Physics - Rotation Question with Solution | TestHub

From the given figure, the axes $c$ and $d$ are mutually perpendicular. Similarly, the axes $a$ and $b$ are mutually perpendicular.

According to the perpendicular axis theorem for a planar body, the moment of inertia about an axis perpendicular to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes in its plane intersecting at the same point.

Applying this theorem to the pairs of perpendicular axes:
$I=I_c+I_d\Rightarrow I=I_3+I_4$
$I=I_a+I_b\Rightarrow I=I_1+I_2$

For a uniform thin disc, the moment of inertia about any diameter is the same due to symmetry. Let this moment of inertia be $I_d$.
Thus, $I_1=I_2=I_3=I_4=I_d$.

Since $I=I_1+I_2=2I_d$, we can also write:
$I=I_1+I_3$ (since $I_1+I_3=I_d+I_d=2I_d=I$).

Answer: $I=I_1+I_2$; $I=I_3+I_4$; $I=I_1+I_3$