Assume that the column is a homogeneous circular cylinder with small cross sections loaded by its own weight. The free end of the column will bend away when its height exceeds a certain critical value. Let x=0 be at the top free end and x=L at the base (assume the column is embedded in concrete).
diagram 1:Edwards Penny p.255
Let θ(x) be the angle between the tangent to the column at the arbitrary point x and the vertical, i.e. the angular deflection of the column.
Question 1. What other variables and constants do we need to consider? (See the appendix for a derivation of our basic equation (1). E=Young's modulus, I= moment of inertia of column, ρ is the linear density)
Question 2. Evaluate γ^{2} E=Young's modulus =2.8x10^{7 }lb/in^{2} for a steel column. I=moment of inertia of a circular disk= .25π r^{4}. ρ=δA, so g ρ=gδA, A is the cross sectional area. Take gδ=.28 lb/in^{3}. Put this into your answer to question 4 to find the "buckling point" or critical height of a flagpole of diameter 1 in.
Question 3. Transform the differential equation (1) to a Bessel differential equation by first using the substitution z=γ^{2/3}x to obtain the differential equation θ''+z θ=0 and then make the substitution θ(z)=u(t)z^{1/2} where t=^{2}/_{3}z^{3/2}. Be careful in using the chain rule. You should get the Bessel differential equation : t^{2}u''+tu'+(t^{2}- 1/9)u=0. Show that the solution to our problem in terms of the Bessel function would be the zeros of J_{-1/3} (2/3 γx^{3/2})
Use Maple (or any other CAS) software to find roots of approximate equation for critical height in Question 2 and to find the zero of the Bessel's function in Question 6. Explain the discrepancy between critical height found in Qiestion2 and Question3.
In Maple use the following commands:
> plot(BesselJ(-1/3,x),x=0..16);
> r1:=RootOf(BesselJ(-1/3,x)=0,x,2);
> h:=solve((2/3)*gam*x^(3/2)=r1,x);