1, 2, 3 = straightforward, intermediate,

challenging

2. What is the angular speed in radians

per second of (a) the Earth in its orbit about
the Sun and (b) the Moon in its orbit about
the Earth?

Answer: (a) 1.99 × 10–7 rad/s; (b) 2.65 × 10–6

rad/s

4. (a) The positions of the hour and Figure P10.12

minute hand on a clock face coincide at 12
o’clock. Determine all other times (up to the 14. A car is traveling at 36.0 km/h on a
second) at which the positions of the hands straight road. The radius of its tires is 25.0
coincide. (b) If the clock also has a second cm. Find the angular speed of one of the
hand, determine all times at which the tires, with its axle taken as the axis of
positions of all three hands coincide, given rotation.
that they all coincide at 12 o’clock.
Answer: 40.0 rad/s
Answer: (a) 12n/11 h; (b) only when straight
up at 12 o’clock 18. A 6.00-kg block is released from A on
the frictionless track shown in Figure
12. The diameters of the main rotor and P10.18. Determine the radial and tangential
tail rotor of a single-engine helicopter are components of acceleration for the block at
7.60 m and 1.02 m, respectively. The P.
respective rotational speeds are 450 rev/min
and 4 138 rev/min. Calculate the speeds of
the tips of both rotors. Compare these
speeds with the speed of sound, 343 m/s.

Answer: 179 m/s for the main rotor, 221 m/s

at the tail. They are respectively 0.522 and
0.644 vsound.

Figure P10.18

Answer: ar = 29.4 m/s2, at = 9.80 m/s2

21. A small object with mass 4.00 kg rad/s. Calculate the ratio of the rotational
moves counterclockwise with constant energy to the translational kinetic energy.
speed 4.50 m/s in a circle of radius 3.00 m Treat the ball as a uniform sphere.
centered at the origin. (a) It started at the
point with cartesian coordinates (3 m, 0). Answer: 1/160
When its angular displacement is 9.00 rad,
what is its position vector, in cartesian unit- 30. Use the parallel-axis theorem and
vector notation? (b) In what quadrant is the Table 10.2 to find the moments of inertia of
particle located, and what angle does its (a) a solid cylinder about an axis parallel to
position vector make with the positive x the center-of-mass axis and passing through
axis? (c) What is its velocity vector, in unit– the edge of the cylinder and (b) a solid
vector notation? (d) In what direction is it sphere about an axis tangent to its surface.
moving? Make a sketch of the position and
velocity vectors. (e) What is its acceleration, Answer: (a) 3/2 MR2; (b) 7/5 MR2
expressed in unit–vector notation? (f) What
total force acts on the object? (Express your 41. A bicycle wheel has a diameter of
answer in unit vector notation.) 64.0 cm and a mass of 1.80 kg. Assume that
the wheel is a hoop with all of its mass
Answer: (a) r = (–2.73i + 1.24j) m; (b) at an concentrated on the outside radius. The
angle of 156° it is in the second quadrant bicycle is placed on a stationary stand on
(between 90 and 180°); (c) v = (–1.85i – 4.10j) rollers, and a resistive force of 120 N is
m/s; (d) 246° from the positive x axis; (e) a = applied tangent to the rim of the tire. (a)
(6.15i – 2.78j) m/s2; (f) ΣF = (24.6i – 11.1j) N What force must be applied by a chain
passing over a 9.00-cm-diameter sprocket if
22. A standard cassette tape is placed in the wheel is to attain an acceleration of 4.50
a standard cassette player. Each side plays rad/s2? (b) What force is required if the
for 30 min. The two tape wheels of the chain shifts to a 5.60-cm-diameter sprocket?
cassette fit onto two spindles in the player.
Suppose that a motor drives one spindle at Answer: (a) 872 N; (b) 1.40 kN
a constant angular speed of ~ 1 rad/s and
that the other spindle is free to rotate at any 44. A mass m1 and a mass m2 are
angular speed. Estimate the order of suspended by a pulley that has a radius R
magnitude of the thickness of the tape. and a mass M (see Fig. P10.43). The cord
has a negligible mass and causes the pulley
Answer: ~ 10–2 cm to rotate without slipping. The pulley
rotates without friction. The masses start
24. The center of mass of a pitched from rest a distance d apart. Treating the
baseball (3.80-cm radius) moves at 38.0 m/s. pulley as a uniform disk, determine the
The ball spins about an axis through its speeds of the two masses as they pass each
center of mass with an angular speed of 125 other.
 is a solid cylinder with a mass of 1 000 kg
and a diameter of 1.00 m. If the bus requires
an average power of 10.0 kW, how long
does the flywheel rotate?

Answer: 10.3 min

56. A flywheel in the form of a heavy

circular disk of diameter 0.600 m and mass
200 kg is mounted on a frictionless bearing.
A motor connected to the flywheel
accelerates it from rest to 1 000 rev/min. (a)
What is the moment of inertia of the
flywheel? (b) How much work is done on it
during this acceleration? (c) When the
angular speed reaches 1 000 rev/min, the
motor is disengaged. A friction brake is
used to slow the rotational rate to 500
rev/min. How much energy is dissipated as
internal energy in the friction brake?
Figure P10.43
Answer: (a) 9.00 kg ∙ m2; (b) 49.3 kJ; (c) –37.0
kJ

Answer:
58. For any given rotational axis, the
radius of gyration K of a rigid body is defined
by the expression K2 = I/M, where M is the
46. A constant torque of 25.0 N · m is total mass of the body and I is its moment
applied to a grindstone whose moment of of inertia. Thus, the radius of gyration is
inertia is 0.130 kg · m2. Using energy equal to the distance between an imaginary
principles, find the angular speed after the point mass M and the axis of rotation such
grindstone has made 15.0 revolutions. that I for the point mass about that axis is
(Neglect friction.) the same as that for the rigid body. Find the
radius of gyration of (a) a solid disk of
Answer: 30.3 rev/s radius R, (b) a uniform rod of length L, and
(c) a solid sphere of radius R, all three of
48. A bus is designed to draw its power which are rotating about a central axis.
from a rotating flywheel that is brought up
to its maximum rate of rotation (3 000 Answer:
rev/min) by an electric motor. The flywheel
 (a) (b)
2. A bowling ball has a mass of 4.00 kg,
a moment of inertia of 1.60 × 10–2 kg · m2,
(c) and a radius of 0.100 m. If it rolls down the
lane without slipping at a linear speed of
4.00 m/s, what is its total energy?
66. A wheel is formed from a hoop and
n equally spaced spokes extending from the Answer: 44.8 J
center of the hoop to its rim. The mass of
the hoop is M, and the radius of the hoop 6. A ring of mass 2.40 kg, inner radius
(and hence the length of each spoke) is R. 6.00 cm, and outer radius 8.00 cm rolls
The mass of each spoke is m. Determine (a) (without slipping) up an inclined plane that
the moment of inertia of the wheel about an makes an angle of θ = 36.9° (Fig. P11.6). At
axis through its center and perpendicular to the moment the ring is at position x = 2.00
the plane of the wheel and (b) the moment m up the plane, its speed is 2.80 m/s. The
of inertia of the wheel about an axis ring continues up the plane for some
through its rim and perpendicular to the additional distance and then rolls back
plane of the wheel. down. It does not roll off the top end. How
far up the plane does it go?
Answer:
(a)

(b)

69. An electric motor can accelerate a

Ferris wheel of moment of inertia I = 20 000
kg · m2 from rest to 10.0 rev/min in 12.0 s.
When the motor is turned off, friction
causes the wheel to slow down from 10.0 to
8.00 rev/min in 10.0 s. Determine (a) the
torque generated by the motor to bring the
wheel to 10.0 rev/min and (b) the power Figure P11.6
that would be needed to maintain this
rotational speed. Answer: 3.19 m

Answer: (a) 2.16 × 103 N ∙ m; (b) 439 W

(from Chapter 11, fifth ed.)