How do you find the moment of inertia of a polygon?

In summary, The conversation is about calculating the moment of inertia for a polygon without decomposing it into triangles. The last example on the Wikipedia page provides the formula for this, but the person is having trouble understanding it. They mention that using polygons with 4 or more sides can complicate things, such as shading and building BSP-trees. However, the person mentions that they are only working with 2D graphics and finding a way to solve the problem for convex polygons will achieve their goal.
  • #1
epaik91
3
0
I'm working on an engine right now, and I'm having trouble calculating the moment of inertia for a polygon. Is there any way to easily do this without decomposing the polygon into triangles?

edit: I've looked at the wikipedia page with examples on the subject (http://en.wikipedia.org/wiki/List_of_moments_of_inertia) and I'm having trouble understanding the last example, which seems to be what I need.
 
Last edited:
Mathematics news on Phys.org
  • #2
That last example is, indeed, the moment of inertia formula for the polygon, and is the final result of having decomposed the polygon into triangles.

I derived it once for myself, I'm sorry that I'm not inm the mood to do it once again.
 
  • #3
Any particular reason your polygons are not triangles in the first place? Polygons of 4 or more sides complicates everything.

4 or more sides means your polygon does not need to have all vertexes on the same plane, which complicates shading.

Building a BSP-tree gets more complicated the more sides your polygons have (you will end up splitting a whole lot).

and so on.

k
 
  • #4
kenewbie said:
Any particular reason your polygons are not triangles in the first place? Polygons of 4 or more sides complicates everything.

4 or more sides means your polygon does not need to have all vertexes on the same plane, which complicates shading.

Building a BSP-tree gets more complicated the more sides your polygons have (you will end up splitting a whole lot).

and so on.

k

I'm sorry if I wasn't clear, but I'm doing everything on a 2D plane. I'm not sure what you mean by stating that the vertexes won't be on the same plane, but I think you may be confusing 3D graphics with my problem (shading?). In addition, finding a way to solve this problem for convex polygons will achieve what I'm going for, which is why I'm avoiding BSP-trees or any other kind of decomposition of the polygon.
 
  • #5
Yep, I was thinking 3D, sorry.

k
 

1. What is the definition of moment of inertia?

The moment of inertia of a polygon is a measure of its resistance to rotational motion, similar to how mass is a measure of its resistance to translational motion. It is calculated by summing the products of each individual mass element and its distance from the axis of rotation squared.

2. How do you find the moment of inertia of a regular polygon?

The moment of inertia of a regular polygon can be calculated using a formula specific to the shape, which takes into account the number of sides, length of sides, and distance from the axis of rotation. For example, the moment of inertia of a regular hexagon is given by (1/2)MR², where M is the mass of the polygon and R is the distance from the center of mass to any side.

3. What is the difference between moment of inertia and moment of inertia tensor?

The moment of inertia tensor is a mathematical representation of the moment of inertia for an object with a continuous mass distribution, such as a polygon. It takes into account the distribution of mass in all three dimensions, while the moment of inertia for a polygon only considers mass elements in a single plane.

4. Can the moment of inertia of a polygon be negative?

No, the moment of inertia of a polygon cannot be negative. It is always a positive value, as it represents the resistance of the object to rotation. However, it can be zero for certain shapes, such as a thin, flat sheet with no thickness.

5. How does the moment of inertia of a polygon affect its rotational motion?

The moment of inertia of a polygon determines how difficult it is to change its rotational motion. Objects with a larger moment of inertia will require more torque to accelerate their rotation, while objects with a smaller moment of inertia will rotate more easily. This property is important in understanding the stability and dynamics of rotating objects.

Similar threads

Replies
2
Views
2K
  • Introductory Physics Homework Help
2
Replies
40
Views
2K
Replies
12
Views
279
  • Classical Physics
2
Replies
49
Views
2K
  • Introductory Physics Homework Help
Replies
13
Views
1K
  • Mechanical Engineering
Replies
15
Views
1K
Replies
10
Views
1K
Replies
9
Views
5K
  • Classical Physics
Replies
5
Views
2K
Back
Top