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# Coefficient of Linear Expansion of a Rod

We require to measure the length of a rod, or the distance between two marks on it, at two known temperatures, say 15°C. and 100°C.

The highest degree of accuracy requires complicated apparatus. The following method is simple, and will give very fair results.

A thick straight rod is taken, about 50 cm. in length, and a glass tube of 4 or 5 cm. bore and somewhat greater length than the rod. The tube is closed with a cork at each end, and through each cork a small piece of glass tubing is passed, and also a thermometer. Two fine scratches are made on the rod, one close to each end, at right angles to its length, and two other scratches, one across each of the former, parallel to the length. The glass tube is clamped in a horizontal position and the rod placed inside it, resting on two pieces of cork or wood in such a manner that the scratches are on the upper surface and can be seen through the glass. The whole should rest on a large stone slab - a stone window-sill serves admirably.

The piece of glass tubing in one of the corks is connected with a boiler from which steam can be passed into the tube, the other communicates with an arrangement for condensing the waste steam.

A pair of reading microscopes are then brought to view the cross-marks on the rod, and are clamped securely to the stone. The microscopes, described in §5, should be placed so that they slide parallel to the length of the rod; this can be done by eye with sufficient accuracy for the purpose.

If microscopes mounted as in §5 are not available, a pair with micrometer eye-pieces, or with micrometer scales in the eye-pieces, may be used.

For convenience of focussing on the rod which is in the glass tube, the microscopes must not be of too high a power. Their supports should be clamped down to the stone at

points directly behind or in front of the position of the microscopes themselves, to avoid the error due to the expansion of the metal slides of the microscopes, owing to change of temperature during the experiment

Call the microscopes A and B; let A be the left-hand one of the two, and suppose the scale reads from left to right. Turn each microscope-tube round its axis until one of the cross-wires in the eye-piece is at right angles to the length of the rod, and set the microscope by means of the screw until this cross-wire passes through the centre of the cross on the rod.

Read the temperature, and the scale and screw-head of each microscope, repeating several times. Let the mean result of the readings be

Temp.: 15°C
A: 5.106 cm.
B: 4.738 cm.

Now allow the steam to pass through for some time; the marks on the copper rod will appear to move under the microscopes, and after a time will come to rest again.

Follow them with the cross-wires of the microscopes and read again. Let the mean of the readings be

Temp. 100°C
A: 5.074 cm.
B: 4.780 cm.

Then the length of the rod has apparently increased by 5.106-5.074+4.780- .738, or 0.074 cm.

The steam will condense on the glass of the tube which surrounds the rod, and a drop may form just over the cross and hide it from view. If this be the case, heat from a small spirit flame or Bunsen burner must be applied to the glass in the neighbourhood of the drop, thus raising the temperature locally and causing evaporation there.

Of course the heating of the rod and tube produces some alteration in the temperature of the stone slab and causes it to expand slightly, thus producing error. This will be very slight, and for our purpose negligible, for the rise of temperature will be small and the coefficient of expansion of the stone is also small.

We have thus obtained the increase of length of the rod due to the rise of temperature of 85°. We require also its original length.

To find this, remove the rod and tube and replace them by a scale of centimetres, bringing it into focus. Bring the cross-wires over two divisions of the scale, say 10 and 60, and let the readings be

A: 4.576 cm.
B: 5.213 cm.

Then clearly the length of the rod at 15° is

50-(5.106-4.576)+(4.738-5.213), or

48.995 cm.

To find the coefficient of expansion we require to know the length at 0°C.; this will differ so little from the above that we may use either with all the accuracy we need, and the required coefficient is 0.074/(85 x 48.995), or 0.0000178.

Experiment. - Determine the coefficient of expansion of the given rod.

Enter results thus:

```Increase of length of rod between 15° and 100°: 0.074 cm.
Length at 15°: 48.995 cm.
Coefficient: 0.0000178
```

Last Update: 2011-03-27