Why is the restoring torque of a wire dominated by the shear modulus and not the stress?

Question

I have been looking into different suspensions for a torsional pendulum setup, and came across this paper by Quinn comparing (round) wires and (ribbon) strips.

You can find the article at doi: 10.1088/0026-1394/34/3/6

In equation (2) in the paper, the torsional constant of a round wire is given as:

$$ c_w = \frac{\pi r^4}{2L}\left(F + \frac{Mg}{\pi r^2}\right) $$

where, $r$ is the radius of the wire, $L$, the length, $M$, the mass of the load, and $F$ is the shear modulus of the suspension material.

They then mention that the stress term ${Mg}/{\pi r^2}$ is always much smaller than $F$ and therefore the restoring torque of a wire is always dominated by the elastic component.

This is the part that I have a bit of trouble understanding. Consider, $M = 2$ kg, $r = 30$ $\mu$m, $F = 53$ MPa $= 5.3 \times 10^{7}$ N/m$^2$

These values were approximated from the experimental values described in the paper

Then, ${Mg}/{\pi r^2} \approx 2 \times 10^{11}$ N/m$^2$

This is clearly larger than the shear modulus term, contrary to the previous statement about the elastic component always being the dominating factor. Can some please explain where I might be going wrong here? Much thanks!

Answer 1

A radius of 30 $\mu$m is very small for a wire to be used in an ordinary experiment, for example in an undergraduate laboratory. You can easily show that if a 2.0 kg load could be hung from it, the tensile stress would be about 7 GN m$^{-2}$. [I don't know how you arrive at your figure of $2\times 10^{11}\text{N m}^{-2}$ for $mg/\pi r^2$; I make it 7 GN m$^{-2}$ for $r=30\ \mu\text {m and}\ M=2.0\ \text{kg}$.] The ultimate tensile stress for copper is around 0.2 GN m$^{-2}$, and of steels, only some special types will withstand 7 GN m$^{-2}$, so you wouldn't normally be able to make such a thin wire support a 2 kg load.

With a more practicable radius of wire – let's say 300 $\mu$m – we have $mg/\pi r^2 = 0.07\ \text{GN m}^{-2}$, and that is pretty negligible compared with a shear modulus of 50 GN m$^{-2}$, the figure for copper.