Question: 29. The moment of inertia of a thin rod about an axis passing through its mid point and perpendicular to the rod is 2400 \mathrm{~g} \mathrm{~cm}^{2}. The length of the 400 \mathrm{~g} \operatorname{rod} is nearly :
(1) 8.5 cm
(2) 17.5 cm
(3) 20.7 cm
(4) 72.0 cm
Answer: Option (1)
Explanation:
The moment of inertia of a thin uniform rod about an axis passing through its midpoint
and perpendicular to its length is given by
I = \frac{1}{12} M L^{2}Given moment of inertia I = 2400 \, \mathrm{g\,cm^{2}}.
Mass of the rod M = 400 \, \mathrm{g}.
Substituting the given values,
2400 = \frac{1}{12} \times 400 \times L^{2}Simplifying,
2400 = \frac{400}{12} L^{2} 2400 = \frac{100}{3} L^{2} L^{2} = \frac{2400 \times 3}{100} L^{2} = 72 L = \sqrt{72} L \approx 8.49 \, \mathrm{cm}Thus, the length of the rod is nearly 8.5 cm.
Hence, the correct answer is Option (1).