Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_171

⚡ Mecademy AIENG정역학 · ch5  Problem Statement A homogeneous solid sphere of radius is resting on the bottom of a tank containing a liquid of density , which is greater than the density of the sphere. As the tank is filled, a depth is reached at which the sphere begins to float. Determine the expression for the density of the sphere. Problem 5/171 (a) Expression for the density of the sphere 1. Formula: At the instant the sphere begins to float, the normal force from the bottom of the tank becomes zero. Therefore, the weight of the sphere must be exactly balanced by the buoyant force provided by the displaced liquid. Where: is the weight of the sphere. is the buoyant force. 2. Substitution: The total volume of the sphere is . The submerged volume is the volume of the spherical segment of height . Using the formula for a spherical cap: Substituting these into the equilibrium equation: 3. Calculation: Cancel common terms () from both sides: r ρ l ρ s h ρ s ρ s W=F B W=ρ V g stotal F = B ρ V g lsubmerged V = total πr 3 43 V submerged h V = submerged (3 r − 3 πh 2 h) ρ πrg = s ( 3 4 3 )ρ (3r−h)g l [ 3 πh 2 ] g,π, 3 1 4ρ r= s 3 ρ h(3r− l 2 h) Solve for : 4. Result: ● Final Conclusion: The density of the sphere is expressed as . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/171 ρ s ρ = s ρ l 4r 3 h(3r−h) 2 ρ = s 4r 3 ρ h(3r−h) l 2 ρ = s ρ l 4r 3 h(3r−h) 2 ρ = s ρ l 4r 3 h(3r−h) 2