Page 225 - Fisika Terapan for Engineers and Scientists

Problems 425

*74. A flywheel rotating freely on a shaft is suddenly coupled by center. The lower part of the spike protrudes 6.0 cm from the
means of a drive belt to a second flywheel sitting on a parallel disk. If you stand this top on its spike and start it spinning at
shaft (Fig. 13.38). The initial angular velocity of the first fly- 200 rev/s, what will be its precession frequency?
wheel is ; that of the second is zero. The flywheels are uni- 80. Suppose that the flywheel of a gyroscope is a uniform disk
form disks of masses M , M and of radii R , R , respectively. of mass 250 g and radius 3.0 cm. The distance of the center
1
2
1
2
The drive belt is massless and the shafts are frictionless. of this flywheel from the point of support is 4.0 cm. What
(a) Calculate the final angular velocity of each flywheel. is the precession frequency if the flywheel is spinning at
(b) Calculate the kinetic energy lost during the coupling 120 rev/s?
process. What happens to this energy? 81. If a bicycle in forward motion begins to tilt to one side, the
torque exerted by gravity will tend to turn the bicycle. Draw a
diagram showing the angular momentum of a (slightly tilted)
R 2 front wheel, the weight of the wheel, and the resulting torque.
In which direction is the instantaneous change in angular
momentum? Will this change make the tilt worse or better?
82. Slow precession can be used to determine a much more rapid
rotational frequency. Consider a top made by inserting a small
pin radially into a ball (a uniform sphere) of radius R 6.0 cm.
R 1 The pin extends 1.0 cm from the surface of the ball and
supports the top. When set spinning, the top is observed to
precess with a period of 0.75 s. What is the rotational
frequency of the top?
*83. The wheel of an automobile has a mass of 25 kg and a diame-
FIGURE 13.38 Two flywheels coupled by a drive belt. ter of 70 cm. Assume that the wheel can be regarded as a uni-
form disk.
*75. A thin rod of mass M and length l hangs from a pivot at its (a) What is the angular momentum of the wheel when the
upper end. A ball of clay of mass m and of horizontal velocity automobile is traveling at 25 m/s (90 km/h) on a straight
v strikes the lower end at right angles and remains stuck road?
(a totally inelastic collision). How high will the rod swing after
(b) What is the rate of change of the angular momentum of
this collision?
the wheel when the automobile is traveling at the same
**76. If the melting of the polar ice caps were to raise the water level speed along a curve of radius 80 m?
on the Earth by 10 m, by how much would the day be length-
(c) For this rate of change of the angular momentum,
ened? Assume that the moment of inertia of the ice in the
what must be the torque on the wheel? Draw a diagram
polar ice caps is negligible (they are very near the axis), and
showing the path of the automobile, the angular-
assume that the extra water spreads out uniformly over the
momentum vector of the wheel, and the torque
entire surface of the Earth (that is, neglect the area of the con-
vector.
tinents compared with the area of the oceans). The moment of
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inertia of the Earth (now) is 8.1 10 kg m . *84. Consider the airplane propeller described in Problem 38 in
Chapter 12. If the airplane is flying around a curve of radius
77. Consider a projectile of mass m launched with a speed v at 500 m at a speed of 360 km/h, what is the rate of change of
0
an elevation angle of 45 . If the launch point is the origin of
the angular momentum of the propeller? What torque is
coordinates, what is the angular momentum of the projectile
required to change the angular momentum at this rate? Draw
at the instant of launch? At the instant it reaches maximum
a diagram showing L, dL dt, and .
height? At the instant it strikes the ground? Is the angular
*85. A large flywheel designed for energy storage at a power plant
momentum conserved in this motion with this choice of 5 2
has a moment of inertia of 5 10 kg m and spins at 3000
origin?
rev/min. Suppose that this flywheel is mounted on a horizon-
tal axle oriented in the east–west direction. What are the mag-
13.4 Torque and Angular Momentum nitude and direction of its angular momentum? What is the
as Vectors rate of change of this angular momentum due to the rotational
motion of the Earth and the consequent motion of the axle of
78. Show that for a flat plate rotating about an axis perpendicular the flywheel? What is the torque that the axle of the flywheel
to the plate, the angular-momentum vector lies along the axis exerts against the bearings supporting it? If the bearings are at
of rotation, even if the body is not symmetric. a distance of 0.60 m from the center of the flywheel on each
79. A child’s toy top consists of a uniform thin disk of radius side, what are the forces associated with this torque?
5.0 cm and mass 0.15 kg with a thin spike passing through its