Meriam, Kraige & Bolton — Statics of Particles: Problem 2_41

⚡ Mecademy AIENG정역학 · ch2  Problem Statement 2/41 In raising the pole from the position shown, the tension in the cable must supply a moment about of . Determine . Problem 2/41 1. Identify Geometry and Coordinates We set up a Cartesian coordinate system with the origin at the pivot point . Based on the diagram: The cable is attached to the pole at point , which is from . The pole is inclined at an angle of with the horizontal. The ground anchor point for the cable is point , located at a horizontal distance of to the left of . Coordinates of point : Coordinates of point : The position vector from to point is: 2. Express Tension as a Vector The tension force acts along the line from to . The vector is: The magnitude of (the length of the cable segment) is: T O72 kN⋅mT O(0,0) P30 mO 60 ∘ Q 12 mO P x = P 30cos60= ∘ 30×0.5=15 m y = P 30sin60= ∘ 30× = 2 3 15 ≈ 325.981 m Q x = Q −12 m y = Q 0 m OP r= OP 15i+ 15 j (m) 3 TPQ PQ = PQr − Q r = P (−12−15)i+(0− 15 )j= 3−27i− 15 j (m) 3 PQ The force vector is given by its magnitude multiplied by the unit vector from to : 3. Calculate the Moment about The moment is the cross product of the position vector and the force vector : Using the property , , and , : The magnitude of the moment is: 4. Solve for Tension Substitute the given moment value : Tension L= = (−27)+(−15 ) 2 3 2 = 729+675 = 1404 6 ≈ 3937.470 m TTPQ T=T = L PQ (−27i− 1404 T 15 j) 3 O M O r OP T M = O r × OP T M = O (15i+ 15 j)× 3 (−27i−15 j)[ 6 39 T 3] i×i=0j×j=0i×j=kj×i=−k