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(B) The moment of inertia of a thin rod of mass $m$ and length $l$ about an axis passing through one of its ends and perpendicular to its length is gi

Vedclass · Accepted Answer

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

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(B) The moment of inertia of a thin rod of mass $m$ and length $l$ about an axis passing through one of its ends and perpendicular to its length is given by $I = \frac{ml^2}{3}$.$1$. For the rod along the $x$-axis: The axis of rotation is the $z$-axis. Since the rod lies in the $xy$-plane and is perpendicular to the $z$-axis at the origin,its moment of inertia about the $z$-axis is $I_x = \frac{ml^2}{3}$.$2$. For the rod along the $y$-axis: The axis of rotation is the $z$-axis. Since the rod lies in the $yz$-plane and is perpendicular to the $z$-axis at the origin,its moment of inertia about the $z$-axis is $I_y = \frac{ml^2}{3}$.$3$. For the rod along the $z$-axis: The rod lies along the axis of rotation itself. Therefore,the distance of every mass element of this rod from the $z$-axis is zero. Thus,$I_z = 0$.Total moment of inertia $I_{	ext{total}} = I_x + I_y + I_z = \frac{ml^2}{3} + \frac{ml^2}{3} + 0 = \frac{2ml^2}{3}$.

Three identical thin rods each of mass $m$ and length $l$ are placed along $x, y$ and $z$-axes respectively. They are placed such that one end of each rod is at the origin $O$. The moment of inertia of this system about the $z$-axis is:

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