A half sfere shell with mass m and radius R rests at ahorizontal table. At the t

Question

A half sfere shell with mass m and radius R rests at ahorizontal table. At the top of the shell there is a hole. Theshell is filled with a liquid with density p1 up to height h<R,The air ha density p. How large i the heigth h,messured from the table, when theshell lifts from the table because of the pressure from theliquid? A half sfere shell with mass m and radius R rests at ahorizontal table. At the top of the shell there is a hole. Theshell is filled with a liquid with density p1 up to height h<R,The air ha density p. How large i the heigth h,messured from the table, when theshell lifts from the table because of the pressure from theliquid?

Explanation / Answer

This is a bouyancy and hydrostatic pressure question. But before I solve it, I must mention that in real life youalso need the density of the spherical shell because that, too, hasa bouyancy in air. In this case, you need to solve for the pressure the fluidexerts upwards on the shell. This will be based on thedifferential pressure on the shell casing in the verticaldirection. We will need to do a little calculus. . Let r = the radius of the hemisphere x = the horizontal distance from the center of thehemisphere y = the height of the hemisphere at "x" h = height of the water column . For any particlar point where h>=y, the force equationupward on the sphere will be F = (h-y) (1 - .a) g * (surface area *cos) Where (surface area * cos) is the projected horizontalarea (essentially a gaussian surface) . But since we are going to do an integration, we can substitute(surface area * cos) with 2x dx, accordingly F = (h-y) (1 - .a) g 2x dx y = (r2 - x2) x.final = (r2 - h2), so ourintegral will be: . F = [from r to x.final] (h-(r2 - x2)) (1 -.a) g 2x dx F =  [from r to x.final] (1 -.a) g 2 (h-(r2- x2)) x dx For the integration, see: wolfram integrator . F =  [from r to x.final] (1 -.a) g 2 * (1/3(r2- x2)3/2 - 1/2 h (r2- x2)) F =  (1 - .a) g 2 *[(1/3(r2 - r2)3/2 - 1/2 h(r2 - r2)) - 1/3(r2- ((r2- h2))2)3/2 + 1/2 h(r2 - ((r2- h2))2)] F =  (1 - .a) g 2 * [+ 1/2 h(r2 - (r2 - h2)) -1/3(r2 - (r2- h2))3/2 ] F =  (1 - .a) g 2 * [+ 1/2 h(h2) -1/3 (h2)3/2 ] F =  (1 - .a) g 2 * (1/2h3 - 1/3 h3 ) Note: This is a simplier answer than I thought :-) . Now, set F = mg mg = F =  (1 - .a) g 2 * 1/6h3 h = (3m/((1 - .a)))1/3 I recommend that you review this thouroughly and particularlook for any errors. It was long and tough and I could haveeasily lost some variables. Good luck ********** I gave this problem to some of my friends to try out (one ofwhom is an ME prof) and they got the same answer that I did. (Some had prettier math using sins and cos instead of(r2 - x2)). Fun problem --pretty answer. F =  (1 - .a) g 2 *[(1/3(r2 - r2)3/2 - 1/2 h(r2 - r2)) - 1/3(r2- ((r2- h2))2)3/2 + 1/2 h(r2 - ((r2- h2))2)] F =  (1 - .a) g 2 * [+ 1/2 h(r2 - (r2 - h2)) -1/3(r2 - (r2- h2))3/2 ] F =  (1 - .a) g 2 * [+ 1/2 h(h2) -1/3 (h2)3/2 ] F =  (1 - .a) g 2 * (1/2h3 - 1/3 h3 ) Note: This is a simplier answer than I thought :-) . Now, set F = mg mg = F =  (1 - .a) g 2 * 1/6h3 h = (3m/((1 - .a)))1/3 I recommend that you review this thouroughly and particularlook for any errors. It was long and tough and I could haveeasily lost some variables. Good luck ********** I gave this problem to some of my friends to try out (one ofwhom is an ME prof) and they got the same answer that I did. (Some had prettier math using sins and cos instead of(r2 - x2)). Fun problem --pretty answer. F =  (1 - .a) g 2 * [+ 1/2 h(r2 - (r2 - h2)) -1/3(r2 - (r2- h2))3/2 ] F =  (1 - .a) g 2 * [+ 1/2 h(h2) -1/3 (h2)3/2 ] F =  (1 - .a) g 2 * (1/2h3 - 1/3 h3 ) Note: This is a simplier answer than I thought :-) . Now, set F = mg mg = F =  (1 - .a) g 2 * 1/6h3 h = (3m/((1 - .a)))1/3 I recommend that you review this thouroughly and particularlook for any errors. It was long and tough and I could haveeasily lost some variables. Good luck ********** I gave this problem to some of my friends to try out (one ofwhom is an ME prof) and they got the same answer that I did. (Some had prettier math using sins and cos instead of(r2 - x2)). Fun problem --pretty answer. F =  (1 - .a) g 2 * [+ 1/2 h(h2) -1/3 (h2)3/2 ] F =  (1 - .a) g 2 * (1/2h3 - 1/3 h3 ) Note: This is a simplier answer than I thought :-) . Now, set F = mg mg = F =  (1 - .a) g 2 * 1/6h3 h = (3m/((1 - .a)))1/3 I recommend that you review this thouroughly and particularlook for any errors. It was long and tough and I could haveeasily lost some variables. Good luck ********** I gave this problem to some of my friends to try out (one ofwhom is an ME prof) and they got the same answer that I did. (Some had prettier math using sins and cos instead of(r2 - x2)). Fun problem --pretty answer. I recommend that you review this thouroughly and particularlook for any errors. It was long and tough and I could haveeasily lost some variables. Good luck ********** I gave this problem to some of my friends to try out (one ofwhom is an ME prof) and they got the same answer that I did. (Some had prettier math using sins and cos instead of(r2 - x2)). Fun problem --pretty answer.

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