# Calculating Moment of Inertia for a Rigid Body

• ehrenfest
In summary, the moment of inertia remains the same for any axis passing through the origin for a rigid body consisting of three thin uniform rods, each with a mass of m and length of 2a, held mutually perpendicular at their midpoints. This can be shown by demonstrating that the tensor of inertia is proportional to the unit matrix, meaning that it will not change under any rotation.
ehrenfest

## Homework Statement

A rigid body consists of three thin uniform rods, each of mass m and length 2a, held mutually perpendicular at their midpoints. Show that the moment of inertia is the same for any axis passing through the origin.

## The Attempt at a Solution

I calculated the principal moments of inertia. To show this, do I need to find a coordinate transformation to take one of the principal axes to an arbitrary axis that goes through the origin and then show that the transformation does not change that element on the diagonal of the inertia tensor? Or is there another way to do this?

anyone?

The three principal moments of inertia are equal.
This means the tensor of inertia is proportional to the unit matrix,
so any rotation will not change the tensor of inertia.
It will still be the unit matrix in any rotated system.

## 1. What is the formula for calculating moment of inertia for a rigid body?

The formula for calculating moment of inertia is I = ∫r²dm, where I is the moment of inertia, r is the distance from the axis of rotation, and dm is the differential mass. This formula is an integral that must be solved using calculus.

## 2. How is the moment of inertia affected by the shape of the rigid body?

The moment of inertia is affected by the shape of the rigid body because it is a measure of the object's resistance to rotational motion. Objects with a larger moment of inertia will be harder to rotate compared to objects with a smaller moment of inertia.

## 3. Can the moment of inertia be negative?

No, the moment of inertia cannot be negative. It is always a positive value because it is a measure of an object's resistance to rotational motion. If the moment of inertia were negative, it would mean that the object is easier to rotate in one direction compared to the other, which is not physically possible.

## 4. How does the distribution of mass affect the moment of inertia?

The distribution of mass affects the moment of inertia because it determines how the mass is distributed relative to the axis of rotation. Objects with a more spread out mass will have a larger moment of inertia compared to objects with a more concentrated mass.

## 5. How do you calculate the moment of inertia for a composite rigid body?

To calculate the moment of inertia for a composite rigid body, you can use the parallel axis theorem. This theorem states that the moment of inertia of a composite object is equal to the sum of the moment of inertia of each individual component plus the moment of inertia of the entire system about a parallel axis passing through the center of mass. This can be expressed as I = Icm + Mh2, where Icm is the moment of inertia about the center of mass, M is the total mass of the object, and h is the distance between the two axes of rotation.

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