Rigid Body Dynamics - Part II

In summary, to find the moment of inertia for the disk about its diameter, you need to calculate the total mass of the disk and then use the formula I = mr^2. For the cone, you can use the same technique to find the moment of inertia by dividing the mass by the surface area and using the formula I = mr^2.
  • #1
Zenshin
5
0
Hello again! About my last problem (cone principal moments of inertia around top vertice), I´ve found the main moments of inertia, but I did so by integrating the moments of inertia dI o a disk rotaing about it´s diameter (1/4 ML^2, by table), and found the correct answer. But I have done this by the way we did it in Basic Physics 1... in Classical Mechanics I, I think there´s a way in which you use "proportion factors" and find the moments of inertia like Ixx =M(Ky x Lambda-y + Kz x Lambda-z), but i could apply this method...Still, I haven´t figured out how to find a h/a (height / base radius) of that cone in which every axis passing through the vertice v is a principal axis... Somehow, I think it´s a eigenvalue and eigenvector problem of the Inertia tensor... Oh, and by the way, does anyone know how to find the moment of inertia of that disk, about it´s diameter?? I got the value from a table, since I could not find the answer myself...(tried something similar to the spherical moment of inertia by integrating a cascade of disks...didn´t work as well with a cascade of bars for the disk though...)


Thanks in advance!
 
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  • #2
To find the moment of inertia of the disk about its diameter, you need to first calculate the total mass of the disk. This can be done by dividing the mass of the disk by its area. Once you have the total mass of the disk, you can use the formula I = mr^2, where 'I' is the moment of inertia, 'm' is the mass, and 'r' is the radius of the disk. From this, you can calculate the moment of inertia for the disk about its diameter. For the cone, you can use the same technique as before to calculate the moment of inertia. You will need to divide the mass of the cone by its surface area and then use the formula I = mr^2, where 'I' is the moment of inertia, 'm' is the mass, and 'r' is the radius of the cone. This will give you the moment of inertia for the cone about its vertice.
 
  • #3
It's great that you were able to find the main moments of inertia by integrating the moments of inertia of a disk rotating about its diameter. This is a valid method and can be used to solve many rigid body dynamics problems. However, as you mentioned, there is another method using "proportion factors" which can also be used to find the moments of inertia. This method involves using the parallel axis theorem and the perpendicular axis theorem to find the moments of inertia about different axes.

To find the moments of inertia of the cone about its principal axes, you can use the eigenvalue and eigenvector approach. This involves finding the eigenvalues and eigenvectors of the inertia tensor, which represents the moments of inertia of the cone. The eigenvalues represent the principal moments of inertia and the corresponding eigenvectors represent the principal axes. By finding the eigenvalues and eigenvectors, you can then use the formula you mentioned to find the moments of inertia about the principal axes.

As for finding the moment of inertia of a disk about its diameter, you can use the parallel axis theorem. This theorem states that the moment of inertia of a body about an axis parallel to its centroidal axis is equal to the moment of inertia about the centroidal axis plus the product of the mass of the body and the square of the distance between the two axes. In this case, the centroidal axis is the axis passing through the center of the disk and the parallel axis is the axis passing through its diameter.

I hope this helps and good luck with your studies!
 

1. What is the difference between a rigid body and a deformable body in dynamics?

A rigid body is a body that maintains its shape and size even when subjected to external forces, while a deformable body can change its shape and size under the influence of external forces. In rigid body dynamics, the body is assumed to be rigid, while in deformable body dynamics, the body is allowed to deform.

2. How do you define the motion of a rigid body in dynamics?

The motion of a rigid body is defined by its translation and rotation. Translation refers to the movement of the body as a whole, while rotation refers to the movement of the body around a fixed point or axis.

3. What are the equations of motion for a rigid body in dynamics?

The equations of motion for a rigid body in dynamics are Newton's second law of motion, which states that the sum of all forces acting on the body is equal to its mass times its acceleration, and the parallel axis theorem, which relates the moment of inertia of a body to its rotation about a different axis.

4. How is angular momentum conserved in rigid body dynamics?

Angular momentum is conserved in rigid body dynamics when there is no external torque acting on the body. This means that the total angular momentum of the body remains constant throughout its motion.

5. What are some real-world applications of rigid body dynamics?

Rigid body dynamics has many applications in engineering and physics, including the design and analysis of structures, vehicles, and machines. It is also used in the study of celestial mechanics, such as the motion of planets and satellites. Additionally, rigid body dynamics is important in the development of video game physics and computer animation.

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