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388 CHAPTER 12 Rotation of a Rigid Body

initial angular velocity? Through what total angle does the 12.5 Kinetic Energy of Rotation;
spool rotate while slowing to a stop? What is the angular Moment of Iner tia
acceleration during this motion?
23. If you lift the lid of a washing machine during the rapid 30. Find the moment of inertia of an orange of mass 300 g and
spin–dry cycle, the cycle stops (for safety), typically after 5.0 diameter 9.0 cm. Treat the orange as a uniform sphere.
revolutions. If the clothes are spinning at 6.0 rev/s initially, 31. The original Ferris wheel built by George Ferris (see
what is their constant angular acceleration during the slowing Fig. 12.20) had a radius of 38 m and a mass of 1.9 10 kg.
6
motion? How long do they take to come to a stop? Assume that all of the mass of the wheel was uniformly
24. A toy top initially spinning at 30 rev/s slows uniformly to a distributed along its rim. If the wheel was rotating at
stop in 25 seconds. What is the angular acceleration during 0.050 rev/min, what was its kinetic energy?
this motion? Through how many revolutions does the top turn
while slowing to a stop?
*25. The rotation of the Earth is slowing down. In 1977, the Earth
took 1.01 s longer to complete 365 rotations than in 1900.
What was the average angular deceleration of the Earth in the
time interval from 1900 to 1977?
*26. An automobile engine accelerates at a constant rate from
200 rev/min to 3000 rev/min in 7.0 s and then runs at con-
stant speed.
(a) Find the angular velocity and the angular acceleration
at t 0 (just after acceleration begins) and at t 7.0 s
(just before acceleration ends).
(b) A flywheel with a radius of 18 cm is attached to the shaft
of the engine. Calculate the tangential and the centripetal
acceleration of a point on the rim of the flywheel at the
times given above.
(c) What angle does the net acceleration vector make with
the radius at t 0 and at t 7.0 s? Draw diagrams show-
ing the wheel and the acceleration vector at these times. FIGURE 12.20 The original Ferris wheel.

12.4 Motion with Time-Dependent
32. What is the moment of inertia of a broomstick of mass
Angular Acceleration 0.50 kg, length 1.5 m, and diameter 2.5 cm about its longitu-
dinal axis? About an axis at right angles to the broomstick,
27. A disk has an initial angular velocity of 8.0 radians/s. At passing through its center?
0
t 0, it experiences a time-dependent angular acceleration
2
4
given by Ct , where C 0.25 radian/s . What is the 33. According to spectroscopic measurements, the moment of
inertia of an oxygen molecule about an axis through the center
instantaneous angular velocity at t 3.0 s? What is the
of mass and perpendicular to the line joining the atoms is
change in angular position between t 0 and t 1.0 s? 46 2
1.95 10 kg m . The mass of an oxygen atom is 2.66
28. A rigid body is initially at rest. Beginning at t 0, it begins 26
10 kg. What is the distance between the atoms? Treat the
rotating, with an angular acceleration given by atoms as pointlike particles.
2
2
{1 [t (4 s )]} for 0 t 2.0 s and 0 thereafter.
0
2
The initial value is 20 radians/s . What is the body’s 34. The moment of inertia of the Earth about its polar axis is
0
2
E
angular velocity after 1.0 s? After a long time? How many rev- 0.331M R , where M is the mass and R the equatorial
E
E
E
radius. Why is the moment of inertia smaller than that of a
olutions have occurred after 1.0 s?
sphere of uniform density? What would the radius of a sphere
*29. A sphere is initially rotating with angular velocity in a vis- of uniform density have to be if its mass and moment of iner-
0
cous liquid. Friction causes an angular deceleration that is pro-
tia are to coincide with those of the Earth?
portional to the instantaneous angular velocity, A ,
35. Problem 41 in Chapter 10 gives the dimensions of a molecule of
where A is a constant. Show that the angular velocity as a
nitric acid (HNO ). What is the moment of inertia of this mol-
function of time is given by 3
ecule when rotating about the symmetry axis passing through
e At the H, O, and N atoms? Treat the atoms as pointlike particles.
0