Radial beam

“ It will therefore be useful for us to examine the strength of a beam when its load is applied otherwise than at the centre. [Pg 199]
406. We shall employ, in the first place, a beam 40" span, 0"·5 broad, and 1" deep; and we shall break it by applying a load simultaneously at two points, as may be most conveniently done by the contrivance shown in the diagram, Fig. 53. a b is the beam resting on two supports; c and d are the points of trisection of the span; from whence loops descend, which carry an iron bar p q; at the centre r of which a weight w is suspended. ”

“ In most balances the beam rests on steel knife edges; but a maker who has lately obtained celebrity makes his supports of pure rock crystal. The steel edges can be seen with the naked eye; the quartz edges cannot be seen even with a magnifying glass. One writer on this subject thinks that with these perfect crystal edges, with an inflexible girder beam, a short beam giving quick vibrations, and a sensitiveness that can be increased by screwing up the centre of gravity, there can hardly be a practical limit to the smallness of the weight that will turn the beam. ”

“ If the motion of the radiometer wheel is an effect of mechanical impulses transmitted in the direction of the beam of light, it was certainly to be expected that the beam would act on the lustrous as well as on the blackened mica surfaces, however large might be the difference in the resultants producing mechanical motion, in consequence of the great absorbing power of the lampblack. ”

“ Give us the real dope on how you're going to do it.”
“Well, with what I picked up from the Lassans, I think I know. They know all about light and mechanics, but they're rotten chemists, and don't realize how good a thing they've got in lots of ways. Now look—if you throw a beam of radiations from a cathode tube into finely divided material you break up some of the atoms. Well, all we have to do is get an extra-powerful cathode tube, break up a lot of atoms, and then deliver the positive charges from them onto whatever we're going for. That would be your gravity beam.” ”

“ We [Pg 222] have shown that when a beam of wood is supporting a weight, the fibres at the bottom of the beam are extended, the tendency being to tear them (Art. 376) . The fibres on the top of the beam are compressed, while the centre of the beam is in its natural state. The condition of strain in a cast iron beam is precisely similar; the bottom portions are in a state of extension, while the top is compressed. If therefore a beam be very thin, the material at the lower part may not be sufficient to withstand the forces of extension, and fracture is produced. ”

“ It consists of an horizontal beam suspended by a glass fibre, and having disks of pith at each end coated with lampblack. The whole is inclosed in a glass case, made of tubes blown together, and by means of the pump the air is entirely removed. In the centre of the horizontal beam is a silvered mirror, and a ray from the electric light is reflected from it on to a scale in front, where it is visible as a small circular spot of light. It is evident that an angular movement of the torsion-beam will cause the spot of light to move to the right or to the left along the scale. ”

“ There is no difficulty in using the ray, since, as it is a condition of space, not a vibration, it is impossible to stop it by any shield. There is only one way to work with it, to create it directionally. We make the field by projecting certain strains along a beam, then once started the field follows that line to a distance dependent on the strength of the generator. ”

“ It is preferable to have a divergent beam from a single source form a well-defined image after refraction. Owing to the successful removal of the disturbing causes it is now possible to allow the radiator to be placed in open space, a plate with a rectangular aperture allowing the radiation to fall on the refracting cylinder along a given direction. The size of the plate is 26 X 15 cm., and the aperture is 7 X 6 cm. (see fig. 1) . The radiator and the receiver are placed on opposite sides of the plate. ”

“ The strength of a beam 40" × 0"·5 × 1" is four times as great as the strength of 40" × 0"·5 × 0"·5, though the quantity of wood is only twice as great in one as in the other. In general we may state that if a beam were bisected by a longitudinal cut, the strength of the beam would be halved when the cut was horizontal, and unaltered when the cut was vertical; thus, for example, two beams of experiment No. 4, placed one on the top of the other, would break with about 40 lbs., whereas if the same rods were in one piece, the breaking load would be nearly 80 lbs. ”

“ If we decompose the light vector into two components, of which one has the direction of the beam, while the other is perpendicular to it, then the latter can be arbitrarily rotated around the beam, and in every direction, the ratio is determined between the two. ”

“ The circular beam may be considered as built up of flat and very thin bands of light, parallel to the edges of the prism, and a simple ray would be formed by one of these flat bands; as the round opening would allow an indefinite number of such rays to enter, each would produce its own spectrum on the screen, and the actual image would be formed of a number of spectra overlapping each other. When the aperture by which the light is admitted consists merely of a narrow slit, or line, parallel to the edges of the prism, we obtain what is termed a pure spectrum. ”

“ If the direction of the force is vertical, and the beam is inclined, then the strain is increased by as much as the diagonal of inclination exceeds the vertical; for example, let one thousand pounds be suspended from the lower end of a beam ten feet long, inclined at an angle of 45°. The diagonal being ten, the vertical will be 7.07 feet, and the strain is increased as follows:—
7.07 to 10 as 1,000 to 1,414 lbs. As the angle of inclination, from the horizontal, increases, the strain from a given load decreases, until the beam is vertical, when a weight acts with its least power. ”

“ The side of the beam, supposing it square, was found by multiplying the diameter of the cylinder by the length of the stroke, and extracting the cube root of the quotient, which will be the depth or thickness of the beam. This rule allows a beam 16 feet long to bend 1/8th of an inch, and a beam 32 feet long to bend 1/4 of an inch. For cast iron beams the square of the diameter of the cylinder, multiplied by the length between the centres, is equal to the square of the depth, multiplied by the thickness. ”

“ Thus, if a beam which is one foot broad and one foot deep support a given weight, then a beam of the same depth and two feet broad will support double the weight; but if a beam be one foot broad and two feet deep it will support four times as much as a beam one foot broad and one foot deep. If a beam one foot square support a given weight, then a beam two feet square will support eight times as much. Also a cylinder of two inches in diameter will support eight times as much as a cylinder one inch in diameter. The appended table gives data bearing on the subject. ”

“ For experimental investigations it is also necessary to have a narrow pencil of electric radiation, and this is very difficult to obtain, unless waves of very short length are used. With large waves diverging in all directions and curling round corners, all attempt at accurate work is futile. For angular measurements it is necessary to direct the electric beam in the given direction along narrow tubes, and receive it in another tube in which is placed the receiver. ”

“ Such accurate adjustments make it possible to carry out some of the most delicate experiments.
The radiator-tube, with the lens and the attached polariser, is capable of rotation. The emergent beam may thus be polarised in a vertical or a horizontal plane. The analyser fitted on to the receiver may also be rotated. The gratings may thus be adjusted in two positions.
(1) Parallel position. (2) Crossed position. In the first position the radiation transmitted through both the gratings, falls on the sensitive surface, and the galvanometer responds. The field is then said to be bright. ”

“ If 3-inch thickness of wrought-iron plates placed at the angle proved better than 41⁄2 inches vertical, the total weight would be about 1570 tons. If the draught of water formed by the apex at f were too great, the inverted pyramid could be truncated say to the ballast-line at e, or any higher elevation, giving sufficient longitudinal stiffness as a beam; but it is obvious that making a perfect apex forms a stronger beam. In any case the angle downwards from the water-line must be preserved. ”

“ The amount of strain caused by any weight applied in a transverse direction, to a beam supported at both ends, is as the breadth, as the length inversely, and as the 125square of the depth. Whatever depression takes place, tends to shorten the upper, and to extend the under-side; whence the fibres of the top part suffer compression, and those of the bottom extension. The amounts of compression and extension must of course be equal, and therefore if any material resists these two strains in a different degree, the number of fibres opposing each will also be different. ”

“ The radiation diagram shown in Figure 1303b depicts relative values of power in the same plane existing at the same distances from the antenna or the origin of the radar beam. Maximum power is in the direction of the axis of the beam. Power values diminish rapidly in directions away from the axis. The beam width is taken as the angle between the half-power points. ”

“ When the spark-gap is vertical, the electric radiation is to a great extent polarised, the vibration being accompanied by electric force in a vertical plane, and magnetic force in a horizontal plane. I shall, for simplicity, consider only the electric vibration. When the partially-polarised radiation is transmitted through a horizontal grating, the emergent beam is completely polarised, the vibration taking place in a vertical plane passing through the axis. ”

“ Other beams ran lengthwise of the body, placed straight, obliquely, diagonally, and in every direction calculated to strengthen and stiffen, and all bound together with rods and bars and bolts. One great beam, reaching from rear part through the body to centre of his forehead, is calculated to sustain fully a ton’s weight, if at any time his great head should need such support. ”

“ I point out that perfect superposition of the two opposite beams T and R is characterized by general extinction of the telescope field for the radiation employed, which is here close to the indigo portion of the mercury arc spectrum. Starting from there, a small rotation ε of the beam splitter around a vertical axis, in the clockwise direction (swing-direction D [2] ) , or counter-clockwise (swing-direction S [3] ) , narrows the dark field into a central vertical fringe bordered by parallel side fringes. ”

“ This subject is to be studied by examining the effect of different deflections upon the fibres of a beam. A beam is always deflected [Pg 183] whatever be the load it supports; thus by looking through the telescope of the cathetometer I can detect an increase of deflection when a single pound is placed in the tray: hence whenever a beam is loaded we must have some deflection. An experiment will show what amount of deflection may be experienced without producing any permanently injurious effect. ”

“ A pillar supporting a weight is under compression. At c the weight tends to bend the beam. The upper part is under tension, the lower part under compression. This is known as beam action, and depends on whether the beam is supported at one end, as shown, or on both ends. ”

“ If the beam be sufficiently loaded, the centre of gravity will at length be raised to the fulcrum S, and the beam will rest indifferently in any position. If more weight be then added, the centre of gravity will be raised above the fulcrum, and the beam will turn over. ”

“ The collapse of a beam by a transverse strain commences by fracture of the fibres on the lower surface, followed by a rupture of all fibres up to a considerable depth. Here we see that by a transverse force the fibres in a beam of 48" × 1" × 1" have been broken by a strain of 140 lbs. (Art. 366) ; but we have already stated (Art. 353) that to tear such a rod across by a direct pull at each end a force of about four tons is necessary. ”

“ Thus we see that the resistance of a beam to a cross strain, as well as to tension and compression, is affected by the incompressibility and inextensibility of the material.
The formula for the dimensions of any beam to support a strain transversely is
S = the ultimate strength in lbs. ”

“ Each end of the beam is sprung obliquely upwards and towards the middle, so as to form a spring through which a pushing screw passes, which serves to vary the distance of the point of support from the fulcrum, and, at the same time, by its oblique action to raise or depress it, so as to furnish a means of bringing the points of support and the fulcrum into a right line. ”

“ The radiator tube was provided with an ordinary lens whose focal distance for electric radiation is about 4 cm. The beam thus rendered approximately parallel fell perpendicularly on the face of the glass prism. The two prisms were made by cutting a cube of glass—an ordinary paper weight—across a diagonal. The size of the cube was 4·5 cm. on each side. [1] One prism was fixed on the spectrometer circle; the other could be moved so as to change the thickness of the interposed air-space between the two sections very gradually. ”