Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_58

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the depth of the square cutout in the uniform hemisphere for which the - coordinate of the mass center will have the maximum possible value. Problem 5/58 (a) Determination of the depth for maximum -coordinate of the mass center 1. Present Final Formula: We treat the body as a composite of a full hemisphere (Body 1) and a square prism cutout (Body 2) which is removed. Let the origin be at the center of the flat circular face and the -axis point into the hemisphere. The - coordinate of the centroid of the composite body is given by: 2. Substitute Values: From the geometry of the problem, the side of the square cutout is . The properties of the individual parts are: Hemisphere (Body 1): , Square Prism (Body 2): , Substituting these into the centroid formula: 3. Partial Operations: To find the maximum value of , we set its derivative with respect to to zero: . Using the quotient rule : Setting the numerator to zero and dividing by : hz hz zz =zˉ V−V 12 V −V 1 zˉ 12 zˉ 2 s= + 2 r = 2 r r V = 1 πr 3 23 =zˉ 1 r 8 3 V = 2 sh= 2 rh 2 =zˉ 2 2 h (h)=zˉ = πr−rh 3 2 32 ( πr )( r )−( r h)( ) 3 2 3 8 3 2 2 h πr−rh 3 2 32 πr− rh 4 1 4 2 1 22 zˉ h = dh dzˉ 0 = ( v u ) ′ v 2 uv−uv ′′ = dh dzˉ = ( πr−rh) 3 2 322 (−rh)( πr−rh)−( πr− rh)(−r) 2 3 232 4 14 2 1222 0 r 2 Combine like terms and divide by : Multiply by 12 to clear fractions: 4. Final Calculation: Solve the quadratic equation using the quadratic formula : Numerical evaluation: (dis