fascinated said:
Any idea for how much sag with piano wire and if these lengths are accessible?
At 3 millimeters, steel wire has a tensile strength around 1500 MPa according to
this manufacturer. Steel has a specific gravity of about 8 (8 kg/liter or 8000 kg/m
3).
We should be able to use these figures to determine both the tensile strength of a 3 millimeter steel wire and the linear density of the same.
1500 MPa is 1500 million Newtons per square meter. We are talking about a circular cross-section with a radius of 1.5 mm. ##a = \pi r^2## so that is 0.000007 square meters (seven millionths of a square meter). Multiplied by 1500 so we have about 10,000 Newtons.
But you are more comfortable with imperial units. So call that one ton. 2000 pounds, more or less.
Meanwhile, a cross section of seven millionths of a square meter multiplied by about 1.5 kilometers should give us one one hundredth of a cubic meter. Or about 80 kg. Call it 160 pounds mass. Which under one gee is also 160 pounds force. (Yay, imperial units can be handy -- once in a blue moon).
So now we have the sine of our sag angle: 160/2000. Multiply by 2500 and we have a sag of 200 feet fopr a Vee shape. Divide by 2 for the catenary curve and we have a sag of 100 feet, give or take.
Really, you should start doing these calculations yourself. You learn more by working problems than by asking for finished answers.
I am a bit surprised that steel does this poorly. Googling at random, I found some
1/8 inch kevlar cord with a polyester jacket. 825 pound breaking strength. I did not see density figures. But 3600 MPa with a specific gravity of 1.4 is a heck of a lot better than steel (1500 and 8 as indicated above).