Resolve moment of inertia at an angle

In summary, the conversation discusses calculating the moment of inertia of a square lamina rotated at an angle $\theta$ about a vertex. The suggestion is made to split the rotated square into two planes and use the parallel axes theorem to find the moment of inertia about the vertex. However, it is mentioned that this method may be off and a triple integral may be necessary. The possibility of using the two planes method is also mentioned.
  • #1
curiousPep
17
1
Initially, I calculate the moment of inertia of of a square lamina (x-z plane). Thr this square is rotated an angle $\theta$ about a vertex and I need to calculate the new moment of inertia about that vertex.

Can I split the rotated square to two squares in the x-z plane and y-z plane to find the matrix of moment of inertia about x,y,z axis and then use the rotated shape and the parallel axes theorem to find the moment of inertia abou the vertex?
What I mean is resolving the the dark shape to two shapes (red and orange outline) and find the individual moment of inertia to find the moment of inertia of the dark shape
 

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  • #3

FAQ: Resolve moment of inertia at an angle

What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is calculated by summing the mass of each particle in an object multiplied by the square of its distance from the axis of rotation.

How is moment of inertia at an angle calculated?

The moment of inertia at an angle is calculated using the parallel axis theorem. This involves adding the moment of inertia of the object about an axis through its center of mass to the product of its mass and the square of the distance between the two axes.

Why is it important to calculate moment of inertia at an angle?

Calculating the moment of inertia at an angle is important in understanding the rotational motion of objects that are not rotating about their center of mass. It helps in predicting how an object will behave when subjected to external torques.

What factors affect the moment of inertia at an angle?

The moment of inertia at an angle is affected by the mass distribution of the object and the distance between the axis of rotation and the object's center of mass. Objects with more mass located farther from the axis of rotation will have a higher moment of inertia.

How is moment of inertia at an angle used in real-world applications?

Moment of inertia at an angle is used in various engineering and physics applications, such as designing rotating machinery, predicting the behavior of vehicles in motion, and understanding the motion of celestial bodies. It is also important in sports, such as in calculating the moment of inertia of a gymnast during a routine.

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