Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_38

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the -coordinate of the mass center of the homogeneous hemisphere with the smaller hemispherical portion removed. Problem 5/38 (a) Determination of the -coordinate of the mass center 1. Formula: For a composite body consisting of two parts where one part is removed (a void), the -coordinate of the centroid (mass center for a homogeneous body) is given by: where: and are the volume and centroid of the large hemisphere. and are the volume and centroid of the removed small hemisphere. For a solid hemisphere of radius , the volume is and the centroid relative to the flat base along the axis of symmetry is . 2. Substitution: Substitute the given geometric values into the volume and individual centroid expressions: Large hemisphere: Small hemisphere: Now substitute these into the composite centroid formula: 3. Calculation: Simplify the numerator and denominator terms: x x x =xˉ V −V 12 V −V 1 xˉ 12 xˉ 2 V 1 xˉ 1 V 2 xˉ 2 rV= πr 3 23 =xˉ r 8 3 R = 1 R⟹V = 1 πR, = 3 23 xˉ 1 R 8 3 R = 2 ⟹ 2 R V = 2 π = 3 2 ( 2 R ) 3 πR, = 12 13 xˉ 2 = 8 3 ( 2 R ) R 16 3 =xˉ πR− πR 3 2 3 12 1 3 πR R− πR R( 3 2 3 )( 8 3 )( 12 1 3 )( 16 3 ) Numerator: Denominator: Final ratio: 4. Result: ● Final Conclusion: The -coordinate of the mass center for the given body is . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/38 Num= πR − 4 1 4 πR = 64 1 4 − πR = ( 64 16 64 1 ) 4 πR 64 15 4 Den= πR − 3 2 3 πR = 12 1 3 − πR = ( 12 8 12 1 ) 3 πR 12 7