Page 191 - Fisika Terapan for Engineers and Scientists

Problems 391

Use these data to calculate the moment of inertia of the Earth *68. Derive the formula for the moment of inertia of a thin hoop of
about its axis. mass M and radius R rotating about a diameter.
61. The drilling pipe of an oil rig is 2.0 km long and 15 cm in *69. Find a formula for the moment of inertia of a uniform thin
diameter, and it has a mass of 20 kg per meter of length. square plate (mass M, dimension l l) rotating about an axis
Assume that the wall of the pipe is very thin. through the center and perpendicular to the plate.
(a) What is the moment of inertia of this pipe rotating about *70. Find the moment of inertia of a uniform cube of mass M and
its longitudinal axis? edge l. Assume the axis of rotation passes through the center
(b) What is the kinetic energy when rotating at 1.0 rev/s? of the cube and is perpendicular to two of the faces.
62. Engineers have proposed that large flywheels be used for the *71. What is the moment of inertia of a thin, flat plate in the shape
temporary storage of surplus energy generated by electric power of a semicircle rotating about the straight side (Fig. 12.28)?
plants. A suitable flywheel would have a diameter of 3.6 m and The mass of the plate is M and the radius is R.
a mass of 300 metric tons and would spin at 3000 rev/min. z
What is the kinetic energy of rotation of this flywheel? Give the
answer in both joules and kilowatt-hours. Assume that the
moment of inertia of the flywheel is that of a uniform disk.
63. An automobile of mass 1360 kg has wheels 76.2 cm in diameter
of mass 27.2 kg each.Taking into account the rotational kinetic
energy of the wheels about their axles, what is the total kinetic
energy of the automobile when traveling at 80.0 km/h? What y
percentage of the kinetic energy belongs to the rotational
motion of the wheels about their axles? Pretend that each wheel
has a mass distribution equivalent to that of a uniform disk. x
*64. The Oerlikon Electrogyro bus uses a flywheel to store energy
for propelling the bus. At each bus stop, the bus is briefly con-
FIGURE 12.28 A semicircle
nected to an electric power line, so that an electric motor on
rotating about its straight edge.
the bus can spin up the flywheel to 3000 rev/min. If the fly-
wheel is a disk of radius 0.60 m and mass 1500 kg, and if the
bus requires an average of 40 hp for propulsion at an average **72. Find the moment of inertia of the thin disk with two semi-
speed of 20 km/h, how far can it move with the energy stored circular cutouts shown in Fig. 12.29 rotating about its axis.
in the rotating flywheel? The disk is made of material of uniform thickness; its mass
*65. Pulsars are rotating stars made almost entirely of neutrons is M.
closely packed together. The rate of rotation of most pulsars
gradually decreases because rotational kinetic energy is gradu- R
ally converted into other forms of energy by a variety of com- r
plicated “frictional” processes. Suppose that a pulsar of mass
30
1.5 10 kg and radius 20 km is spinning at the rate of 2.1
2
rev/s and is slowing down at the rate of 1.0 10 15 rev/s .
What is the rate (in joules per second, or watts) at which the
rotational energy is decreasing? If this rate of decrease of the
h
energy remains constant, how long will it take the pulsar to
come to a stop? Treat the pulsar as a sphere of uniform density.
FIGURE 12.29 Disk with
66. For the sake of directional stability, the bullet fired from a rifle
two semicircular cutouts.
is given a spin angular velocity about its axis by means of spiral
grooves (“rifling”) cut into the barrel. The bullet fired by a
Lee–Enfield rifle is (approximately) a uniform cylinder of **73. A cone of mass M has a height h and a base diameter R. Find
length 3.18 cm, diameter 0.790 cm, and mass 13.9 g. The its moment of inertia about its axis of symmetry.
bullet emerges form the muzzle with a translational velocity of **74. Derive the formula given in Table 12.3 for the moment of
3
628 m/s and a spin angular velocity of 2.47 10 rev/s. What inertia of a sphere.
is the translational kinetic energy of the bullet? What is the
rotational kinetic energy? What fraction of the total kinetic
energy is rotational?
*67. Find a formula for the moment of inertia of a thin disk of
mass M and radius R rotating about a diameter.