Can the Same Equation for Moment of Inertia Apply to Non-Uniform Cylinders?

[rclakmal]

second moment of area ?

Homework Statement

In finding the metacentric height of a floating body ( G-center of gravity ,M-metacentric point ,B-buoyancy point )
so if the object is stable then M should be above G .
GM=BM-BG

i have learned that BM=I/V (where I second moment of are about the horizontal center line ,and V is submerged volume)
so if the object is a cylinder i can find I using Pi*d^(4)/64 ...but can i use the same equation if the cylinder is not a uniform one .


And is it the same thing meant by
1)second moment of area
2)Moment of inertia
Or are they two different values ?
Can they be connected using a another varaible ?

Homework Equations

The Attempt at a Solution

 

[Hao]

i have learned that BM=I/V (where I second moment of are about the horizontal center line ,and V is submerged volume)
so if the object is a cylinder i can find I using Pi*d^(4)/64 ...but can i use the same equation if the cylinder is not a uniform one .

The second moment of area is similar to the moment of inertia except for one extremely important point:

The moment of inertia includes an integral over density, $$\rho$$.
$$I_{inertia} = \int \rho r^2 dA$$.

The second moment of area does not.
$$I_{area} = \int r^2 dA$$.

Hence, for a uniform object, $$I_{inertia} = \rho I_{area}$$.

Therefore, the second moment of area is independent of the uniformity of the body.

 

[tiny-tim]

moment of inertia ~ second moment of area

Hi Hao!

You're right, and you're wrong!

For all normal physicists, you're right …

but some structural engineers use the name https://www.physicsforums.com/library.php?do=view_item&itemid=31" for the second moment of area, so they can be the same (and in this case I suspect they are)!