Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_55

⚡ Mecademy AIENG정역학 · ch7  Problem Statement Two semicylindrical shells with equal projecting rectangles are formed from sheet metal, one with configuration (a) and the other with configuration (b). Both shells rest on a horizontal surface. For case (a) determine the maximum value of for which the shell will remain stable in the position shown. For case (b) prove that stability in the position shown is not affected by the dimension . Problem 7/55 (a) Maximum value of for stability in configuration (a) For a body with a curved base rolling on a flat horizontal surface, the equilibrium position is stable if the center of mass of the body is located below the center of curvature of the base at the point of contact. We define a coordinate system with the origin at the center of the semicylindrical shell, with the positive -axis pointing upwards. 1. Formula: The vertical position of the center of mass relative to is given by: For stability, we require . 2. Substitution: For the semicylindrical shell: The mass is proportional to the arc length, . Its centroid is located at from the center. For the two vertical rectangles: Each has height , so the total mass is proportional to . Their centroid is at the midpoint of their height, which is above the horizontal diameter (the level of ). 3. Calculation: Simplify the numerator: Set the condition for stable equilibrium: h h h GC C y y G C y = G = m ∑ i m y ∑ ii m +m shellrects m y +m y shellshellrectsrects y < G 0 m ∝ shell πry = shell −