The moment of inertia ($I$) of a thin rod of mass $M$ and length $L$ about an axis passing through its mid-point and perpendicular to its length is given by the formula: $I = \frac{ML^2}{12}$ Given that: $I = 2400\text{ g cm}^2$ $M = 400\text{ g}$ Substituting the given values into the formula: $2400 = \frac{400 \times L^2}{12}$ $L^2 = \frac{2400 \times 12}{400}$ $L^2 = 6 \times 12 = 72$ $L = \sqrt{72} \approx 8.485\text{ cm}$ Therefore, the length of the rod is nearly $8.5\text{ cm}$.