23. [Fig. 13] represents a coil of flat rope whose greater diameter D and smaller diameter d are to be determined. The area of the hub about which the rope is to coil is (¼)πd², while the area included by the outer coil of rope is (¼)πD² hence, the area of annular space occupied by the rope is

(¼)πD² - (¼)πd² = (¼)π(D² - d²).

Such values for D and d must be chosen that the equation of moments in [Art. 22] is satisfied, while the area (¼)π(D²-d²) must correspond to the space occupied by the given rope when rolled.

Fig. 13

Illustration.—2,000 feet of rope ½ inch thick requires

2,000 × 12
————— = 12,000
2

square inches in which to be coiled. To satisfy the equation of moments, D must equal 3.5 d; hence, to satisfy both these conditions

(¼)π[(3.5d)² - d²] = 12,000;