Three thin uniform rods each of mass M and le | vedclass

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(B) The system consists of three rods placed along the $x$,$y$,and $z$ axes.Let $I_x$,$I_y$,and $I_z$ be the moments of inertia of the individual rods

Vedclass · Accepted Answer

$\frac{2ML^2}{3}$

Vedclass · Answer

(B) The system consists of three rods placed along the $x$,$y$,and $z$ axes.Let $I_x$,$I_y$,and $I_z$ be the moments of inertia of the individual rods about the $z$-axis.$1$. For the rod along the $z$-axis: The rod lies on the axis of rotation,so its moment of inertia about the $z$-axis is $I_1 = 0$.$2$. For the rod along the $x$-axis: The rod is perpendicular to the $z$-axis at one end. The moment of inertia of a rod of mass $M$ and length $L$ about an axis passing through one end and perpendicular to the rod is $I_2 = \frac{ML^2}{3}$.$3$. For the rod along the $y$-axis: Similarly,the rod is perpendicular to the $z$-axis at one end. The moment of inertia is $I_3 = \frac{ML^2}{3}$.The total moment of inertia of the system about the $z$-axis is $I = I_1 + I_2 + I_3 = 0 + \frac{ML^2}{3} + \frac{ML^2}{3} = \frac{2ML^2}{3}$.

Three thin uniform rods each of mass $M$ and length $L$ are placed along the three axes of a Cartesian coordinate system with one end of all the rods at the origin. The moment of inertia of the system of the rods about the $z$-axis is:

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