# Homework Help: Stability of object

1. Mar 5, 2016

### foo9008

1. The problem statement, all variables and given/known data
is it necessary to find the meta center for this question ? the formula of meta center is I /Vd , the I ( moemtn of inertia ) for rectangle is (ab^3) / 12 , but what is a ? what is b ? i'm confused now

2. Relevant equations

3. The attempt at a solution
ρ
gVsubmerged = weight(579N)
1025(9.81)Vsubmerged = weight(579N)
Vsubmerged = 0.058m^3

0.058m^3 = (1.2x0.9x hsubmerged)
hsubmerged =0.054m

center of buoyancy = 0.054/2 = 0.027m
metacenter = I / Vd = [(ab^3) / 12 ] /Vd

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2. Mar 5, 2016

3. Mar 5, 2016

### haruspex

For a rectangle measuring a by b, the second moment of area about an axis through the mid points of the sides of length b is $\frac 1{12}ab^3$.

4. Mar 11, 2016

### foo9008

what is the value of a and b ? is the second moment about x -axis(Ixx) , right ? IMO , a is 0.9m , b is 1.2m , am i correct ?

Last edited: Mar 11, 2016
5. Mar 11, 2016

### haruspex

In principle, you should consider each axis as a potential axis for tipping. However, it is obvious that one of ab3, a3b is the larger. Which gives the greater danger of tipping, the larger or the smaller?

6. Mar 11, 2016

### foo9008

how to know which axis is more danger of tipping ?

the hydrostatic pressure act on the top and the bottom surface of the object , so i am consdiering the moment about x of tipping

7. Mar 11, 2016

### haruspex

Is tipping more likely with a high metacentre or a low one?

8. Mar 11, 2016

### foo9008

low metccenter , right ?

9. Mar 11, 2016

### haruspex

10. Mar 11, 2016

### foo9008

low metacenter means the M is below G , right ? so , the GM is negative , so the object is unstable ?

11. Mar 11, 2016

### foo9008

in the link , it shows that the high metacenter of object enable the object to have higher stability against overturning, so the low metecanter of the object less stable , right ?

12. Mar 11, 2016

### haruspex

Yes.
To get a low metacentre, does "I" need to be large or small? If a>b, which of ab3 and a3b will give that less stable i?

13. Mar 12, 2016

### foo9008

to get low metacenter , the I has to be small. since the water pressure act on the bottom of the box , the b is 1.2m , a is 0.9m , am i right ?

14. Mar 12, 2016

### haruspex

It depends how you are defining a and b. Is that using the a3b form or the ab3 form?

15. Mar 12, 2016

### foo9008

I am using
ab3

16. Mar 12, 2016

### haruspex

Does a=.9, b=1.2 give a lower I than the other way around?

17. Mar 12, 2016

### foo9008

a=1.2 , b = 0.3 , i would get lower metacenter , but the formula is yp = yc + Ixx / (ycA) , Ixx is the second moment of inertia about x-axis . Ir cant be changed , right ?

18. Mar 12, 2016

### haruspex

The object in question could tip about either of two axes. If you fix one direction as x and apply that formula you will only be considering one axis of tipping.

19. Mar 12, 2016

### foo9008

why not 3 axes? since the object is 3d ( as shown in the diagram )

20. Mar 12, 2016

### foo9008

for a = 1.2 , b = 0.3 , i will get metacenter = 1.25m , but , which value to choose ? 1.25m or 2.22m( a = 0.3, b = 1.2 ) ?
if metacenter = 1.25m or 2.22m , G located at 0.86m +0.3m =1.16 m , object is stable

21. Mar 12, 2016

### haruspex

Yes, it's 3D, and you are right that in principle it could tip over any axis. But it is fairly obvious that the most likely tip is about the natural axes. Feel free to analyse the general case.
To check for stability, consider the worst case, i.e. the least metacentric height.
The metacentric height you calculated, 1.25m, is the height from the centre of buoyancy, not from the base.
But that only makes it even more stable than you concluded.

22. Mar 12, 2016

### foo9008

Last edited: Mar 12, 2016
23. Mar 12, 2016

### foo9008

i thought when we consider the center of pressure , we only need to consider the Ixx ( second moment of inertia about x -axis) , y -axis is the vertical axis ..... this is the first question i ever encountered that we need to find the second moment of inertia about 2 axes

24. Mar 12, 2016

### foo9008

Why do we wanna consider 2 axes for rotation of object?
Normally we consider only 1 axis of rotation of object, right?

25. Mar 12, 2016

### SteamKing

Staff Emeritus
You're getting confused here.

We're talking about the stability of this platform. The center of pressure, etc. has no bearing on this analysis.

The location of the metacenter M depends on the inertia of the waterplane cut by the hull of the platform. Using b = 0.3 would make no sense, since this dimension represents the depth of the platform's hull.

The Ixx used in the equation for yp represents something else and is not applicable here.

For the stability of this platform, BM = I / V, but I can be calculated about an axis which either runs parallel to the length of the platform or parallel to the breadth of the platform, which ever choice leads to the least value of I. V = volume of displacement and remains the same in any event.