The rotor shaft of the helicopter is subjected to the tensile force and torque shown when the rotor blades provide the lifting force to suspend the helicopter at midair. If the shaft has a diameter of 6 in., determine the principal stresses and maximum inplane shear stress at a point located on the surface of the shaft.
Definition 1.1. Given two integers a and d with d non-zero, we say that d divides a (written d | a) if there is an integer c with a = cd. If no such integer exists, so d does not divide a, we write d - a. If d divides a, we say that d is a divisor of a. Proposition 1.2.1: Assume that a, b, and c are integers. If a | b and b | c, then a | c. Proposition 1.3. Assume that a, b, d, x, and y are integers. If d | a and d | b, then d | ax + by. Corollary 1.4. Assume that a, b, and d are integers. If d | a and d | b, then d | a + b and d | a − b. Proposition 1.4. Let a, b, c ∈ Z be integers. a) If a | b and b | c, then a | c. b) If a | b and b | a, then a = ±b. c) If a | b and a | c, then a | (b + c) and a | (b − c). Prime: A prime number is an integer p ≥ 2 whose only divisors are 1 and p. A composite number is an integer n ≥ 2 that is not prime. Ex: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The Division Algorithm: Let a and b be integers with b > 0. Then there exist unique integers q (the quotient) and r (the remainder) so that a = bq + r with 0 ≤ r < b. The greatest common devisor: Assume that a and b are integers and they are not both zero. Then the set of their common divisors has a largest element d, called the greatest common divisor of a and b. We write d = gcd (a, b). 12: 1, 2, 3, 4, 6, and 12. 18: 1, 2, 3, 6, 9, and 18. Then {1, 2, 3, 6} is the set of common divisors of 12 and 18. Gcd(12,18) = 6 Proposition 1.10: If a and b are integers with d = gcd(a, b), then a b gcd , =1
