Long wire problem: Deriving an expression from the Bessel Function

[fuzzeleven]

A straight wire clamped vertically at its lower end stands vertically if it is short, but bends under its own weight if it is long. It can be shown that the greatest length for vertical equilibrium is l, where kl^(3/2) is the first zero of J_-1/3 and k=4/3r²*√(ρg/∏Y) where r is the radius, ρ is the linear density, g is the acceleration of gravity, and Y is the Young modulus. Find l for a steel wire of radius 1 mm; for a lead wire of the same radius.


What I've done so far is plug -1/3 in for p to simplify the Bessel function, and then set that equal to zero. What I really am interested in is how to derive the expression with the Bessel function, but I really don't know where to start. Any suggestions on how to tackle this? Much appreciated.

 

[rude man]

According to a calculator I found on-line, the first zero of J_-1/3 = 1.866350858873895. So
1.866350858873895 = kl^3/2
You know the other parameters so just solve for l.

PS - I can't vouch for the calculator. It's at http://cose.math.bas.bg/webMathematica/webComputing/BesselZeros.jsp

I suppose in a way that's cheating but that's what you'd do in real life - use all available data. I don't know of an equation that would yield the zeros explicitly.

I don't understand what you tried to do, why you did it, and what is p?

 

[LawrenceC]

For the zeroes, you might try the Handbook of Mathematical Functions by Abramowitz and Stegun. The have polynomial respresentations for Bessel functions that are good to 8 or 9 significant figures over designated ranges. You can find the zeroes of the polynomials by Newton's method or other root finding methodology. I cannot look in the book because I no longer have access to a copy but have used their polynomials to represent products of Bessel functions (orthogonality considerations) with good accuracy.