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# Young's Modulus

To determine Young's Modulus for copper, two pieces of copper wire seven or eight metres in length are hung from the same support One wire carries a scale of millimetres fixed to it so that the length of the scale is parallel to the wire. A vernier is fixed to the other wire,1 by means of which the scale can be read to tenths of a millimetre. The wire is prolonged below the vernier, and a scale pan attached to it; in this weights can be placed. The wire to which the millimetre scale is attached should also carry a weight to keep it straight. Let us suppose that there is a weight of one kilogramme hanging from each wire.

Measure by means of a measuring tape or a piece of string the distance between the points of suspension of the wires and the zero of the scale. Let this be 716.2 centimetres. Now put into the pan a weight of 4 kilogrammes, and read the vernier. Let the reading be 2.56 centimetres. The length of the wire down to the zero of the vernier is therefore 718.76 centimetres.

Now remove the 4 kilogramme weight from the pan. The vernier will rise relatively to the scale, and we shall obtain another reading of the length of the wire down to the zero of the vernier. Let us suppose that the reading is 0.23 centimetre. The length of the wire to which the millimetre scale is attached is unaltered, so that the new length of the wire from which the 4 kilogramme weight has been removed is 718.53 centimetres.

Thus, 4 kilogrammes stretches the wire from 718.53 centimetres to 718.76 centimetres. The elongation, therefore, is 0.23 centimetre, and the ratio of the stretching force to the extension per unit length is

4x718.53/0.23, or 12500 kilogrammes approximately.

We require the value of Young's Modulus for the material of which the wire is composed. To find this we must divide the last result by the sectional area of the wire.

If, as is usual, we take one centimetre as the unit of length, the area must be expressed in square centimetres.

Thus, if the sectional area of the wire experimented on above be found to be 0.01 square centimetre (see §3), the value of the modulus for copper is

12500/0.01, or 1250000 kilogrammes per square centimetre.

The modulus is clearly the weight which would double the length of a wire of unit area of section, could that be done without breaking it.

Thus, it would require a weight of 1,250,000 kilogrammes to double the length of a copper wire of one square centimetre section.

The two wires in the experiment are suspended from the same support. Thus, any yielding in the support produced by putting on weights below or any change of temperature affects both wires equally.

It is best to take the observations in the order given above, first with the additional weight on, then without it, for by that means we get rid of the effect of any permanent stretching produced by the weight.

The wire should not be loaded with more than half the weight required to break it. A copper wire of 0.01 sq. cm. section will break with a load of 60 kgs. Thus, a wire of 0.01 sq. cm. section may be loaded up to 30 kgs. The load required to break the wire varies directly as the cross-section.

To make a series of determinations, we should load the wire with less than half its breaking strain, and observe the length; then take some weights off - say 4 or 5 kgs. if the wire be of about 0.01 sq. cm. section, and observe again; then take off 4 or 5 kgs. more, and observe the length; and so on, till all the weights are removed.

The distance between the point of support and the zero of the millimetre scale, of course, remains the same throughout the experiment. The differences between the readings of the vernier give the elongations produced by the corresponding weights.

The cross-section of the wire may be determined by weighing a measured length, if we know, or can easily find, the specific gravity of the material of which the wire is made. For, if we divide the weight in grammes by the specific gravity, we get the volume in cubic centimetres, and dividing this by the length in centimetres, we have the area in square centimetres.

It may more readily be found by the use of Elliott's wire-gauge (see §3).

Experiment. - Determine the modulus of elasticity for the material of the given wire.

Enter results thus:

 Length of unstretched wire 718.53 cm. Extension per kilogramme (mean of 4 observations) 0.075 cm Cross-section 0.01 sq.cm

Value of E 1,250,000 kilogrammes per. sq. cm.

 1 We believe that we are indebted indirectly to the Laboratory of King's College, London, for this elegant method of reading the extension of a wire.

Last Update: 2011-03-27