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424 CHAPTER 13 Dynamics of a Rigid Body

(a) What total angular momentum about the axis of the Earth of inertia remains constant during this movement. The
do all these automobiles contribute together? Assume that moment of inertia of the rowboat about the vertical axis is
2
2
the automobiles travel at an average latitude of 40 . 20 kg m and that of the woman is 0.80 kg m .
(b) How much will the rate of rotation of the Earth change *72. Two automobiles both of 1200 kg and both traveling at
because of the action of these automobiles? Assume that 30 km/h collide on a frictionless icy road. They were initially
the axis of the Earth remains fixed. The moment of inertia moving on parallel paths in opposite directions, with a
37
2
of the Earth is 8.1 10 kg m . center-to-center distance of 1.0 m (Fig. 13.36). In the colli-
sion, the automobiles lock together, forming a single body of
*67. Two artificial satellites of equal masses are in circular orbits of
wreckage; the moment of inertia of this body about its center
radii r and r around the Earth. The second has an orbit of
1 2 3 2
larger radius than the first (r r ). What is the speed of of mass is 2.5 10 kg m .
2 1
each? What is the angular momentum of each? Which has the (a) Calculate the angular velocity of the wreck.
larger speed? Which has the larger angular momentum? (b) Calculate the kinetic energy before the collision and after
*68. Consider the motion of the Earth around the Sun. Take as the collision. What is the change of kinetic energy?
origin the point at which the Earth is today and treat the
Earth as a particle.
1.0 m
(a) What is the angular momentum of the Earth about this
origin today?
(b) What will be the angular momentum of the Earth about FIGURE 13.36 Two automobiles collide.
the same origin three months from now? Six months from
*73. In one experiment performed under weightless conditions in
now? Nine months from now? Is the angular momentum
Skylab, the three astronauts ran around a path on the inside
conserved?
wall of the spacecraft so as to generate artificial gravity for
*69. The friction of the tides on the coastal shallows and the ocean
their bodies (Fig. 13.37). Assume that the center of mass of
floors gradually slows down the rotation of the Earth. The
each astronaut moves around a circle of radius 2.5 m; treat the
period of rotation (length of a sidereal day) is gradually
astronauts as particles.
increasing by 0.0016 s per century. What is the angular
2
deceleration (in radians/s ) of the Earth? What is the rate of (a) With what speed must each astronaut run if the average
normal force on his feet is to equal his normal weight (mg)?
decrease of the rotational angular momentum? What is the
rate of decrease of the rotational kinetic energy? The moment (b) Suppose that before the astronauts begin to run, Skylab is
2
of inertia of the Earth about its axis is 0.331M R , where M floating in its orbit without rotating. When the astronauts
E E E
is the mass of the Earth and R its equatorial radius. begin to run clockwise, Skylab will begin to rotate coun-
E
terclockwise. What will be the angular velocity of Skylab
*70. Phobos is a small moon of Mars. For the purposes of the fol-
lowing problem, assume that Phobos has a mass of 5.8 10 15 when the astronauts are running steadily with the speed
kg and that it has a shape of a uniform sphere of radius 7.5 calculated above? Assume that the mass of each astronaut
3
3
10 m. Suppose that a meteoroid strikes Phobos 5.0 10 m is 70 kg and that the moment of inertia of Skylab about
2
5
its longitudinal axis is 3 10 kg m .
off center (Fig. 13.35) and remains stuck. If the momentum
13
of the meteoroid was 3 10 kg m/s before impact and the (c) How often must the astronauts run around the inside if
mass of the meteoroid is negligible compared with the mass of they want Skylab to rotate through an angle of 30 ?
Phobos, what is the change in the rotational angular velocity
of Phobos? 7.5 km
5.0 km
meteoroid
FIGURE 13.35 A meteoroid strikes Phobos.

*71. A woman stands in the middle of a small rowboat. The row-
boat is floating freely and experiences no friction against the
water. The woman is initially facing east. If she turns around
180 so that she faces west, through what angle will the row-
boat turn? Assume that the woman performs her turning FIGURE 13.37 Three astronauts about to start running
movement at constant angular velocity and that her moment around inside Skylab.