Three rods each of length L and mass M are placed along X, Y and Z-axes in such a way that one end of each of the rod is at the origin. The moment of inertia of this system about Z axis is

To find the moment of inertia of the system consisting of three rods about the Z-axis, we calculate the moment of inertia of each individual rod about the Z-axis and then sum them up.

Let the three rods be placed as follows:
1. Rod 1 along the X-axis, with one end at the origin (0, 0, 0) and the other end at (L, 0, 0).
2. Rod 2 along the Y-axis, with one end at the origin (0, 0, 0) and the other end at (0, L, 0).
3. Rod 3 along the Z-axis, with one end at the origin (0, 0, 0) and the other end at (0, 0, L).

Let's find the moment of inertia of each rod about the Z-axis (Z-axis is the axis of rotation):

1. For Rod 3 (along the Z-axis):
Since this rod lies entirely along the Z-axis (axis of rotation), the perpendicular distance of every mass element of this rod from the Z-axis is zero. Therefore, its moment of inertia about the Z-axis is:

$I_{\mathrm{z3}}=0$

2. For Rod 1 (along the X-axis):
This rod lies along the X-axis with one of its ends at the origin. The Z-axis is perpendicular to this rod and passes through one of its ends (the origin). The moment of inertia of a uniform rod of mass $M$ and length $L$ about an axis passing through one of its ends and perpendicular to its length is given by:

$I_{\mathrm{z1}}=\frac{M{L}^2}{3}$

3. For Rod 2 (along the Y-axis):
Similarly, this rod lies along the Y-axis with one end at the origin. The Z-axis is perpendicular to this rod and passes through one of its ends (the origin). Thus, its moment of inertia about the Z-axis is also:

$I_{\mathrm{z2}}=\frac{M{L}^2}{3}$

Total Moment of Inertia of the System:
Since moment of inertia is a scalar quantity, we add the individual moments of inertia:

$I_{\mathrm{total}}=I_{\mathrm{z1}}+I_{\mathrm{z2}}+I_{\mathrm{z3}}$

$I_{\mathrm{total}}=\frac{M{L}^2}{3}+\frac{M{L}^2}{3}+0=\frac{2M{L}^2}{3}$

Therefore, the moment of inertia of the system about the Z-axis is $2M{L}^2/3$.