Page 190 - Fisika Terapan for Engineers and Scientists

390 CHAPTER 12 Rotation of a Rigid Body

*54. Find the moment of inertia of the flywheel shown in *57. Derive a formula for the moment of inertia of a uniform
Fig. 12.23 rotating about its axis. The flywheel is made of spherical shell of mass M, inner radius R , outer radius R ,
1
2
material of uniform thickness; its mass is M. rotating about a diameter.
*58. Find the moment of inertia of a flywheel of mass M made by
cutting four large holes of radius r out of a uniform disk of
R
3 R radius R (Fig. 12.26). The holes are centered at a distance R/2
4 from the center of the flywheel.
1 R
4 45°

45°
R R
FIGURE 12.23 A flywheel.

1 R
*55. A solid cylinder capped with two solid hemispheres rotates 2 2
about its axis of symmetry (Fig. 12.24). The radius of the r
cylinder is R, its height is h, and the total mass (hemispheres
included) is M. What is the moment of inertia?
z
FIGURE 12.26 Disk with four holes.

*59. Show that the moment of inertia of a long, very thin cone
(Fig. 12.27) about an axis through the apex and perpendicular
3
to the centerline is Ml 2 , where M is the mass and l the
5
height of the cone.
z
h
y

y
x
l
R
x
FIGURE 12.24 A solid cylinder
capped with two solid hemispheres.
FIGURE 12.27 A long, thin cone
rotating about its apex.
*56. A hole of radius r has been drilled in a circular, flat plate of
radius R (Fig. 12.25). The center of the hole is at a distance d
*60. The mass distribution within the Earth can be roughly
from the center of the circle. The mass of this body is M. Find
approximated by several concentric spherical shells, each of
the moment of inertia for rotation about an axis through the
constant density. The following table gives the outer and the
center of the circle, perpendicular to the plate.
inner radius of each shell and its mass (expressed as a fraction
of the Earth’s mass):

R
FRACTION
SHELL OUTER RADIUS INNER RADIUS OF MASS
1 6400 km 5400 km 0.28
d
2 5400 4400 0.25
r
3 4400 3400 0.16
4 3400 2400 0.20
5 2400 0 0.11
FIGURE 12.25 Circular plate with a hole.