Question

Three identical thin rods, each of length L an mass m, are welded perpendicular to one another as shown in Figure P10.23. The assembly is rotated about an axis that passes through the end of one rod and is parallel to another Determine the moment of inertia of this structure. (Answer in terms of m and L.) Axis of rotationn Figure P10.23

Answer 1

Mass of each rod is and length .

Total mass of the three rods is $m+m+m=3m$

The axis of rotation is parallel to Y axis, and is at a distance of $\frac{L}{2}$ from the center (origin) of system of rods where the total mass of the system is located.

Moment of inertia of the rod along X-axis about Y-axis (passing through origin) is $I_1=\frac{m L^2}{12}$

Moment of inertia of the rod along Y-axis about Y-axis is $I_2=0$

Moment of inertia of the rod along Z-axis about Y -axis is $I_3=\frac{m L^2}{12}$

Y-axis is at a distance of    $x=\frac{L}{2}$ from the given axis of rotation,

By parallel axis theorem,

Moment of inertia of the system of rods = Moment of inertia of system about Y-axis + (mass of system)(distance between origin and axis of rotation )²

$I=\left ( \frac{mL^2}{12}+0+\frac{mL^2}{12} \right )+3m\left ( \frac{L}{2} \right )^2$

$I= \frac{11}{12}mL^2$