Meriam, Kraige & Bolton — Statics of Particles: Problem 2_42

⚡ Mecademy AIENG정역학 · ch2  Problem Statement 2/42 The lower lumbar region A of the spine is the part of the spinal column most susceptible to abuse while resisting excessive bending caused by the moment about A of a force F. For given values of F, b, and h, determine the angle which causes the most severe bending strain. Problem 2/42 (a) Determination of Angle for Maximum Moment ● Calculation Process 1. Identify the Geometry and Coordinate System: Establish a Cartesian coordinate system with the origin at point (the lower lumbar region). The position vector from to the point of application of the force (the hands) is given by: where is the horizontal distance and is the vertical distance. The force has a magnitude and is directed at an angle with the vertical. Expressing in component form: (Note: The choice of sign for components depends on the assumed direction of the force; however, the magnitude of the moment—which determines the "severity"—will lead to the same optimal angle regardless.) 2. Establish the Moment Equation: The moment of the force about point is the cross product of the position vector and the force vector: Evaluating the cross product: The magnitude of the moment, which causes the bending strain, is: θ θ AA F r=bi+hj bhF FθF F=Fsinθi−Fcosθj M A A M = A r×F=(bi+hj)×(Fsinθi−Fcosθj) M = A b(−Fcosθ)−h(Fsinθ)=−F(bcosθ+hsinθ) M=∣M ∣= A F(bcosθ+hsinθ) 3. Maximize the Moment Magnitude: To find the angle that causes the most severe bending strain, we maximize the m