Measurement of Young'' $ Modulus. 409 



In the bifilar oscillation the effect of flexure is negligible. 



In the rolling oscillation when the centre of gravity is at 

 the same level as the points of attachment of the wires there 

 is a quick oscillation due entirely to the resistance to flexure 

 of the wires. 



In the pitching oscillation both stretching and flexure of 

 the wires are involved. The influence of flexure alone may 

 be ascertained from the rolling oscillation, and by allowing 

 for this in the pitching oscillation the stretch modulus may 

 be deduced. 



This method has the disadvantage of requiring the moments 

 of inertia of the needle to be separately determined. This, 

 however, can be avoided by the following modification of 

 the experiment. 



If the centre of gravity of the needle is raised above the 

 level of the points of attachment of the wires the period of the 

 rolling oscillation is lengthened ; and by suitably adjusting 

 the height of the centre of gravity this period may be made 

 infinite. In that case the effect of gravity exactly counteracts 

 the effect of flexure for a small rolling displacement. This 

 being so, if we set the needle to pitch, the effects of gravity 

 and of the flexure will still exactly counteract each other, and 

 the resistance to stretching of the wires will alone control 

 and determine the period of the pitching. 



In adjusting the apparatus for this experiment it is neces- 

 sary to take care that the moments of inertia involved in the 

 bifilar and pitching oscillations are equal. This may be 

 secured by fixing at each end of the bar of the needle a cross 

 consisting of four equal screws at right angles to the length 

 of the needle, two horizontal and two vertical. Nuts on 

 these screws afford a convenient means of adjusting the 

 position of the centre of gravity and the moments of inertia. 



For the determination of the stretch modulus, however, 

 this adjustment of the centre of gravity is not necessary. 

 The double observation which eliminates the effect of torsion 

 eliminates at the same time the effect of flexure of the wires. 



The equation of motion of the pitching oscillation may be 

 written 2 



m^e= -(~ +/)#, 



where fd expresses the couple due to flexure, this couple 

 being independent of the distance between the wires. The 

 frequencies of pitching with the wires 2c and 2d apart 

 respectivelv are given bv 



PJiil. Mag. S. 6. Vol. 4. No. 21. Sept. 1902. 2 E 



