Area of the Cross-Section of a Cylindrical Tube

The area of the cross-section 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 emery-powder 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 air-syringe. 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.

The tube is then to be filled with pure⁽²⁾ mercury; this is best done by immersing it in a trough of mercury of the necessary length. [A deep groove about half an inch broad cut in a wooden beam makes a very serviceable trough for the purpose.] When the tube is quite full, close the ends with the forefinger of each hand, and after the small globules of mercury adhering to the tube have been brushed off, allow the mercury to run into a small beaker, or other convenient vessel, and weigh it. Let the weight of the mercury be w. Measure the length of the tube by the calipers or beam-compass, and let its length be l. Look out in the table (33) the density of mercury for the temperature (which may be taken to be that of the mercury in the trough), and let this be ρ. Then the volume v of the mercury is given by the equation

and this volume is equal to the product of the area A of the cross-section 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 cross-section, 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 l₁ l₂, l₃ . . . . 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 cross-section 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.

1	For this and a great variety of similar purposes an aspirating pump attached to the water-supply of the laboratory is very convenient.
2	A supply of pure mercury may be maintained very conveniently by distillation under very low pressure in an apparatus designed by \Veinhold (see Carl's Rep. vol. 15, and Phil. Mag., Jan., 1884).