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Area of the CrossSection of a Cylindrical Tube
The area of the crosssection of a narrow tube is best determined indirectly from a measurement of the volume of mercury contained in a known length of the tube. The principle of the method is given in Section 9. The tube should first be ground smooth at each end by rubbing on a stone with emerypowder and water, and then very carefully cleaned, first with nitric acid, then with distilled water, then with caustic potash, and finally rinsed with distilled water, and very carefully dried by passing air through it, which has been dried by chloride of calcium tubes.^{(1)} The different liquids may be drawn up the tube by means of an airsyringe. If any trace of moisture remain in the tube, it is very difficult to get all the mercury to run out of it after it has been filled.
and this volume is equal to the product of the area A of the crosssection and the length of the tube. Hence
If the length be measured in centimetres and the weight in grammes, the density being expressed in terms of grammes per c.c., the area will be given in sq. cm. The length of the mercury column is not exactly the length of the tube, in consequence of the fingers closing the tube pressing slightly into it, but the error due to this cause is very small indeed. This gives the mean area of the crosssection, and we may often wish to determine whether or not the area of the section is uniform throughout the length. To do this, carefully clean and dry the tube as before, and, by partly immersing in the trough, introduce a thread of mercury of any convenient length, say about 5 centimetres long. Place the tube along a millimetre scale, and fix it horizontally so that the tube can be seen in a telescope placed about six or eight feet off. By slightly inclining the tube and scale, adjust the thread so that one end of it is as close as possible to the end of the tube, and read its length in the telescope. Displace the thread through 5 cm. and read its length again; and so on, until the thread has travelled the whole length of the tube, taking care that no globules of mercury are left behind. Let l_{1} l_{2}, l_{3} . . . . be the successive lengths of the thread. Then run out the mercury into a beaker, and weigh as before. Let the weight be w, and the density of the mercury be ρ. Then the mean sectional areas of the different portions of the tube are
The mean of all these values of the area should give the mean value of the area as determined above. The accuracy of the measurements may thus be tested. On a piece of millimetre sectional paper of the same length as the tube mark along one line the different points which correspond to the middle points of the thread in its different positions, and along the perpendicular lines through these points mark off lengths representing the corresponding areas of the section, using a scale large enough to show clearly the variations of area at different parts of the length. Join these points by straight lines. Then, the ordinates of the curve to which these straight lines approximate give the crosssection of the tube at any point of its length. Experiment.  Calibrate, and determine the mean area of the given tube. Enter the result thus :  [The results of the calibration are completely expressed by the diagram.]
Mean of the five determinations for calibration 0.409 sq. mm.


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