Help! Moment of inertia??

[genralz]

Does a square have a greater moment of intertia? or a circle?

[mooeypoo]

Do you have the equation/method of calculating moment of inertia?

It will help you figuring this out; it's pretty straight-forward method, but I don't want to just give you a solution.. If you want to post the equation and start the method, we'll surely help out.

[Bignose]

Not to be too pedantic, but it definitely depends on the size of the circle and the square, as well. It very much depends on whether you are talking about the circle that circumscibes the square, or the circle the inscribes the square. Or maybe the circle with the same area as a given square (assuming constant density)? Each of these has a potentially different answer, the specific situation needs to be made clearer.

 

And, what will help answer the question is the equation of moment of inertia... like moo said.

[Kedas]

A square or circle doesn't have a volume/mass therefore it has no moment of inertia.

used if you want to know how much energy is in there.

 

You mean

http://en.wikipedia.org/wiki/List_of_area_moments_of_inertia

"The second moment of area, also known as the area moment of inertia or second moment of inertia "

used if you want to know how easy it will bend and where.

[Bignose]

Kedas, while physically being merely a 2-D object a circle or square wouldn't have any mass in the real world, it doesn't prevent us from mathematically talking about a circle or square's MOI. If you want to start down that road, why not just say that a circle or square can't have an MOI because there is no such thing as a perfect circle or perfect square?

 

Mathematically we can assign an area density or mass per unit area and make it mathematically have mass. Just like mathematically we can and do use point masses and point charges all the time, despite there being no such thing in the real world. It is a simplifying assumption made primarily to make the math easier and get a result that is going to be very close to the real world. To a very large extent, it doesn't really matter at all that no 2-D object has mass, or that there is no such thing as a perfect circle, mathematically such objects do exists and we can perform mathematics on them.

[Kedas]

Kedas, while physically being merely a 2-D object a circle or square wouldn't have any mass in the real world, it doesn't prevent us from mathematically talking about a circle or square's MOI. If you want to start down that road, why not just say that a circle or square can't have an MOI because there is no such thing as a perfect circle or perfect square?

 

Mathematically we can assign an area density or mass per unit area and make it mathematically have mass. Just like mathematically we can and do use point masses and point charges all the time, despite there being no such thing in the real world. It is a simplifying assumption made primarily to make the math easier and get a result that is going to be very close to the real world. To a very large extent, it doesn't really matter at all that no 2-D object has mass, or that there is no such thing as a perfect circle, mathematically such objects do exists and we can perform mathematics on them.

 

From what I read genralz isn't interested in math for the math.

The simple fact is that the MOI formula requires a mass to have some meaning.

Yes you can make up an mass that has no meaning, I don't care.

 

genralz, in case you mean MOI of a cylinder or cubiod etc.

here are the formula's.

http://en.wikipedia.org/wiki/List_of_moments_of_inertia

[Bignose]

But I'm not just "making up a mass that has no meaning". Consider if I made a square and a circle out of very thin sheet metal. Sure, technically, that is a very squat cylinder or a parallelepiped with a very thin depth, but if you use an MOI calculation for a 2-D object with a given mass per unit area, the answer between the 3-D and the 2-D calculations will be negligibly different -- implying that the 2-D approximation is very, very good.

 

Let's even look at your list of moments of inertia. Specifically, the one for a solid cylinder:

 

[math]I_z = \frac{m r^2}{2}[/math] where [math]I_z[/math] is the MOI about the axis, [math]m[/math] is the mass of the cylinder, and [math]r[/math] is the radius. Note that the height, [math]h[/math] isn't even part of the equation! That means that it doesn't matter if the height is 1 cm, 1 million cm, or even infinite or zero centimeters! The answer is the same!

 

Even further proving my point is that the formula for the cuboid and the formula for the "thin rectangular plane" are the same! The depth doesn't matter at all! You don't even have to perform the 3-D calculations to get the same answer as the 2-D ones, meaning that you don't have to do the extra work of the full 3-D calculation.

 

And, the fact is, you can define densities not just as mass per unit volume. In the sheet metal example above, it is perfectly reasonable to define a mass per unit area. For a long thin rod or chain, it is perfectly reasonable to define a mass per unit length -- and is done so often to predict the shape a hanging chain or rope will take. Sure, mathematically, a line is only 1 dimensional, but the math of that 1 dimensional object can be used to predict with incredible accuracy the behavior of a true 3 dimensional object such as a rope or chain. Just like the math for a 2-D object like a circle or square can be used to predict very accurately the behavior of a true 3-D object like a circle or square cut out of sheet metal. Of course in reality there is that 3rd dimension, but it isn't important for the math. It is definitely possible to find the moment of inertia of a circle or square.

[D H]

The problem with this thread is not that the concept of moment of inertia doesn't make sense for a square or a circle. As Bignose just pointed out, there's nothing wrong with the concept. The problem is that the question as posed in the original post is ill-formed. The moment of inertia of a circular manhole cover about an axis normal to and running through the center of the cover is a lot greater than the moment of inertia of a small square of plastic.

 

I'll re-ask the question. If you take two equal masses of the same material and form them into a thin circular disc and a thin square plate, which will have the greater moment of inertia? The answer is the square. The circle is the shape that minimizes the moment of inertia.

[Kedas]

Not sure why you talk about 2D formula's, there are none for MOI.

You know very well that it is not possible to have a mass without a volume.

Maybe on some quantum level but I'm sure these formulas don't apply there.

Even if the height is only dh it would still be a cylinder.

And simplifying the formula for certain applications is ok for me but that still doesn't mean that a circle has a MOI.

Unless you like to change the definition of a circle that it has a height.

 

What you are saying is as interesting as saying that a circle (r>0,L=0) does or doesn't have an electrical resistance. if there is no distance to travel there no meaning in the word electrical resistance.

[Bignose]

Kedas,

 

One more time, while I agree that in physical reality there is no such thing as a solely 2-D object, that in no way whatsoever hinders the development of the mathematics of a 2-D object. The definition of MOI involves a triple integral over 3 space dimensions. Mathematically, if the object is only 2-D, then it acts just like it is a Dirac delta function in that third dimension.

 

And, one more time, treating an object that is very nearly purely 2-D, as in a 1m circle made of sheet metal, that is approximating that very small thickness of the sheet metal as a delta function, can be very accurate. The difference between the two is almost negligible. It is a difference between mathematics and physical reality, but very often the mathematical shortcut is exceptionally good.

 

Other examples is treating the atoms of an ideal gas a point particles yields the perfect gas law, which can be very accurate under specific conditions. When launching a probe from the Earth to Mars, the gravitational influence of Pluto isn't negligible, but treating Pluto as a point mass is accurate enough. The real art of physics is very often in finding the approximations that make the math significantly simpler, but don't affect the answer a great deal. Physics is replete with examples if you just look for them.

[Pete]

A square or circle doesn't have a volume/mass therefore it has no moment of inertia.

That the question is regards to a plate can be understood from the context in which the question was asked. In this case we know its really a very thin plate that is being asked about. This notion arises in other areas of physics. For example: in many physics applications one deals with things whose thickness is small in comparison to its area like a very thin piece of paper. One then assigns a surface mass density rather than a volume mass density. The same thing holds for a long thin rod. In that case one uses the approximation of a linear mass density.

 

You'll notice that this never came up in another thread on a similar topic here. See http://www.scienceforums.net/forum/showthread.php?t=24161

 

Notice that the moments of inertia given in this table

http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.inertia.html

 

are with respect to geometric objects e.g. rectangular plane