long. The flexure is proportional to the cube, the breaking weight inversely as the square of the length.

In the same way the effect of applying a load at different points, or the deflection at different points with a given load, may be measured. A long beam may also be supported at several points, and the effect of a moving load noted. To measure the strength of a beam built in at one or both ends, clamp it at those points between two similar beams, one above, the other below, and fasten them by bolts or clamps to a fixed upright. Having thus fully determined the strength of the single beams, they should be examined when combined. Join two beams together so as to form a T, and measure the strain necessary to pull the vertical one off. Different kinds of joints may thus be compared. Some form of truss should now be built, and its strength

tested. Let the king-post, Fig. 33, be the form selected, and make the span AB 40 inches. Cut two rods a little longer than this, and bore the holes A, B and D. Cut two more rods CD with holes distant 10 inches, and attach them to the others by bolts at D. Add AC

A

Fig. 33.

and CB, and connect the two trusses thus formed by cross pieces at A, B, C and D, so that they shall be ten inches apart. A small bridge is thus made whose stiffness is remarkably great. Such a structure should bear a weight of fifty, or even a hundred pounds, with little flexure. Measure the deflection at different points under varying loads, trying the effect of applying the latter at the centre, on one side, or distributing it uniformly.

The force acting on any beam is readily determined by replacing it by a spring balance, with a screw and nut which may be made to take up the whole strain, without distorting the structure. If the force is one producing compression, it is best to elongate the beams, as at CE, and insert the balance between C and E. Compare the various results with those obtained by computation. It must be remembered that this method of connecting the beams of a truss by bolts is not employed in actual practice, but it is very convenient, and sufficiently strong for a model. If preferred,

6

proper tools may be supplied, and the student may frame his truss as in a real bridge or roof. Further, any beam, as CD, subjected only to tension, may be replaced by a wire.

The rods will also be found useful for a great variety of purposes. Thus one of the easiest ways to make large screens is to fasten four of them together at the ends, and attach thick paper to them by double-pointed tacks. Again, a convenient way to make a large rectangular box to cut off light, or to protect an instrument from dust, is to connect twelve of these rods at the ends by

AP

Fig. 34.

slipping them into corner pieces of tin, Fig. 34, and covering them with black paper. In the same way all the principles of framing may be taught, and quite complicated structures built. It is well to have some of the latter loaded until they break, to determine the weakest points. They are then easily repaired by inserting new pieces of wood. Another very good object to construct and test is a suspension bridge. Use two stout wires or chains for the suspension cables, and build the roadway of the rods, hanging it from the chains by wires with screw threads cut on their ends, so that their lengths may be adjusted by nuts. Test the strain on the chains by inserting a spring balance like that known as the German icebalance, and measure the deflections of the different parts under varying loads.

38. LAWS OF TORSION.

Apparatus. Let AB, Fig. 35, represent the bar whose torsion is to be measured. The further end B is firmly fastened to a piece of wood C, which can turn around the axis of the beam, but may be clamped in any position to the semicircle immediately behind it. A in the same way is attached to D, which carries two brass rods, acting like the dog of a lathe. EF is a long rod mounted on an axle, which may be turned by placing weights in the scale-pan G. The latter is supported by a cord passing over a curved block at the end of E, so that the moment of the weight, or its tendency to twist the bar is unchanged, whatever the position of EF. To measure the angle of torsion, two mirrors are attached to AB, and the scales H, H' reflected in them are viewed by telescopes I, I'. By making H, H' arcs of circles, with centres at A and B, the angle of torsion may be obtained directly.

Experiment. To measure the torsion of the beam AB, attach it to the stand, as in the figure, placing the mirrors at a distance apart equal to the length to be examined. Focus the telescopes I, I' so that the scales H, H' shall be distinctly visible, and read the position of the cross-hairs. Place a

weight in the scale-pan G which will twist both the mirrors A and B, deviating the former the most. See how much each scale-reading has changed; their difference measures the angular

B

G

Fig. 35.

twist of AB. This may be reduced to degrees from the radius of curvature of the scale, and the magnitude of its divisions, recollecting that the motion of the mirror is only one-half that of the reflected ray. From the theory of torsion the following laws are readily deduced. The angle of torsion is proportional to the moment of the deflecting force. To prove this law, measure the torsion with several different weights in G, and see if the angles are proportional to the weight. The distance of the point of application of G from the axis may also be varied, and it will be found that the torsion is proportional to the product of this distance multiplied by G. The torsion is proportional to the length of the bar. Prove this by varying the distance between the mirrors, leaving the bar unchanged. By using a variety of bars it may be proved that in those having similar sections the torsion is proportional to the fourth power of the similar dimensions. In rectangular bars of breadth b, and depth d, it is proportional to bd (b2 + d2), and in tubes to (4) calling r and the outer and inner radii. In practice, owing to the warping of the surfaces, these formulæ undergo slight modifications.

70

39. FALLING BODIES.

Apparatus. This consists of two parts, a chronograph capable of measuring very minute intervals of time, as hundredths of a second, and the arrangement represented in Fig. 36, for making and breaking an electric circuit when the body falls. A ball A is attached to the spring D by a short thread and wire. Burning the thread the ball is released, and the spring rising allows the current to pass from the battery B, to C, D, E, F, and the chronograph G. This marks the beginning of the time to be recorded. Its end is shown by the breaking of the circuit, which occurs at F when A strikes E. To have the current broken during this time, instead of closed, it is merely necessary to place the points E and F below the springs. Another method of releasing the ball is to put an iron pin in its upper side, and support it by an electromagnet. The instant the current is broken it will fall. One of the best forms of chronograph for this purpose is that devised by Hipp, in which a reed making a thousand vibrations a second replaces the pendulum of an ordinary clock. It therefore ticks a thousand times a second, and measures small intervals of time with great precision. Very good results may be obtained with a common marine clock, removing the hairspring and replacing it by an elastic bar of steel. An electro-magnet is placed close to the pallet, so that when the current passes the spring is bent. The clock, therefore, starts the instant the circuit is broken, and stops as soon as it is closed.

2h

7, equals √27,

h,

B

Com

Experiment. The time of falling of any body through a height in which g 9.80 metres. Measure the time of fall through various heights by altering the length of the wire AD, noting the height in each case with care, repeating several times and taking the mean. pare this with the result given by theory. In vacuo the time would be independent of the material or magnitude of the ball. In air, however, this is not the case, owing to the resistance. The latter may therefore be determined by measuring the time of fall of bodies having the same form but different weights. This apparatus may also be applied to the measurement of the velocity of curve-motion and personal equation, or the coefficient of friction

E

Fig. 36.

of a body may be found with great accuracy by measuring its time of descent down an inclined plane.

40. METRONOME PENDulum.

Apparatus. A light deal rod is provided, to which two leaden weights may be attached at any point by set screws. A knife-edge passes through the centre of the rod, so that it may be swung like a pendulum. Either weight may also be attached to the knife-edge by a fine wire. A millimetre scale is used to measure the distance of the weights from the knife-edge, and a bracket fastened to the wall and carrying a steel plate serves as a support.

Experiments. First, to prove the law of the single pendulum. Attach the heavier weight to the knife-edge by the wire, using as great a length as possible. Measure the time in seconds of making 100 single, or 50 double vibrations, also the distance from the centre of the lead to the knife-edge. Repeat with several lengths of wire.

Now compare the observed time with that given by the formula in which g 9.80 m., and 7 is the measured length. Repeat one of these measurements with the lighter weight, using the same value of l. It should give the same result as the other. Next place both weights on the deal rod at opposite ends. Measure, as before, their time of vibration, also their distances from the knife-edge. Compute the time of vibration as above, merely subw'l' 2 + w'l'' 2 stituting for the value w'l' + w''l''' weights, and l', '" their distances from the knife-edge. If greater accuracy is required, a third term must be introduced for the weight of the rod. Repeat with various positions of the two weights, and compare the results with theory.

in which w', w" are the

41. BORDA'S PENDULUM.

Apparatus. In Fig. 37, CD is the pendulum, formed by attaching a ball of lead C, to a wire nearly four feet long, and supporting it on a knife-edge D. A sheet of platinum is fastened below the ball, so that when at rest it dips edgewise into a mercury cup, making electrical connection with the battery B. E is a clock whose pendulum F dips in a second mercury cup. When both pendulums are at rest the current passes from B through C, D, E