Eduprobe
Question:
A uniformly tapering conical wire is made from a material of Young's modulus Y and has a normal, unextended length L. The radii, at the upper and lower ends of this conical wire, have values R and 3R, respectively. The upper end of the wire is fixed to a rigid support and a mass M is suspended from its lower end. The equilibrium extended length, of this wire, would equal to:
L(1+13MgπYR²)
L(1+29MgπYR²)
L(1+19MgπYR²)
L(1+23MgπYR²)
Solution:
Take a cross section at distance x from the bottom, hence at this point, r = 3R − x/L 2R. Mgπr² = Y dy/dx = Mgπ(3R − x/L 2R)² where dy is elongation of the portion dx. Integrating this equation for x in 0 to L, we get, y = L(1+13MgπYR²)