Meriam, Kraige & Bolton — Statics of Particles: Problem 2_105

⚡ Mecademy AIENG정역학 · ch2  Problem Statement 2/105 A force F is applied to the surface of the sphere as shown. The angles and locate point P, and point M is the midpoint of ON. Express F in vector form, using the given x-, y-, and z-coordinates. Problem 2/105 ● 1. Define the Coordinates of Key Points Based on the diagram, let be the radius of the sphere centered at the origin . The points are defined as follows: Point : Located on the surface of the sphere in the -plane at an angle from the positive -axis. Point : Midpoint of the segment . Point : Located on the surface of the sphere with radius , azimuthal angle , and elevation angle (angle from the -plane). ● 2. Determine the Direction Vector The force is applied at point and its line of action passes through points and . From the diagram, the force vector is directed from toward . We first find the relative position vector: θφ RO(0,0,0) Nxyθ x r = N Rcosθi+Rsinθj+0k MON r = M r = 2 1 N cos θ i+ 2 R sin θ j+ 2 R 0k PRθ φxy r = P (Rcosφcosθ)i+(Rcosφsinθ)j+(Rsinφ)k MP FPMP MP = MPr− P r M = MPRcosφcosθ− cosθ i+ ( 2 R )Rcosφsinθ− sinθ j+ ( 2 R )(Rsin = MPRcosφ− cosθi+cosφ− sinθj+sinφk[( 2 1 )( 2 1 )] ● 3. Calculate the Magnitude of Since : Using the identity : ● 4. Express Force in Vector Form The force vector is the product of its magnitude and the unit vector in the direction of : Simplify the fraction: ✨ Final Answer Summary MP ∣∣= MPRcosφ− cosθ + Rcosφ− sinθ +[ R sinφ] [( 2 1 )] 2 [( 2 1 )] 2 2 ∣∣= MPR cosφ− (cos θ +sin θ