Meriam, Kraige & Bolton — Internal Forces and Moments: Problem 7_37

⚡ Mecademy AIENG정역학 · ch7  Problem Statement The body consisting of the solid hemisphere (radius and density ) and concentric right- circular cone (base radius , height , and density ) is resting on a horizontal surface. Determine the maximum height which the cone may have without causing the body to be unstable in the upright position shown. Evaluate for the case where (a) the hemisphere and cone are made of the same material, (b) the hemisphere is made of steel and the cone is made of aluminum, and (c) the hemisphere is made of aluminum and the cone is made of steel. Problem 7/37 (a) General derivation and same material case 1. Formula: For stability of a body resting on a spherical surface, the center of mass must be located at or below the center of curvature of the contact surface. Let the origin be at the center of the common base , with the -axis positive upwards. The position of the composite center of mass is: For stability, we require , so the limit for maximum height occurs when: 2. Substitution: Identify mass and center of mass for the hemisphere (part 1) and cone (part 2): Hemisphere: , , Cone: , , Substituting into the stability condition: 3. Calculation: Simplify the equation: rρ 1 rhρ 2 h G O Oy =yˉ = m ∑ i m y ∑ ii m +m 12 m y +m y 1122 ≤yˉ0h m y + 11 m y = 22 0 V = 1 πr 3 23 y = 1 − r 8 3 m = 1 ρ V 11 V = 2 πrh 3 12 y = 2 h 4 1 m = 2 ρ V 22 ρ πr− r + 1 ( 3 2 3 )( 8 3 )ρ πrh h = 2 ( 3 1 2 )( 4 1 )0 − ρ πr + 4 1 1 4 ρ πrh = 12 1 2 22 0 Divide both sides by :