Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_121

⚡ Mecademy AIENG정역학 · ch6  Problem Statement A semicylinder of uniform density rests on a horizontal surface and is subjected to a force applied as shown. If is slowly increased and kept normal to the flat surface, plot the tilt angle as a function of up to the point of slipping. Determine the tilt angle and the corresponding value for which slipping occurs. The coefficient of static friction is . Problem 6/121 (a) Relationship between the tilt angle and applied force ● Calculation Process 1. Present Final Formula: For a uniform semicylinder of radius , the distance from the geometric center of the flat side to the center of mass is given by . Considering the moment equilibrium about the instantaneous contact point on the horizontal surface, and knowing that the force is applied at the edge (distance from ) and remains normal to the flat surface, the moment equation is: 2. Substitute Values: Substitute into the moment equation: 3. Partial Operations: Simplify the equation by dividing through by : Rearrange to solve for : P P θPθ max P max 0.35 θP r OG=rˉ 3π 4r C Pr O M = ∑ C rP(1−sinθ)−Wsinθ=rˉ0 =rˉ 3π 4r rP(1−sinθ)=W sin θ( 3π 4r ) r P(1−sinθ)= sin θ 3π 4W sinθ P=Psinθ+ sin θ = 3π 4W P+ sin θ( 3π 4W ) sinθ= P+ 3π 4W P 4. Final Calculation: Taking the inverse sine, we obtain the functional relationship for the tilt angle: where . ● Final Conclusion: The tilt angle as a function of applied force is . As increases, increases from zero. (b) Determine the tilt angle and correspo