Moment of inertia (Perpendicular axis theorem)

In summary, the conversation involved deriving the moment of inertia about the axis of symmetry with height h and discussing the use of the perpendicular axis theorem to find the moment of inertia perpendicular to the axis of symmetry. The solution for Ix and Iy was also derived, leading to confusion about the application of the theorem as it only applies to 2D lamina.
  • #1
andyonassis
4
1
Homework Statement
Find the moment of inertia of frustum perpendicular to axis of symmetry about the top with radius r and bottom with radius R
Relevant Equations
I = (3m/10)[(R^5-r^5)/(R^3-r^3)] (about axis of symmetry)
So i derived the moment of inertia about the axis of symmetry (with height h) and I am confused about the perpendicular axis theorem.
The problem ask to find the moment of inertia perpendicular to axis of symmetry
conical-frustum.jpg

So the axis about h, i labelled as z, the two axis that are perpendicular to z, i labelled as x and y.
I already found $$I_{z} = (3m/10)\frac{(R^5-r^5)}{(R^3-r^3)]}$$
So according to perpendicular axis theorem,
$$I_z = I_x + I_y$$
and because of symmetry $$I_x = I_y$

So i derived a solution for Ix and Iy
$$Ix = I_y = (3m/20)\frac{(R^5-r^5)}{(R^3-r^3)]}$$
I'm confused as the theorem also says $$I_x = I_y+I_z$$ and $$I_y = I_x+I_z$$
 
Last edited:
Physics news on Phys.org
  • #2
Perpendicular axis theorem only applies to 2D lamina!
 
  • Like
Likes andyonassis and Hamiltonian

FAQ: Moment of inertia (Perpendicular axis theorem)

1. What is the moment of inertia?

The moment of inertia is a measure of an object's resistance to rotational motion about a specific axis. It is also known as rotational inertia.

2. How is the moment of inertia calculated?

The moment of inertia can be calculated by summing the products of each particle's mass and its squared distance from the axis of rotation.

3. What is the Perpendicular Axis Theorem?

The Perpendicular Axis Theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two perpendicular axes lying within the plane of the object.

4. How is the Perpendicular Axis Theorem applied?

The Perpendicular Axis Theorem is applied by breaking down a complex object into simpler shapes, calculating the moments of inertia for each shape, and then using the theorem to find the total moment of inertia about the perpendicular axis.

5. What are some real-world applications of the Perpendicular Axis Theorem?

The Perpendicular Axis Theorem is commonly used in engineering and physics to calculate the moment of inertia for various objects, such as rotating machinery, bridges, and vehicles. It is also used in sports, such as figure skating and diving, to analyze the rotational motion of athletes.

Back
Top