Answered: Find the height of the metacenter from…

Name: Find the height of the metacenter from the top of the beaker
Uploaded: 2024-03-20T06:56:55.000Z
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Description: Find the height of the metacenter from the top of the beaker

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Find the height of the metacenter from the top of the beaker

Expert Solution

Step 1 Metacentric height calculation

case1: Cylinder is vertical (projected area is circular, view from free surface)

Metacentric height, (GM)

$G M = \frac{I_{m i n}}{V} - B G w h e r e B G - d i s \tan c e b e t w e e n c e n t r o i d a n d c e n t e r o f b u y o u a n c y I_{m i n} - s e c o n d a r e a m o m e n t V - v o l u m e o f t h e s u b m e r g e d o b j e c t I_{m i n} = I_{X X} = I_{Y Y} = \frac{π}{64} d^{4} = \frac{π}{64} {(2 R)}^{4} = \frac{π}{4} R^{4} V = \frac{π}{4} \times {(2 R)}^{2} \times 2 h = 2 π R^{2} h since BG = 0 in case of submerged body G M = \frac{\frac{π}{4} R^{4}}{2 π R^{2} h} - 0 G M = \frac{R^{2}}{8 h} f r o m t o p G l i e s a t \bar{h} \bar{h} = h + \frac{2 h}{2} = 3 h t h e r e f o r e p o i n t M l i e s a t \bar{h} - G M 3 h - \frac{R^{2}}{8 h} (A n s)$

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