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 given by $I_{center} = \frac{1}{12}ML^2$, where $M$ is the mass of the rod and $L$ is its length. For the axis through the 30-cm mark, we use the parallel axis theorem: $I_{30cm} = I_{center} + Md^2$, where $d$ is the distance from the center to the 30-cm mark (20 cm or 0.2 m). Thus, $I_{30cm} = \frac{1}{12}ML^2 + M(0.2)^2$. The ratio of $I_{30cm}$ to $I_{center}$ includes the term $\frac{1}{12}ML^2$ in both, which cancels out, leaving the ratio dependent on the additional $M(0.2)^2$ term for $I_{30cm}$. This results in a ratio greater than 1, and through calculation, it can be shown that the ratio is 1.5, considering the distribution of mass and the use of the parallel axis theorem.

A Slender Uniform Rod 100