Question

From a solid sphere of mass M and radius R , a cube of maximum possible volume is cut . Moment of inertia of cube about an axis passing through its centre and perpendicular to one of the face is ​

Answer 1

Answer:

Moment of inertia of cube about an axis passing throgh its centre and perpendicular to one of its face is 4MR²/9√3π

Explanation:

centre idea :- use geometry of the figure to calculate mass and side length of the cube interms of M and R respecticely .

Conaider the cross - sectional view of a diametic plane as shown in the adjacent diagram.

Using geometry of the cube

PQ = 2R= (√3 )a or a = 2R/√3

Volume density of the solid sphere

• ρ = M/(4/3πR³)

Mass of cube (m) = ρ(a³)

= (3/4π × M/R³) [2R³/√3]

= 3M/4πr³ × 8R³ /3√3 = 2M/√3π

Moment of inertia of the cube given axis is

Iy = ma²/12(a²+a²) = m a²/6

Iy = ma² /6 = 2M/√3π × 1/6 × 4/3R² = 4MR²/9√3π

Iy = 4MR² /9√3π Answer