A bar in tension has a circular cross section and includes a conical portion of length l , as Shown, the task is to find the spring rate of the entire bar. Equation (4–4) is useful for the outer portions of diameters d1 and d2, but a new relation must be derived for the tapered section. If α is the apex half-angle, as shown, show that the spring rate of the tapered portion of the shaft is