column bending under its own weight: A uniform thin column of length L, positioned vertically with one end embedded in the ground, will deflect, or bend away,from the vertical under the influence of its own weight when its length or height exceeds a certain critical value.It can be shown that the angular of the deflection θ(x) of the column from the vertical at a point p(x) is a solution of the boundary-value problem:
EI (d^2 θ)/(dx)^2 + δg(L-x)θ=0 , θ(0)=0 , θ^' (L)=0,
where E is young's modulus, I is the cross-sectional momment of inertia,δ is the constant linear density, and x is the distance along the column measured from its base.
(a) restate the boundary-value problem by making the change of variable t=L-x. Then use the result of the problem earlier in this exercise set to express the general solution of the differential equation in terms of bessel funtion.
(b) use the general solution found in part (a) to find a solution of the BVP and an equation which defines the critical length L, that is, the smallest value of L for which the column will start to bend.
(c) with the aid of a CAS, find the critical length L of a solid steel rod of radius r=0.05in, δg=0.28 A Ib/in,E=2.6 x 10^7 Ib/in^2, A = πr^2, and I= (1/4)πr^4.