(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 = Mk 2 . Determine the radius of gyration for each of the objects (hoop, cylinder, sphere, etc.) shown in Fig. 10–20.
(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 = Mk 2 . Determine the radius of gyration for each of the objects (hoop, cylinder, sphere, etc.) shown in Fig. 10–20.
(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. 10–20.
(b) On an old-fashioned rotating piano stool, a woman sits holding a pair
of dumbbells at a distance of 0.60 m from the axis of rotation of the
stool. She is given an angular velocity of 3.00 rad/s, after which she
pulls the dumbbells in until they are only 0.20 m distant from the
axis. The woman's moment of inertia about the axis of rotation is
5.00 kg-m² and may be considered constant. Each dumbbell has a
mass of 5.00 kg and may be considered a point mass. Ignore friction.
(a) What is the initial angular momentum of the system? (b) What
is the angular velocity of the system after the dumbbells are pulled in
toward the axis? (c) Compute the kinetic energy of the system before
and after the dumbbells are pulled in. Account for the difference, if
any
(II) A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0.80 rad/s . Its total moment of inertia is 1360 kg.m2 Four people standing on the ground, each of mass 65 kg, suddenly step onto the edgeof the merry-go-round. (a) What is the angular velocity of the merry-go-round now? (b) What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?
(c) A uniform rod has a moment of inertia for rotation around its long axis that is (1/2)MR². A cylindrical rod of uniform density is located with its center at the origin, and its axis along the z axis. Its radius is 0.03
m, its length is 0.5 m, and its mass is 2 kg. It makes one revolution every 0.03 seconds. If you stand on the +z axis and look toward the origin at the rod, the rod spins clockwise.
What is the rotational angular momentum of the rod?
Irot =
kg · m2/s
What is the rotational kinetic energy of the rod?
Krot =
Chapter 10 Solutions
Physics for Scientists and Engineers with Modern Physics
Essential University Physics: Volume 2 (3rd Edition)
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