With reference to Fig. of a cube of edge a and mass m, state whether the following are true or false. (O is the centre of the cube.)

Options

A. The moment of inertia of cube about x-axis is,
`I_(z) = I_(x) + I_(y)`
B. The moment of inertia of cube about z' is, `I_(z')`
`= I_(z) + (ma^(2))/(2)`
C. The moment of inertia of the about z'' is
`= I_(z) + (ma^(2))/(2)`
D. `I_(x) = I_(y)`

Answer

B, D

Explanation

Choice (a) is false, as theorem of perepndicular axes applies only to a plane lamina.
Now, Z axis is parallel to Z' axis and disatnce between them `= (a sqrt(2))/(2) = (a)/(sqrt(2))`. Therefore, according to the
theorem of parallel axes, `I_(z') = I_(z) + m((a)/(sqrt(2)))^(2) = I_(z) + (ma^(2))/(2)`
Choice (b) is true.
Again, choice (c) is false as Z'' axis is not parallel to Z-axis.
Choice (d) is true as from symmetry, we find that `I_(x) = I_(Y)`.