Moment of inertia for door hinges

[DanicaK]

How do we find out that the moment of inertia about the hinges of a door with a width w is I=M w²/3.
I find it like this in a book and also in table for the moments of inertia of homogeneous rigid objects with different geometries (about a long thin rod with rotation axis trough end).
Thank you.

 

[Doc Al]

If you understand how the moment of inertia of the thin rod is found, then just realize that the door can be treated like a bunch of rods. (Imagine the door sliced into thin slits.)

 

[DanicaK]

OK, but i don't understand neither how the moment of inertia of a thin rod is found. Can you explain me please

 

[Doc Al]

Take a look at these examples (click on the figures for details): Common Moments of Inertia

 

[asteriatic]

The moment of inertia about the hinges of a door can be calculated using the formula I = Mw^2/3, where M is the mass of the door and w is its width. This formula is derived from the general formula for moment of inertia, which takes into account the distribution of mass and the distance of the mass from the axis of rotation. In the case of a door, the majority of the mass is concentrated along the width, which is why the width is used in the formula. This formula is also commonly used for calculating the moment of inertia for other homogeneous rigid objects with different geometries, such as a long thin rod. It is important to note that this formula assumes a uniform distribution of mass, so it may not be accurate for doors with irregular shapes or varying mass distributions. Other factors, such as the material and thickness of the door, may also affect the moment of inertia and should be taken into consideration when calculating it.