Rotational Inertia Practice

Sample questions

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1. Sample 1difficulty 2/5

   Of the following objects (all same mass $M$ and radius $R$), which has the <strong>largest</strong> moment of inertia about its symmetry axis?

   • A

     Thin hoop / ring: $MR^2$

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   • B

     Solid disk/cylinder: $\frac{1}{2}MR^2$

   • C

     Solid sphere: $\frac{2}{5}MR^2$

   • D

     Thin rod about its center, length $R$: $\frac{1}{12}MR^2$

   Why

   The hoop concentrates all mass at the maximum distance from the axis, giving the largest $I = MR^2$.

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2. Sample 2difficulty 4/5

   A thin uniform rod of mass $M$ and length $L$ has moment of inertia $\tfrac{1}{12}ML^2$ about its center. What is its moment of inertia about an axis perpendicular to the rod through one end?

   • A

     $ML^2$

   • B

     $\tfrac{1}{3}ML^2$

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   • C

     $\tfrac{1}{12}ML^2$

   • D

     $\tfrac{1}{6}ML^2$

   Why

   Parallel-axis: $I = I_{cm} + Md^2 = \tfrac{1}{12}ML^2 + M(L/2)^2 = \tfrac{1}{3}ML^2$.

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3. Sample 3difficulty 4/5

   A solid disk and a hoop of equal mass and radius roll down identical inclines from rest. Which reaches the bottom first?

   • A

     Depends on incline angle

   • B

     Hoop

   • C

     Same

   • D

     Disk (smaller $I/MR^2$ → less rotational KE → more translational KE → faster)

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   Why

   Disk $I = \frac{1}{2}MR^2$; hoop $I = MR^2$. The disk converts more PE to translational KE (less to rotation), so it accelerates faster.

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