Meriam, Kraige & Bolton — Statics of Particles: Problem 2_129

⚡ Mecademy AIENG정역학 · ch2  Problem Statement 2/129 The force F acts along an element of the right circular cone as shown. Determine the equivalent force–couple system at point O. Problem 2/129 ● Coordinate Analysis First, we define the coordinates of the two key points along which the force acts. Let the apex of the cone be point and the point on the base circumference be point . From the given diagram: The apex is located on the -axis at a height . Thus, its coordinates are . The base of the cone lies in the -plane with center at the origin . The radius of the base is given as . Point is on the base circumference at an angle from the positive -axis. Its coordinates are: So, . ● Force Vector Expression 1. Determine the displacement vector from to : F AB AzhA(0,0,h) xyO(0,0,0) R= 2 h Bθx x = B Rcosθ= cos θ 2 h y = B Rsinθ= sin θ 2 h z = B 0 B cosθ, sinθ,0( 2 h 2 h ) AB = AB(x − B x )i+ A (y − B y )j+ A (z − B z )k A = AB cosθ−0 i+ ( 2 h ) sinθ−0 j+ ( 2 h )(0−h)k= cos θ i+ 2 h si 2 h 2. Calculate the distance : 3. Express force as a vector: ● Equivalent Couple at Point O The equivalent couple is the moment of force about point . We can use the position vector of point relative to : . 1. Calculate the cross product: 2. Using unit vector cross product rules (, , ): The equivalent force–couple system at point is: Resultant Force: Equivalent Couple: ✨ Final Answer Summary ∣∣ AB d= = cosθ + sinθ +(−h) ( 2 h ) 2 ( 2 h ) 2 2 (cos θ +sin θ )+ h 4 h 2 2 2 2 F F=F = d AB cosθi+ sinθj−hk