A baton consists of a nearly massless rod of length L and two spheres attached to the ends of the rod,whose masses are each M and whose centers are separated by L.The spheres have radius R.If the sphere radius R is not negligible compared to the length L,what is the moment of inertia of the baton about an axis perpendicular to its length and passing through its center? A)2MR² B)2ML² C)2ML² + 2MR² D)(1/2)ML² E)(1/2)ML² + 2MR²

Question

Quizplus · Accepted Answer

The moment of inertia about the center of mass of each sphere, using the parallel-axis theorem, is given by:

$I_{sphere}=\frac{2}{5}MR^2+MR^2=\frac{7}{5}MR^2$

The moment of inertia of the rod about its center of mass is given by:

$I_{rod}=\frac{1}{12}ML^2$

Using the parallel-axis theorem, the moment of inertia of the rod about an axis passing through its center and perpendicular to its length is:

$I_{rod,perp}=\frac{1}{12}ML^2+\frac{1}{4}ML^2=\frac{1}{3}ML^2$

The total moment of inertia about the perpendicular axis passing through the center is given by:

$I_{total}=I_{rod,perp}+2I_{sphere}=\frac{1}{3}ML^2+2\cdot\frac{7}{5}MR^2=\frac{1}{2}ML^2+2MR^2$

Therefore, the correct choice is (E).

A Baton Consists of a Nearly Massless Rod of Length