Two identical uniform solid spheres are attached by a solid uniform thin rod, as

Question

Two identical uniform solid spheres are attached by a solid uniform thin rod, as shown in the figure. The rod lies on a line connecting the centers of mass of the two spheres. The axes A, B, C, and D are in the plane of the page (which also contains the centers of mass of the spheres and the rod), while axes E and F (represented by black dots) are perpendicular to the page. (Figure 1).

Rank the moments of inertia of this object about the axes indicated. Rank from largest to smallest. To rank items as equivalent, overlap them.

Explanation / Answer

Rank the moments of inertia of this object about the axes indicated. Rank from largest to smallest. To rank items as equivalent, overlap them.

Moments of inertia depend on the masses and the perpendicular distance from the axis of rotation. Mathematically it can be expressed as

I=mr2
The factor that effect the moment of inertia is the distance of the center of mass of the objects in question to the axis of rotation, so in this case

Axes C and F would represent the largest moments of inertia because the center of mass of the rotating body is furthest from these two axes of rotation. However, C and F would

have equal moments of inertia because they have equal distance from the center of mass to their respective axis of rotation.

The next largest would be B because it is the next closest distance from the center of mass.

A and E would come next in order because although they are exactly between the two rotating masses, they still generate a moment of inertia related to (y/2)^2

Finally, D has the lowest moment of inertia because the perpendicular distance from the axis of rotation never exceeds theradius of one of the balls. So in order from highest to

lowest:
(C and F), B, (A and E), D

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