The wrench (R, K) along the axis by which the spring can be held in the form (α′, r′) is given by the equations
| R = A | sin α′ | ( | cos² α′ | − | cos² α | ) − C | cos α′ | ( | sin α′ cos α′ | − | sin α cos α | ), |
| r′ | r′ | r | r′ | r′ | r |
| K = A cos α′ ( | cos² α′ | − | cos² α | ) + sin α′ ( | sin α′ cos α′ | − | sin α cos α | ). |
| r′ | r | r′ | r |
When the spring is slightly extended by an axial force F, = −R, and there is no couple, so that K vanishes, and α′, r′ differ very little from α, r, it follows from these equations that the axial elongation, δx, is connected with the axial length x and the force F by the equation
| F = | Eπc4 | sin α | δx | , | ||
| 4r² | 1 + σ cos² α | x |
and that the loaded end is rotated about the axis of the helix through a small angle
| 4σFxr cos α | , |
| Eπc4 |
the sense of the rotation being such that the spring becomes more tightly coiled.
66. A horizontal pointer attached to a vertical spiral spring would be made to rotate by loading the spring, and the angle through which it turns might be used to measure the load, at any rate, when the load is not too great; but a much more sensitive contrivance is the twisted strip devised by W.E. Ayrton and J. Perry. A very thin, narrow rectangular strip of metal is given a permanent twist about its longitudinal middle line, and a pointer is attached to it at right angles to this line. When the strip is subjected to longitudinal tension the pointer rotates through a considerable angle. G.H. Bryan (Phil. Mag., December 1890) has succeeded in constructing a theory of the action of the strip, according to which it is regarded as a strip of plating in the form of a right helicoid, which, after extension of the middle line, becomes a portion of a slightly different helicoid; on account of the thinness of the strip, the change of curvature of the surface is considerable, even when the extension is small, and the pointer turns with the generators of the helicoid.
If b stands for the breadth and t for the thickness of the strip, and τ for the permanent twist, the approximate formula for the angle θ through which the strip is untwisted on the application of a load W was found to be