Question

A cubic frame is made of 12 rods each of mass m and length ℓ.

• A. The moment of inertia of the cube about its face diagonal (BD) is .
• B. The moment of inertia of the cube about an axis passing through the center of cube and perpendicular to one of the face (HI) is .
• C. The moment of inertia of the cube about one of the side of cube (AB) is .
• D. The moment of inertia of the cube about an axis passing through the center of the face and the midpoint of the one of the side (JK) is .

Answer 1

A

Answer 2

The moment of inertia of a thin rod of mass $m$ and length $\ell$ about an axis:

1. Through its center and perpendicular to its length: $I_c = \frac{1}{12}m\ell^2$
2. Through one end and perpendicular to its length: $I_e = \frac{1}{3}m\ell^2$
3. Along its length: $I_{along} = 0$

Let the cube edges be along the x, y, and z axes, with one corner at the origin (0, 0, 0). The length of each edge is $\ell$. The vertices are (0,0,0), ($\ell$,0,0), (0,$\ell$,0), (0,0,$\ell$), ($\ell$,$\ell$,0), ($\ell$,0,$\ell$), (0,$\ell$,$\ell$), ($\ell$,$\ell$,$\ell$).

(A) Moment of inertia about face diagonal BD. The moment of inertia is $\frac{26m\ell^2}{3}$.

Therefore, the correct answer is A.