The Moment of Inertia of a Hollow Cubical Box

The moment of inertia of an object is a measure of its resistance to rotational motion about a particular axis. It depends on the mass distribution of the object and the axis of rotation. In the case of a hollow cubical box, we can calculate its moment of inertia about an axis passing through the centers of two opposite faces.

Definition of Moment of Inertia
The moment of inertia of an object can be defined as the sum of the products of each particle's mass and its squared distance from the axis of rotation. Mathematically, it can be expressed as:

I = Σ(m * r^2)

where I is the moment of inertia, Σ denotes the sum, m represents the mass of each particle, and r is the distance of each particle from the axis of rotation.

Moment of Inertia of a Hollow Cubical Box
To calculate the moment of inertia of a hollow cubical box about an axis passing through the centers of two opposite faces, we need to consider the distribution of mass in the box.

Mass Distribution
In a hollow cubical box, the mass is distributed uniformly along its edges. Each edge of length a has a mass m = ρ * a, where ρ is the mass per unit length.

Calculating the Moment of Inertia
To calculate the moment of inertia, we need to consider the contributions of all the particles in the box.

1. Divide the Box into Smaller Particles
We can divide the box into smaller particles, considering each edge as a separate particle. There are a total of 12 edges in a cubical box.

2. Determine the Distance of Each Particle from the Axis of Rotation
The distance of each particle from the axis of rotation passing through the centers of two opposite faces is a/2.

3. Calculate the Moment of Inertia for Each Particle
Using the formula I = Σ(m * r^2), we can calculate the moment of inertia for each particle. Since the mass distribution is uniform, the mass of each particle is the same, i.e., m = ρ * a.

4. Sum Up the Moments of Inertia
Finally, we sum up the moments of inertia for all the particles to obtain the total moment of inertia of the hollow cubical box about the given axis.

Final Result
The moment of inertia of a hollow cubical box about an axis passing through the centers of two opposite faces is given by the sum of the moments of inertia for each particle. This can be calculated using the formula I = Σ(m * r^2), considering the mass distribution and distances of particles from the axis of rotation.

By following these steps, you can determine the moment of inertia of the hollow cubical box accurately.

5ma square/18