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This same apparatus may be applied to illustrate the case of a body with one point fixed, acted on by parallel forces, as, for example, the lever, by using a stand H with two pins, between which the beam may turn. This stand is also useful in finding the point of application of the resultant in the above cases.

26. CENTRE OF GRAVITY.

Apparatus. Several four-sided pieces of cardboard (not rectangles) and a plumb line, made by suspending a small leaden weight by a thread, from a needle with sealing wax head.

Experiment. Make four holes in the cardboard, two AB, Fig. 22, close to two adjacent corners, the others in any other part not too near the centre. Pass the needle through A and support the cardboard by it; the thread will hang vertically downwards, and the centre of gravity must lie somewhere in this line, or it would not be in equilibrium. Mark a point on this line as low down as possible, and connect it with the pin hole. Do the same with B; the intersection of the two is the centre of gravity. Turn the cardboard over and repeat with the other holes. This gives two determinations of the centre of gravity. To see if the two points are opposite one another, prick through one and see if the hole coincides with the other. By suspending at any other points, the same result should be obtained. Be careful that the holes are large enough to enable the card to swing freely.

A

B

G'

E

Divide into two trian

Bisect AD in E, and

Next, lay the card down on your note book and mark the four points A, B, C, D. Connecting them with lines gives a duplicate of the cardboard. On this construct the centre of gravity geometrically. gles by connecting AC. CD in F. The centre of gravity of ACD must lie in AF, also in CE, hence at G. Obtain G' by a similar construction with ABC. The centre of gravity of the whole figure must lie in GG'. Make a second construction by connecting BD, making the triangles ABD and BCD; the intersection of GG' and its corresponding line gives the centre of gravity. Lay the piece of cardboard on the figure and prick

Fig. 22.

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through the two centres of gravity previously found. They should agree closely with that found geometrically.

27. CATENARY.

Apparatus. A chain three or four yards long, each link of which is a sphere, known in the trade as a ball link chain. Every tenth link should be painted black, and the fiftieths red. A horizontal scale ABC, Fig. 23, attached to the wall, also a number of pins to which the chain may be fastened by short wire hooks, and its length altered at will. A graduated rod BD is used to measure the vertical height of any point of the chain.

B

-

G

F

E

Experiment. First, to determine the average length of the links. Let the chain hang vertically from A, measure the length of each hundred links, and take their mean. A simple proportion gives the number of links to which AC is equal. Suspend the chain at A and C, making the flexure at the centre about half a foot. Measure it exactly, and increase the original length 10 links. at a time to 100. Increase it also by 17 links, by 63 and by 48, and measure as before. Write the re

ED

Fig. 23.

sults in a column and take the first, second and third differences of the first measurements. Now obtain by interpolation the three values for 17, 63 and 48 links, and compare with their measured values.

Next suspend the chain as in ADE, and measure the deflection at intervals of five inches horizontally. This is best done by passing a pin through the graduated rod at the zero point, letting it hang vertically, then measuring by it. Taking differences as before, those of the first order will be at first negative, then increase until they become positive. Where the first difference is zero, is evidently the lowest point of the curve. By the method of inverse interpolation find this point, treating the first differences as if they were the original variable, and recollecting that each difference belongs approximately to the point midway between the

two terms from which it was obtained. Thus the difference obtained from the 5 and 10 inches corresponds to 7. Obtain this point also by measurement, by laying off BF equal to CE, prolonging EF to G and measuring GF. AC minus one-half GE will equal the required distance. Repeat with several points below E, and compare with the computed position of the lowest point.

28. CRANK MOTION.

Apparatus. A steel scale AB, Fig. 24, divided into millimetres, slides in a groove so that its position may be read by an index E. It is connected by the rod AD to the arm of the protractor, whose centre is C. On turning CD, which carries a vernier F, AB moves backwards and forwards. Several holes are cut in AD so that its length may be altered at will.

Experiment. Make AD as long as possible. Measure CD by turning it until D is in line with C and A, and read E; then turn it

F

E

A

B

Fig. 24.

180°, and read again. One-half the difference of these readings equals CD. Next, to find the reading of the vernier when CD and DA are in line. Make ACD about 90° and read E and F. Turn CD until the reading of E is again the

same and read F. The mean of these two readings gives the required point. Repeat two or three times, and take the mean.

Let AB represent the piston rod of an engine, and CD the crank attached to the fly-wheel. The problem is to determine the relative positions of these two, during one revolution. Bring D in line with CA, and move it 10° at a time through one revolution, reading E in each case. Do the same, using a shorter connecting rod, so that AD shall be about two or three times CD. To compare these results with theory, first suppose the rod CD infinitely long. The distance of AB from the mean position will then always equal CD X cos ACD. This is readily computed from the accompanying table of natural cosines. If, as is most convenient, CD is made just equal to 1 decimetre, the distances are given directly in the second column of the table by moving the

Angle. Cosine.

decimal point two places to the right. Compare these results. with your observations. Construct a curve in which abscissas represent the computed positions of AB, and ordinates the difference between the observed and computed results, enlarging the scale ten times. If a smooth curve is thus obtained it is probably due to the short length of AD. The correction due to this is readily proved to be AD ✔AD2 — CD2 sin2 ACD, or calling the ratio AD÷CD=n, it is ¿D (n — √n2 — sin2A CD). Compute this correction for every 30°, knowing that sin2 30° = .25, sin2 60° = .75. The points thus obtained should lie on the residual curve found above. Do the same with the shorter arm AD.

29. Hook's UNIVERSAL JOINT.

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Apparatus. A model of this joint with graduated circles attached to its axles. The latter should be so connected that they may be set at any angle.

Experiment. Set the axes at an angle of 45°, and bringing one index to 0°, the reading of the other will be the same.

Now move

the first 5° at a time to 180°, and read the other in each position. Record the results in columns, giving in the first the reading of one index, in the second that of the other, and in the third their difference, which will be sometimes positive, and sometimes negative. Construct a curve with abscissas taken from the first column, and ordinates from the third, enlarging the latter ten times. It shows how much one wheel gets behind, or in advance of, the other. To compare this result with theory, let Fig 25 represent a plan of the joint, AC and CB being the two axes. Describe a sphere with their intersection C as a centre. The great circle CD is the path described by the ends of one hook, CE that described by the other. D and E must, by the construction of the apparatus, always be 90° apart. Then in the spherical

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triangle CDE we have given DE = 90°, ECD = 45°, the angle between the axes, and one side as CD, and we wish to compute

CE. But by spherical trigonometry, tang CE tang CD cos ECD. Substituting in turn CD = 5°, 10°, 15°, 20°, &c., we compute the corresponding angle through which the second wheel has been turned. Construct a second curve on the same sheet as the other, using the same scale. Their agreement proves the correctness of both.

Experiments like Nos. 28 and 29 may be multiplied almost indefinitely. Thus various forms of parallel motion, the conversion of rotary into rectilinear motion by cams, link motion, gearing, and, in fact, almost all mechanical devices for altering the path of a moving body may be tested and compared with theory.

30. COEFFICIENT OF FRICTION.

Apparatus. AB, Fig. 26, is a board along which a block Cis drawn by a cord passing over a pulley D, and stretched by weights placed in the scale pan E. The friction is produced between the surfaces of C and AB, which should be made so that they may be covered with thin layers of various substances as different kinds of wood, iron, brass, glass, leather, &c. C is made of such a shape that by turning it over the area of the surface in contact may be altered. The pressure on C and the tension of the cord may also be varied at will, by weights.

D

E

A

B

Experiment. Weights are added to E in regular order, as when weighing, and the tension in each case compared with the friction of C. Friction may be of two kinds; first, that required to start a body at rest, called the friction of repose, and secondly, the friction of motion, or that produced when the bodies are moving. To measure the friction of repose, see if the weight is capable of starting the body when at rest, if so, stop it and repeat, varying the weight until a tension is obtained sufficient sometimes to start it and sometimes not. This friction is very irregular, varying with different parts of every surface, and with the time during which the two substances have been in contact. It is but little used practically, since the least jar converts it into the friction of motion. The latter is much less than the friction of repose, and more uni

Fig. 26.

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