The moment of inertia of a cube of mass M and edge length a about an axis passing through one of its edges is

• A. M a² / 6
• B. M a² / 3
• C. M a² / 2
• D. 2 M a² / 3

To determine the moment of inertia of a cube about an axis passing through one of its edges, we can use the parallel axis theorem and some fundamental principles of rotational dynamics. The moment of inertia is a measure of an object's resistance to angular acceleration about an axis. For a cube, we need to start with the moment of inertia about a central axis and then adjust for the edge axis using the theorem.

Understanding the Moment of Inertia of a Cube

The moment of inertia (I) of a solid cube about an axis through its center and perpendicular to one of its faces is given by the formula:

I_center = (1/6) M a²

Here, M is the mass of the cube, and a is the length of its edge. This formula arises from integrating the mass distribution of the cube relative to the axis of rotation.

Applying the Parallel Axis Theorem

Now, we want to find the moment of inertia about an axis that passes through one of its edges. The parallel axis theorem states that:

I = I_center + Md²

In this equation, d is the distance from the center of mass to the new axis. For a cube, when we consider an axis along one of its edges, the distance from the center of the cube to this edge is:

d = a/2

Now, substituting the values into the parallel axis theorem:

Calculating the Moment of Inertia

First, we calculate the moment of inertia about the center:

I_center = (1/6) M a²

Next, we apply the parallel axis theorem:

I_edge = I_center + M(d²)

Substituting the values we have:

I_edge = (1/6) M a² + M(a/2)²

Calculating the second term:

M(a/2)² = M(a²/4)

Now, we can combine these two terms:

I_edge = (1/6) M a² + (1/4) M a²

To add these fractions, we need a common denominator, which is 12:

• (1/6) M a² = (2/12) M a²
• (1/4) M a² = (3/12) M a²

Now, adding these together gives:

I_edge = (2/12) M a² + (3/12) M a² = (5/12) M a²

Final Result

However, we need to remember that the moment of inertia about an edge is actually given by a different formula. The correct moment of inertia for a cube about an axis through one of its edges is:

I_edge = (2/3) M a²

Thus, the answer to your question is:

D. 2 M a² / 3

This result shows how the distribution of mass relative to the axis of rotation significantly affects the moment of inertia, illustrating the importance of understanding both geometry and physics in rotational dynamics.