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# Five solids are shown in cross section in Fig. The cross sections have equal heights and equal maximum widths. The solids have equal masses. Which one has the largest rotational inertia about a perpendicular axis through the center of mass? Which has the smallest?

Kevin Nash
askIITians Faculty 332 Points
6 years ago
Rotational inertia is defined as the distribution of mass around the axis of rotation of the object. The object containing mass farther away from the axis of rotation will have large rotational inertia.
Assume that the geometric center of the objects is the axis of rotation.
Amongst the cross section area of hoop, cylinder and sphere, the rotational inertia for cylinder will be equal to the rotational inertia of sphere, which will be greater than the rotational inertia of the hoop. This can be seen from the fact that hoop does not has mass around near to the axis of rotation, and only has some far away.
One can see the difference by superimposing the objects on one another. Therefore amongst hoop, cylinder and sphere, the rotational inertia of sphere and cylinder is greater than hoop and is equal to one another.

The highlighted area shows the extra area in the sphere/cylinder relative to hoop.
Superimposing the cylinder/sphere on the triangle one can see that the cylinder sphere has larger area than the triangle and therefore have larger rotational inertia than it. the figure below shows the superposition of sphere and triangle, and the extra area possessed by the sphere is highlighted.

Therefore the sphere/cylinder has larger rotational inertia than the triangle.
The figure below shows the superimposed cylinder/sphere against the cube. It can be seen from the figure that the cube occupies larger area relative to the sphere/cylinder and the extra area of the cube relative to sphere/cylinder is highlighted in the figure.

Therefore, the cube will have larger rotational inertia than every other given object.