the whirling table, fig. 18. A spi- ral spring which moves a registering index is fixed to one end of the horizontal rod. The adjustments may be so made as to cause a ball to revolve in any desired circle with any de- sired velocity. The applications of these prin- ciples are of daily occurrence. A horse or a carriage running in a circle exerts a centrifu- gal force requiring an inclination of the body toward the centre of the circle to counteract the tendency to be thrown over. The proper angle of inclination is found as follows : Sup- pose a horse to be running in a circle whose centre is c, fig. 19, and whose radius is a c. Draw the perpendicular a b to represent the weight of the horse, and let b d, paral- F, O< 19. lei with ac, repre- sent the centrifugal force; then a d will be the resultant, and the proper angle of incli- nation will be d a 5, whose tangent = . For tangent b a d =^-- = = . An inclination oa w rg is imparted to railway carriages when trav- ersing curves, by giving such an elevation to the rail on the outer curve of the track that the cross section of the latter shall be perpen- dicular to the required inclination or line of direction. Example: On a railway track 4 ft. 8 in. wide, how much elevation should be given to the outer rail on a curve of 600 ft. radius for a velocity of 30 ft. c per second ? Taking the equation = ^- the value becomes 90 - = ji^- . Therefore the outer rail must be raised -^~- of 56 in., or 2-6 in. When any body is rotated it has a tendency to revolve on its shortest axis, in consequence of the greater momentum in the par- ticles furthest from the centre of motion. When a body having the form shown in fig. 20 is turned on its longer axis by means of a string suspended from c, if the body is perfectly regular and the geomet- rical axis perfectly coincides with the axis of motion, it will not change its posi- tion ; but as such coincidence never exists, the body will on being rotated begin to change its axis of rotation, and when sufficient speed is attained, the increased momentum resulting Fio. 20. from the change of position will cause the body to assume a position at right angles to its first position, and revolve about its shorter axis. The oblate spheroidal figure of the earth and other heavenly bodies is due to the action of centrifugal force. (See HYDROMECHANICS.) IV. THE PENDULUM. A simple pendulum may be defined as a body whose weight is con- fined to a point, and which, suspended from a fixed point, vibrates in an arc. A simple pendulum can only exist in theory. A sin- gle vibration of a pendulum is the distance through which it oscillates from the point at which it begins to descend on one side of the vertical, as at , fig. 21, to the point on the opposite side of the ver- tical, as at 5, where its mo- tion is arrested by the ac- tion of gravity. Its passage from a to b and back to a is called a double vibration. All pendulums are com- pound because, having ex- tension, their different par- ticles are at different dis- tances from the centre of motion, and therefore tend to vibrate in different times, because the time of vibra- tion is increased by increas- ing the length of the pendu- lum. For small arcs the times of vibrations are the same ; beyond certain limits increasing the arc increases the time. These facts were first ascertained by Galileo about 1585, when making use of the pendulum for counting time in astro- nomical observations. It has been demonstra- ted by mathematicians that if a pendulum vi- brates in a small circular arc, the ratio of the time of one vibration to the time in which a body would fall half the length of the pen- dulum is equal to the ratio of the circumfer- ence of a circle to its diameter. Therefore, according to equation (1), substituting I for , and letting t denote the time of one vibra- tion, we have t : A/ :: * : 1, or t = K.A/. From this equation it will be observed that the time of vibration of a pendulum varies as the square root of its length. Squaring both mem- bers of the equation, t* = > or I == |. A half-seconds pendulum is therefore found as A seconds pendulum is 39'1 in., and a two-sec- onds pendulum 156-4 in. When the arc of vibration is 1 on each side of the vertical, tho daily retardation compared to the vibration in an arc which causes no retardation is If second. When the arc is 2, the loss is 6f seconds; when 3, it is 15 seconds; the formula for esti- mating the retardation being |D a , where D represents the number of . degrees the pendu- lum describes on each side of the vertical. The