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• #What is the second moment of inertia of a circle full

In the sheet metal example above, it is perfectly reasonable to define a mass per unit area.

#What is the second moment of inertia of a circle full

Note that the height, h 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!Įven 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.Īnd, the fact is, you can define densities not just as mass per unit volume.

I_z = \frac where I_z is the MOI about the axis, m is the mass of the cylinder, and r is the radius. Specifically, the one for a solid cylinder: Let's even look at your list of moments of inertia. 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. Consider if I made a square and a circle out of very thin sheet metal. Genralz, in case you mean MOI of a cylinder or cubiod etc.īut I'm not just "making up a mass that has no meaning". Yes you can make up an mass that has no meaning, I don't care. The simple fact is that the MOI formula requires a mass to have some meaning. 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.įrom what I read genralz isn't interested in math for the math. 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. 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. Mathematically we can assign an area density or mass per unit area and make it mathematically have mass. 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? The moment of inertia of more complex body is then defined as the sum of the moments of inertia of all the individual elements, \(I=\sum_0^k = \approx 0.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. The moment of inertia of a point mass is often stated, without any justification, as being the mass of the particle \(m\) multiplied by the square of its distance \(k\) from the axis of rotation, or \(mk^2\). It is the rotational equivalent of inertia or mass in systems involving linear acceleration. The ‘moment of inertia’ of a rotating body is the body's resistance to angular acceleration. Moment of Inertia Copyright © David Boettcher 2006 - 2021 all rights reserved. Bocks and Rams: IWC and Stauffer Trademarks.