Page 223 - Fisika Terapan for Engineers and Scientists

Problems 423

velocities of the three smallest orbits are, respectively, 0.529 R its radius. Calculate the angular momentum of rotation of
S
10 6 10 6
10 m, 2.18 10 m/s; 2.12 10 m, 1.09 10 m/s; the Sun. Calculate the total orbital angular momentum of all
10 5
and 4.76 10 m, 7.27 10 m/s. For each of these orbits the planets; make the assumption that each planet moves in a
calculate the orbital angular momentum of the electron, with circular orbit of radius equal to its mean distance from the
the origin at the center. How do these angular momenta Sun listed in Table 9.1. What percentage of the angular
compare? momentum of the Solar System is in the rotational motion of
the Sun?
52. A high-speed meteoroid moves past the Earth along an
(almost) straight line. The mass of the meteoroid is 150 kg, 60. Suppose we measure the speed v and the radial distance r of a
1
1
its speed relative to the Earth is 60 km/s, and its distance of comet when it reaches perihelion. Use conservation of angular
4
closest approach to the center of the Earth is 1.2 10 km. momentum and conservation of energy to determine the speed
and the radius at aphelion.
(a) What is the angular momentum of the meteoroid in the
reference frame of the Earth (origin at the center of the 61. A playground merry-go-round is rotating at 2.0 radians/s.
Earth)? Consider the merry-go-round to be a uniform disk of mass
20 kg and radius 1.5 m. A 25-kg child, moving along a radial
(b) What is the angular momentum of the Earth in the refer-
line, jumps onto the edge of the merry-go-round. What is its
ence frame of the meteoroid (origin at the center of the
new angular velocity? The child then kicks the ground until
meteoroid)?
the merry-go-round (with the child) again rotates at 2.0 radi-
53. A train of mass 1500 metric tons runs along a straight track at
ans/s. If the child then walks radially inward, what will the
85 km/h. What is the angular momentum of the train about a
angular velocity be when the child is 0.50 m from the center?
point 50 m to the side of the track, left of the train? About a
62. The moment of inertia of the Earth is approximately 0.331
point on the track? 2 18
M R . If an asteroid of mass 5.0 10 kg moving at 150
E
E
54. The electron in a hydrogen atom moves around the nucleus
km/s struck (and stuck in) the Earth’s surface, by how long
under the influence of the electric force of attraction, a
would the length of the day change? Assume the asteroid was
central force pulling the electron toward the nucleus.
traveling westward in the equatorial plane and struck the
According to the Bohr theory, one of the possible orbits
Earth’s surface at 45 .
of the electron is an ellipse of angular momentum 2h with
63. In a popular demonstration, a professor rotates on a stool at
a distance of closest approach (1 222 3)a and a distance
0
of farthest recession (1 222 3)a , where h and 0.50 rev/s, holding two 10-kg masses, each 1.0 m from the
0
a are two atomic constants with the numerical values axis of rotation. If she pulls the weights inward until they are
0
2
1.05 10 34 kg m /s (“Planck’s constant”) and 5.3 10 11 m 10 cm from the axis, what is the new rotational frequency?
Without the weights, the professor and stool have a moment
(“Bohr radius”), respectively. In terms of h and a , find the
0 2 2
speed of the electron at the points of closest approach and of inertia of 6.0 kg m with arms extended and 4.0 kg m with
arms pulled in.
farthest recession; then evaluate numerically.
64. In a demonstration, a bicycle wheel with moment of inertia
55. According to a simple (but erroneous) model, the proton is a 2
27 0.48 kg m is spun up to 18 radians/s, rotating about a vertical
uniform rigid sphere of mass 1.67 10 kg and radius
15 axis. A student holds the wheel while sitting on a rotatable
1.0 10 m. The spin angular momentum of the proton is
35 stool. The student and stool are initially stationary and have a
5.3 10 J s. According to this model, what is the angular 2
moment of inertia of 3.0 kg m . If the student turns the bicy-
velocity of rotation of the proton? What is the linear velocity
cle wheel over so its axis points in the opposite direction, with
of a point on its equator? What is the rotational kinetic
what angular velocity will the student and stool rotate? For
energy? How does this rotational energy compare with the
2 simplicity, assume the wheel is held overhead, so that the stu-
rest-mass energy mc ?
dent, wheel, and stool all have the same axis of rotation.
56. What is the angular momentum of a Frisbee spinning at
65. A very heavy freight train made up of 250 cars has a total mass
20 rev/s about its axis of symmetry? Treat the Frisbee as a uni-
of 7700 metric tons. Suppose that such a train accelerates from
form disk of mass 200 g and radius 15 cm.
0 to 65 km/h on a track running exactly east from Quito,
57. A phonograph turntable is a uniform disk of radius 15 cm and
Ecuador (on the equator). The force that the engine exerts on
mass 1.4 kg. If this turntable accelerates from 0 rev/min to
the Earth will slow down the rotational motion of the Earth.
78 rev/min in 2.5 s, what is the average rate of change of the
By how much will the angular velocity of the Earth have
angular momentum in this time interval?
decreased when the train reaches its final speed? Express your
58. The propeller shaft of a cargo ship has a diameter of 8.8 cm, a answer in revolutions per day. The moment of inertia of the
2
length of 27 m, and a mass of 1200 kg. What is the rotational Earth is 0.33M R .
E E
kinetic energy of this propeller shaft when it is rotating at 8
66. There are 1.1 10 automobiles in the United States, each of
200 rev/min? What is the angular momentum?
an average mass of 2000 kg. Suppose that one morning all
59. The Sun rotates about its axis with a period of about 25 days. these automobiles simultaneously start to move in an eastward
2
Its moment of inertia is 0.20M R , where M is its mass and
S S S direction and accelerate to a speed of 80 km/h.