The moment of inertia of a thin rod about an axis passing through its mid point and perpendicular to the rod is 2400 g cm². The length of the 400 g rod is nearly:

To find the length of the thin rod, we can use the formula for the moment of inertia of a uniform thin rod about an axis passing through its midpoint (center of mass) and perpendicular to its length.

The moment of inertia $I$ of a thin rod of mass $M$ and length $L$ about this axis is given by:
$I=\frac{1}{12}M{L}^2$

We are given the following values:
Moment of inertia, $I=2400{\text{g cm}}^2$
Mass of the rod, $M=400\text{g}$

We need to find the length $L$. We rearrange the formula to solve for $L$:
$L^2=\frac{12I}{M}$

Substituting the given values into the equation:
$L^2=\frac{12\times 2400}{400}$

Simplifying the expression:
$L^2=12\times 6$
$L^2=72{\text{cm}}^2$

Taking the square root of both sides to find $L$:
$L=\sqrt{72}\text{cm}$
$L\approx 8.485\text{cm}$

Comparing this value with the options, the length is approximately 8.5 cm.

Therefore, the correct option is 8.5 cm.