Three identical thin rods each of length L and mass m are welded perpendicular to one another as shown in the figure….

Q: Three identical thin rods each of length L and mass m are welded perpendicular to one another as shown in the figure. The assembly is rotated about an axis that passes through the end of one rod and is parallel to another. Find moment of inertia of this structure.

Numerical

(a) $\frac{11}{12}m L^2 $

(b) $\frac{9}{16}m L^2 $

(c) $\frac{8}{11}m L^2 $

(d) $ 2 m L^2 $

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Ans: (a)
Sol: Moment of Inertia of the rod which is along X-axis ,

$\displaystyle I_x = \frac{m L^2}{3}$

Moment of Inertia of the rod which is along Y-axis ,

$\displaystyle I_y = \frac{m L^2}{3}$

Moment of Inertia of the rod which is along Z-axis ,

$\displaystyle I_z = m(\frac{L}{2})^2 = \frac{m L^2}{4} $

Moment of Inertia of this Structure will be ,

$\displaystyle I = I_x + I_y + I_z $

$\displaystyle I = \frac{m L^2}{3} + \frac{m L^2}{3} + \frac{m L^2}{4} $

$\displaystyle I = m l^2 (\frac{1}{3} + \frac{1}{3} + \frac{1}{4}) $

$\displaystyle I = \frac{11}{12} m l^2 $