You build a wheel out of a thin circular hoop of mass 5 kg and radius 3 m, and two thin rods of mass 2 kg and 6 m in length, as shown below. Calculate the system's moment of inertia about a central axis, perpendicular to the hoop.

(II) A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.380 kg. Calculate

(a) its moment of inertia about its center,

(II) Four equal masses M are spaced at equal intervals, ℓ, along a horizontal straight rod whose mass can be ignored. The system is to be rotated about a vertical axis passing through the mass at the left end of the rod and perpendicular to it.

(a) What is the moment of inertia of the system about this axis?

(II) An oxygen molecule consists of two oxygen atoms whose total mass is 5.3 x 10⁻²⁶ and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 x 10⁻⁴⁶ kg • m². From these data, estimate the effective distance between the atoms.

(II) When discussing moments of inertia, especially for unusual or irregularly shaped objects, it is sometimes convenient to work with the radius of gyration, k. This radius is defined so that if all the mass of the object were concentrated at this distance from the axis, the moment of inertia would be the same as that of the original object. Thus, the moment of inertia of any object can be written in terms of its mass M and the radius of gyration as I = Mk² . Determine the radius of gyration for each of the objects (hoop, cylinder, sphere, etc.) shown in Fig. 10–21.

A composite disc is built from a solid disc and a concentric, thick-walled hoop, as shown below. The inner disc (solid) has mass 4 kg and radius 2 m. The outer disc (thick-walled) has mass 5 kg, inner radius 2 m, and outer radius 3 m. Calculate the moment of inertia of this composite disc about a central axis perpendicular to the discs.

Three small objects, all of mass 1 kg, are arranged as an equilateral triangle of sides 3 m in length, as shown. The left-most object is on (0m, 0m). Calculate the moment of inertia of the system if it spins about the (a) X axis; (b) Y axis.