A special cable is designed to be hung vertically and support a load W at a distance L below the support. The cable is to be made of a material having a Young's modulus Y and a weight w per unit volume. The cable is tapered so that the cross sectional area A varies with height z above the weight. The cable material can safely support the load for a stretch of no more than 1%. Determine the function A(z) to support a maximum load.
Work:
L2-L1 = change in length of cable
A = cross sectional area of cable, which varies with height z
stress = Young's modulus * (L2-L1/L1)
F/A = stress, (L2-L1/L1) = strain
F/A = Y*(L2-L1/L1)
F/(pi*r^2) = Y*(L2-L1/L1)
F = Y*(L2-L1/L1)*(pi*r^2)
A(z) = (integral) F dz
A(z) = (integral) Y*(L2-L1/L1)*(pi*r^2) dz
A(z) = Y*(L2-L1/L1)*(pi*r^2) (integral) dz
A(z) = Y*(L2-L1/L1)*(pi*r^2)*z
Solution:
A(z) = Y*(L2-L1/L1)*(pi*r^2)*z