A dumbbell-shaped object is composed by two equal masses,m,connected by a rod of negligible mass and length r.If I₁ is the moment of inertia of this object with respect to an axis passing through the center of the rod and perpendicular to it and I₂ is the moment of inertia with respect to an axis passing through one of the masses,it follows that A) I₁ = I₂. B) I₁ I₂. C) I₂ I₁.

Question

A dumbbell-shaped object is composed by two equal masses,m,connected by a rod of negligible mass and length r.If I₁ is the moment of inertia of this object with respect to an axis passing through the center of the rod and perpendicular to it and I₂ is the moment of inertia with respect to an axis passing through one of the masses,it follows that A) I₁ = I₂. B) I₁ > I₂. C) I₂ > I₁.

Quizplus · Accepted Answer

The moment of inertia depends on the distribution of mass relative to the axis of rotation. For $I_{\text{center}}$, both masses contribute equally since the axis is equidistant from them. For $I_{\text{end}}$, one mass is at the axis (contributing 0 to the moment of inertia) and the other is twice as far from the axis compared to the $I_{\text{center}}$ case, leading to a larger moment of inertia due to the $r^2$ dependence in the moment of inertia formula ($I = mr^2$). Therefore, $I_{\text{end}} > I_{\text{center}}$.

A Dumbbell-Shaped Object Is Composed by Two Equal Masses,m,connected by a Rod