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CHANGE OF VOLUME BY TENSION.

to lengthen the tube. Read each mark in its new position together with the water level, and so proceed, taking a number of readings under different tensions.

Construct a curve in which abscissas represent readings of the water level, and ordinates the changes which take place in the length of the rubber, draw also other curves, in which abscissas represent the water level, and ordinates the change of lengths of each section of the tube. To do this it is most convenient to make a table giving the readings of each point, a second giving the difference of reading of each two consecutive marks, and a third giving the increase of length they undergo when the cord is stretched. The first of the curves will be nearly a straight line, and the tangent of the angle it makes with the horizontal line, or the ratio of its vertical to its horizontal progression multiplied by b, gives the ratio of the increase of volume compared with the increase of length. In the same way by the other curves, determine the change in length of each section of the tube compared with the whole change.

Fig. 30.

Next measure the height of each marked point of the rubber rod, also its diameter at these points. Stretch it and measure again, and take four series of observations in this way. Now construct curves, in which abscissas represent scale readings, and ordinates alterations in thickness as given by the gauge. The scale for the latter must be greatly enlarged. Measure the area enclosed by this space, and reduce it to square millimetres, allowing for the change of scale. Multiplying this area by four times the thickness gives approximately the diminution in volume due to the contraction in the centre. If the rod is much altered in form, the change in cross-section may be obtained more accurately by taking the difference of the squares of the thickness before and after extension. Using them as ordinates of the curve the volume is given by the enclosed area. Construct such a curve for each extended position of the rod, and compare the decrease of volume thus found with the increase due to the change of length, or the product of the cross-section by the change of reading of the lower index mark.

35. DEFLECTION OF BEAMS. I.

Apparatus. In Fig. 31, AB is a rectangular bar of steel resting on two knife-edges, with a load applied to its centre by a weight placed in the scale-pan D. To measure the flexure, a micrometer screw Cis placed over the bar, and turned until its point touches the latter. E is a galvanic battery having one pole connected with the bar, the other with C through an electro-magnet F. When the screw touches the bar the circuit is completed, and the magnet draws down its armature with a click. This gives a very accurate test of the exact position of the screw when contact takes place. The length of the beam may be altered by changing the position of A and B. C and D are also movable, and a set of weights is provided to vary the deflection. To ensure contact the wire should be soldered in position, and the point of C tipped with a piece of sheet platinum. A convenient size of bar for laboratory purposes is about half a metre long, a centimetre wide, and half a centimetre thick, using weights from 100 to 2000 grammes. Instead of the electro-magnet F, a galvanometer may be used if preferred.

A

E

B

Experiment. Set A and B 50 cm. apart, and C and D midway between them. Turn C until it touches the bar, when instantly a current will pass from the battery E through the magnet F, making a click, or if a galvanometer is used, swinging the needle to the right or left. Read the position of the micrometer screw, taking the whole number of turns from the index on one side, and the fraction from the graduated circle. Add 1000 grammes to D, and bring the screw again in contact. The difference in the readings gives the deflection with great accuracy. The general formula for the deflection a of a beam of length 7, breadth b, and

W18
Dbd3

D

Fig. 31.

depth d, under a load W, is a=bds, in which D is the modu

lus of transverse elasticity, or the weight required to make a equal one, when b, d and l, all equal unity. From this formula the following laws may be deduced. The deflection, when small, is proportional to the weight applied.

1st.

Measure the deflec

tion for every hundred grammes, from zero to two kilogrammes, and see if the beam returns to its original position when the load is removed. If not, the change is the permanent set. Construct a curve with abscissas proportional to the load, and ordinates to the corresponding deflection. Evidently, according to the law, this should be a straight line, and the near agreement proves conclusively its correctness. 2d. The deflection is proportional to the cube of the length. Measure the deflection with a load of 2 kgs., changing the length of beam 5 cm. at a time, from 50 cm. to 0, keeping C and D always at the middle of the beam. Care must be taken in each case to first measure the micrometer-reading when no load is applied, as this point will vary, owing to irregularities in the bar or stand. To compare the results with theory, construct a curve in which abscissas represent lengths of the beam, and ordinates deflections. According to the law this should be a cubic parabola, having the equation y a23. To find the value of a, suppose one of the earlier readings gave a deflection of 5.8 mm., for a length of 40 cm.; then 5.8 = a 403, or a = .00009. Substituting this value in the equation, construct the curve y .00009 3 on the same sheet with the experimental curve, and see if they agree. To find the value of D, draw a line nearly coincident with the first series of observations. Deduce from it the increase of deflection for each added kilogramme. Substitute this value for a in the formula, making W= 1, and giving 1, b and d, their proper values, found by measuring the bar. D is now the only unknown quantity, and may be obtained by solving the equation.

If desired, bars of different materials may be provided, and the modulus of each determined by measuring the deflection with a given load, and substituting these values in the formula. The law that the deflection is inversely proportional to the breadth, may be proved with bars alike except in breadth, and the law of the thickness in a similar manner. The form of the beam when bent is found by shifting the micrometer C and measuring the deflection at various points between A and B, and the effect of a change in position of the load by moving D. The same apparatus may be applied to the case of a beam supported at more than two points, or to beams built in at one or both ends. This experiment may also be almost indefinitely extended by using circular and triangular bars, hollow

or solid, also those of a T or I shape, or, in fact, girders of any form.

36. DEFLECTION OF BEAMS. II.

A.

Apparatus. AB, Fig. 32, is the bar to be tested, which may be one of the square rods described in the next experiment. It is clamped at one end by placing it between two similar rods, one of which is nailed to the wall, and the other pressed down upon it by two or three clamps, like those of a quilting frame, as at C. small mirror may be placed upon it at any point, and the deflection measured by reading in it the reflection of a scale F by the telescope E. The beam is bent by weights placed in the scale-pan D. By attaching a finely divided scale to A, the absolute deflection may be read by the telescope E, using it like a cathetometer.

A

Experiment. Clamp AB so that its length shall be 50 cm., and place the mirror at its end. Place the telescope E opposite A and focus it, so that on looking through it the image of the scale F shall be distinctly seen reflected in the mirror. Read the position of the cross-hair in the telescope to tenths of a division of the scale.

Now place a kilogramme.

Fig. 32.

D

in D. The beam is at once bent, and although the mirror moves but little, the scale-reading is greatly altered. The work described under the last experiment may be repeated with this apparatus. Or to vary it, let the following measurements be made. Place 2 kgs. in D, and take the scale-readings when placed at distances of 5, 10, 15, 20, &c., centimetres from B. Again, take the scale-reading very carefully when no load is applied. Now place in D as great a weight as the beam will safely bear, and take readings every half minute for five or ten minutes. Then remove the load and see if the beam returns to its original position. The permanent set of the beam may be studied in this manner.

To compare the results with theory the scale-readings must be reduced to angular deviations by the formula given on page 24. For small deflections, however, it is sufficient to divide the change

in scale-reading by twice the distance EA, to obtain the tangent of the angle through which the mirror moves. In deducing the form of an elastic beam by Analytical Mechanics, we have

d2y

EI = P(1-x), the moment of the deflecting force P. In dx2

this E is the modulus of elasticity, I the moment of inertia of the cross-section, y the deflection of any point at a distance from the end, and the length of the beam. Integrating, we obtain

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Apparatus. A number of deal rods, as nearly alike as possible, half an inch square and five or six feet long. These are to form the beams or units of which all the trusses are to be composed. They may be connected by clamps like those used for quilting frames, by boring holes in them and fastening by wire, or better still by using small carriage-bolts, 1" in diameter. A scale-pan and set of weights serve to apply a strain not exceeding one or two hundred pounds to any part of the truss to be tested. By attaching a fine scale the deflection of any point may be read by a telescope mounted like a cathetometer. The strain on any portion may be determined by inserting a small spring balance, as will be described below.

Experiment. This apparatus may be procured at very small expense, while with it almost all the laws of elasticity may be proved, and the strength of a great variety of trusses for bridges. and roofs tested. Although the following work resembles that of Experiment 35, yet its importance, and the different method of measurement employed, justifies its repetition.

The flexure of a beam is proportional to the load. Set two knifeedges 40 inches apart, cut a rod a little longer than this, and lay it on them. Attach a fine scale to its centre point, focus the telescope on it and record the reading. Add weights, a pound at a time, until the beam breaks. The increase of reading in each case over that given at first, is proportional to the load. It must be noticed, however, that when the beam is much bent a new law holds. Repeat these measurements with rods 30, 20, and 10 inches

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