Moment of inertia: A slender uniform rod 100.00 cm long is used as a meter stick. Two parallel axes that are perpendicular to the rod are considered. The first axis passes through the 50-cm mark and the second axis passes through the 30-cm mark. What is the ratio of the moment of inertia through the second axis to the moment of inertia through the first axis? A) I₂/I₁ = 1.5 B) I₂/I₁ = 1.7 C) I₂/I₁ = 1.9 D) I₂/I₁ = 2.1 E) I₂/I₁ = 2.3

Question

Quizplus · Accepted Answer

The moment of inertia of a slender rod about an axis through its center (the 50-cm mark) is $I_{center} = \frac{1}{12}ML^2$, where $L$ is the length of the rod. For an axis not at the center, the parallel axis theorem is used: $I = I_{center} + Md^2$, where $d$ is the distance from the center axis to the new axis. For the second axis (at the 30-cm mark), $d = 20$ cm or $0.2$ m. Thus, the moment of inertia through the second axis is $I_{30cm} = \frac{1}{12}ML^2 + M(0.2)^2$, and through the first axis (center), it remains $I_{50cm} = \frac{1}{12}ML^2$. The ratio $I_{30cm}/I_{50cm}$ accounts for the added $M(0.2)^2$ term in the numerator, leading to a ratio greater than 1. Using the parallel axis theorem and calculating the ratio, considering the added distance squared term, results in a ratio of 1.5, which accounts for the increased moment of inertia due to the axis being off-center by 20 cm.

Moment of Inertia: a Slender Uniform Rod 100