![derivation of moment of inertia of a circle derivation of moment of inertia of a circle](https://i.ytimg.com/vi/idKWlkgKXzM/hqdefault.jpg)
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? Using the trigonometric identity: sin 2 θ = 1-cos 2 θ / 2 we calculate the integral.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.
![derivation of moment of inertia of a circle derivation of moment of inertia of a circle](https://thecuriousastronomer.files.wordpress.com/2015/11/pagesscreensnapz001.jpg)
Now in order to find the second moment of inertia: Using this we can find the first moment of inertia about the x-axis. The centroid of this elemental area from x-axis = y sin θģ. It is given as:ĪBCD is a sector with area = (r⋅d θ) ⋅ dr = r ⋅ drd θ Now we have to determine the differential area by finding the area of the element. Here in order to derive the moment of inertia of a semicircle we define the coordinates using the polar system. The macro-scale area density is given in this equation: dm/da= σĭerivation of Formula for Moment of Inertia of Semicircleġ. Area density (σ) is an intensive property, meaning that it does not depend on the amount of the material, and also as long as the mass is uniform, its area density is the same whether you have chosen the entire semicircle or a small strip of differential width. Use the concept for area density, which is mass divided by area. Add all of the individual strips using integral Calculus. Now writing an expression for the area density for the whole semicircle and then the tiny strips of differential widths. Selecting and defining a tiny strip of mass with differential width. M.O.I relative to the origin, Jo = Ix + Iy = ⅛ πr 4 + ⅛ πr 4 = ¼ πr 4 Now to find the moment of inertia of the semicircle we will take the sum of both the x and y-axis.
#Derivation of moment of inertia of a circle full#
Thus, M.O.I will be half the moment of inertia of that of a full circle.
![derivation of moment of inertia of a circle derivation of moment of inertia of a circle](https://i.ytimg.com/vi/FmrNcFwpToo/maxresdefault.jpg)
Here, the semi-circle rotating about an axis is symmetric, and therefore, we consider these values equal. Thus similarly for the semicircle, the moment of inertia of the x-axis is equal to that of the y-axis. Now we need to pull out the area of a circle which gives us: M.O.I relative to the origin, Jo = Ix + Iy = ¼ πr 4 + ¼ πr 4 = ½ πr 4 We know that for a full circle because of complete symmetry and uniform area distribution, the moment of inertia relative to the x-axis is equal to that of the y-axis. Further to determine the moment of inertia of the semi-circle, we will take the sum of both the x and y-axis. The moment of inertia of the semicircle is generally expressed as I = πr 4 / 4.Here in order to find the value of the moment of inertia of a semicircle, we have to first derive the results of the moment of inertia full circle and basically divide it by two to get the required result of that moment of inertia for a semicircle. The moment of inertia of the semicircle is generally expressed as I = πr 4 / 4 Similarly larger the moment of inertia of the body more difficult is to stop its rotational motion. For example, if the body is at rest the larger the moment of inertia of the body the more difficult it is to put the body into the rotational motion. In other words, the moment of inertia is the measurement of resistance of the body to a change in its rotational motion. The moment of inertia plays the same role in rotational motion as the mass does in the translational motion.