Final answer:
The higher failure rate of flexible nails is associated with weights that exceed the elastic capabilities of the nail, as determined by its shear modulus and material properties. When nails are subjected to weights greater than what they can support, they bend more and the failure rate increases.
Step-by-step explanation:
When considering the link between weight and the failure rate of flexible nails, physics principles related to stress, strain, and material properties are essential. The shear modulus is a measure of a material's ability to resist shearing when a force is applied. In the context of hanging a picture, a nail's slight flexure can be a result of the shearing effect due to the picture's weight. The elasticity of the nail depends on this modulus and consequently on the material it's made of.
To find the mass of the picture that a steel nail can support without bending excessively as illustrated in Figure 5.17, where the nail bends only 1.80 µm, we must know the shear modulus of the nail. In general, nails that are subject to higher weights than they can support have a greater probability to fail or bend. Assuming the shear modulus and other variables such as the initial length of the nail (Lo) are known to two significant figures, calculations can be made to determine the maximum supported weight before the risk of failure increases.
If the referenced discussion implies that the nail is holding a 'fairly massive' picture with only a minute deflection, then a heavier weight would likely increase the deflection beyond the proportional elastic limit and lead to a higher rate of failure. The effectiveness of a nail in supporting weight relies heavily on avoiding weights that would cause excessive deflection or bending, leading to potential failure.