To summarize the moment of inertia of an object about a given we shall call the z-axis, has the following properties (1) The moment of inertia =∑m+n=(2+)咖m (2)If the object is made of a number of parts, each of whose moment of inertia is known, the total moment of inertia is the sum of the moments of inertia of (3)The moment of inertia about any given axis is equal to the moment of inertia about a parallel axis through the CM plus the total mass times the square of the distance from the axis to the CM (4)If the object is a plane figure, the moment of inertia about an axis perpendicu- ar to the plane is equal to the sum of the moments of inertia about mutually perpendicular axes lying in the plane and intersecting at the The moments of inertia of a number of elementary shapes having uniform mass densities are given in Table 19-1, and the moments of inertia of some other objects, which may be deduced from Table 19-1, using the above properties, are given in Table 19-2 Table 19-1 Obiect Thin rod, length L ⊥ rod at center ML2/12 Thin concentric circular ring, radii L ring at center M(2+r2)/2 Sphere, radius r through center 2Mr2/5 Table 19-2 Z-aXIS Rect sheet, sides a,bat center Ma2/12 Rect sheet, sides a, b l sheet at f(a2+b2)/12 center Thin annular ring MG2+r2)/4 Rect parallelepiped, c, through M(a2+b2)/12 sides a, b c Rt circ. cyl, radius, through M2/2 r, length L Rtcirc. cyl, radius, through M(2/4+L2/12) r, length L center 19-4 Rotational kinetic energy and an. go on to discuss dynamics further. In the analogy between linear motion and angular motion that we discussed in Chapter 18, we used the work theorem, but we did not talk about kinetic energy. what is the kinetic energy of a rigid body, rotating about a certain axis with an angular velocity w? We can im- mediately guess the correct answer by using our analogies. The moment of inertia corresponds to the mass, angular velocity corresponds to velocity, and so the kinetic energy ought to be ilo, and indeed it is, as will now be demonstrated Suppose the object is rotating about some axis so that each point has a velocity hose magnitude is wri, where ri is the radius from the particular point to the axis