Chapter Problems
Equations provided for the AP Exam
1.
(I) Express the following angles in radians: (a)
30º, (b) 57º, (c)
90º, (d) 360º, and (e)
420º. Give as numerical values and as fractions of
2. (I) Eclipses happen on Earth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular
diameters (in radians) of the Sun and the Moon, as seen on Earth.
3. (I) A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges at an angle θ (Fig. below) of 1.4x10-6 rad.
What diameter spot
will it make on the Moon?
4. (I) The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s.
What
is the angular acceleration as the blades slow down?
5.
(II) A child rolls a ball on a level floor 3.5 m to another child. If
the ball makes 15.0 revolutions, what is its diameter?
6.
(II) A bicycle with tires 68 cm in diameter travels 8.0 km. How many
revolutions do the wheels make?
7. (II) (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angular velocity in rads.
(b)
What are the linear speed and acceleration of a point on the edge of the
grinding wheel?
8. (II) A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38).
(a) What is the linear speed of a child seated 1.2 m from the center?
(b)
What is her acceleration (give components)?
9. (II) Calculate the angular velocity of the Earth
(a) in its orbit around the Sun, and
(b)
about its axis.
10. (II) What is the linear speed of a point
(a) on the equator,
(b) on the Arctic Circle (latitude 66.5º N), and
(c) at a
latitude of
45.0º
N, due to the Earth’s rotation?
11.
(II) How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from
the axis of rotation is to experience an acceleration of 100,000
g’s?
12. (II) A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determine
(a) its angular acceleration, and
(b)
the radial and tangential components of the linear acceleration of a point
on the edge of the wheel 2.0 s after it has started accelerating.
13. (II) A turntable of radius R1 is turned by a circular rubber roller of radius R2 in contact with it at their outer edges. What is the ratio of their angular velocities, ω1/ω2
14. (III) In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly.
At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval.
The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, and
(b)
the radial and tangential components of the linear acceleration of a point
on the skin of the ship 5.0 min after it started this acceleration.
8–2 and 8–3 Constant
Angular Acceleration; Rolling
15.
(I) A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s.
Through how many revolutions did it turn in this time?
16. (I) An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant, and
(b)
the total number of revolutions the engine makes in this time.
17. (I) Pilots can be tested for the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching
its final speed.
(a) What was its angular acceleration (assumed constant), and
(b)
what was its final angular speed in rpm?
18. (II) A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have
traveled in
this time?
19. (II) A cooling fan is turned off when it is running at 850 rev/min. it turns 1500 revolutions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?
(b)
How long did it take the fan to come to a complete stop?
20. (II) A small rubber wheel is used to drive a large pottery wheel, and they are mounted so that their circular edges touch.
The small wheel has a radius of 2.0 cm and accelerates at the rate of 7.2 rad/s2 and it is in contact with the pottery wheel (radius 25.0 cm)
without slipping. Calculate
(a) the angular acceleration of the pottery wheel, and
(b)
the time it takes the pottery wheel to reach its required speed of 65 rpm.
21. (II) The tires of a car make 65 revolutions as the car reduces its speed uniformly from 95 km/hr to 45 km/hr. The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires?
(b)
If the car continues to decelerate at this rate, how much more time is
required for it to stop?
22. (I) A 55-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?
(b)
How could she
exert more torque?
23. (I) A person exerts a force of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exerted
(a) perpendicular to the door, and
(b)
at a 45º angle to the face of the door?
24.
(II) Calculate the net torque about the axle of the wheel shown in
the figure below. Assume that a friction torque of
0.40 m N
opposes the motion.
25. (II) Two blocks, each of mass m, are attached to the ends of a massless rod which pivots as shown below. Initially the rod is held in the
horizontal position and then released.
Calculate the magnitude and direction of the net torque on this
system.
26. (II) The bolts on the cylinder head of an engine require tightening to a torque of 88 m N.
a. If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?
b. If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench as shown
in the figure below.
8–5 and 8–6 Rotational
Dynamics
27.
(I) Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m
when the axis of rotation is through its center.
28.
(I) Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter.
The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be
ignored (why?).
29. (II) A small 650-gram ball on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle, and
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia
and air resistance.
30. (II) A potter is shaping a bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands
and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity
of
the wheel is 1.6 rev/s, and the moment of inertia
of the wheel and the bowl
is
0.11 kg m2
31. (II) Calculate the moment of inertia of the array of point objects shown in the figure below about the axes described
Assume m = 1.8 kg, M = 3.1 kg, and the objects are wired together by very light, rigid pieces of wire. Also assume the array is
rectangular and is split through the middle by the horizontal axis.
(a) the vertical axis, and
(b) the horizontal axis.
(c)
About which axis would it be harder to accelerate this array?
32. (II) An oxygen molecule consists of two oxygen atoms whose total mass is 5.3x10-26 kg and whose moment of inertia about an axis perpendicular to the line joining the two
atoms,
midway between them, is
1.9x10-46
kg.m2.
From these data,
estimate the effective distance between the atoms.
33. (II) To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown the figure below.
If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the
required steady force of each rocket if the satellite is to reach 32 rpm in
5.0 min?
34. (II) A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculate (a) its moment of inertia about its center, and (b) the applied torque
needed to accelerate it from rest to 1500 rpm in 5.00 s
if it is known to slow down from 1500 rpm to rest in 55.0 s.
35.
(II) A softball player swings a bat, accelerating it from rest to
in a time of 0.20 s.
Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute
the torque the
player applies to one end of it.
36. (II) A teenager pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s.
Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge.
Calculate the torque required to produce
the acceleration, neglecting frictional torque. What force is required at
the edge?
37. (II) A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.20 mN.
If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,
and how long will it take?
38. (II) The forearm in the figure below accelerates a 3.6-kg ball at 7.0 m/s2 by means of the triceps muscle, as shown. Calculate
(a) the torque needed, and
(b) the
force that must be exerted by the triceps muscle. Ignore the mass of the
arm.
39. (II) Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45.
The ball is
accelerated uniformly from rest to
in 0.350 s, at which
point it is released. Calculate
(a) the angular acceleration of the arm, and
(b)
the force required of the triceps muscle. Assume that the forearm has a mass
of 3.70 kg and rotates like a uniform rod about an axis at its end.
40. (II) A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8–46.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.
(b) How much torque
must the motor apply to bring the blades up to a speed of
in 8.0 s?
41.
(III) An Atwood’s
machine consists of two masses,
and
which are connected by
a massless inelastic cord that passes over a pulley, Fig. 8–47. If the
pulley has radius R
and moment of
inertia I about its axle,
determine the acceleration of the masses
and
and compare to the
situation in which the moment of inertia of the pulley is ignored.
[Hint:
The tensions
and
are not equal. We
discussed this situation in Example 4–13, assuming
for the pulley.]
42.
(III) A hammer thrower accelerates the hammer
from rest within four
full turns (revolutions) and releases it at a speed of
Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,
(b) the (linear) tangential acceleration,
(c) the centripetal acceleration just before release, (d) the net force being exerted on the hammer by the athlete just before release, and (e) the angle of this force with respect to
the radius of the circular motion.
43.
(I) A centrifuge rotor has a moment of inertia of
-3.75x10-2
kg m2. How much energy is
required to bring it from rest to 8250 rpm?
44.
(II) An automobile engine develops a torque of
280 mN at 3800 rpm. What is
the power in watts and in horsepower?
45.
(II) A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 m/s. Calculate its total kinetic energy.
46. (II) Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms,
(a) that due to its daily rotation about its axis, and
(b)
that due to its yearly revolution about the Sun. [Assume the Earth is a
uniform sphere with
mass = 6.0x1024
kg
and
radius = 6.4x106
m, and is 1.5x108
km
from the Sun.]
47. (II) A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s?
Assume it is a solid cylinder.
48. (II) A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without slipping down a 30.0º incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom.
(b) What is the ratio of translational to rotational ke at the bottom? Avoid putting in numbers until the end so you can answer:
(c)
do your answers in (a) and (b)
depend on the radius of the sphere or its mass?
49.
(III) Two masses,
and
are connected by a
rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform
cylinder of radius 0.260 m and mass 7.50 kg.
Initially,
is on the ground and
rests 3.00 m above the
ground. If the system is now released, use conservation of energy to
determine the speed of
just before it strikes
the ground. Assume the pulley is frictionless.
50. (III) A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its lower end does not slip.
What will be the speed of the upper
end of the pole just before it hits the ground? [Hint:
Use conservation of energy.]
51.
(I) What is the angular momentum of a 0.210-kg ball rotating on the end of a
thin string in a circle of radius 1.10 m at an angular speed of
52.
(I) (a) What is the
angular momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18
cm when rotating at 1500 rpm? (b)
How much torque is required to stop it in 6.0 s?
53.
(II) A person stands, hands at his side, on a platform that is
rotating at a rate of
If he raises his arms
to a horizontal position, Fig. 8–48,
the speed of rotation decreases to
(a)
Why? (b) By what factor has his
moment of inertia changed?
54. (II) A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position.
If she makes 2.0 rotations in 1.5 s when in
the tuck position, what is her angular speed
when in the straight
position?
55.
(II) A figure skater can increase her spin rotation rate from an
initial rate of 1.0 rev every 2.0 s to a final rate of
If her initial moment
of inertia was
what is her final
moment of inertia? How does she physically accomplish this change?
56.
(II) A potter’s wheel is rotating around a vertical axis through its
center at a frequency of
The wheel can be
considered a uniform disk of mass 5.0 kg and diameter 0.40 m.
The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel
after the clay sticks to it?
57.
(II) (a) What is the
angular momentum of a figure skater spinning at
with arms in close to
her body, assuming her to be a uniform cylinder with a height of 1.5 m, a
radius of 15 cm, and a mass of 55 kg? (b)
How much torque is required to slow her to a stop in 5.0 s, assuming she
does not move her arms?
58.
(II) Determine the angular momentum of the Earth (a)
about its rotation axis (assume the Earth is a uniform sphere), and (b)
in its orbit around the Sun (treat the Earth as a particle orbiting the
Sun). The Earth has
and
and is
from the Sun.
59.
(II) A nonrotating cylindrical disk of moment of inertia
I is dropped onto an identical
disk rotating at angular speed
Assuming no external
torques, what is the final common angular speed of the two disks?
60.
(II) A uniform disk turns at
around a frictionless
spindle. A nonrotating rod, of the same mass as the disk and length equal to
the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49.
They then both turn around the spindle with their centers superposed. What
is the angular frequency in
of the combination?
61.
(II) A person of mass 75 kg stands at the center of a rotating
merry-go-round platform of radius 3.0 m and moment of inertia
The platform rotates
without friction with angular velocity
The person walks
radially to the edge of the platform. (a)
Calculate the angular velocity when the person reaches the edge. (b)
Calculate the rotational kinetic energy of the system of platform plus
person before and after the person’s walk.
62.
(II) A 4.2-m-diameter merry-go-round is rotating freely with an
angular velocity of
Its total moment of
inertia is
Four people standing
on the ground, each of mass 65 kg, suddenly step onto the edge of the
merry-go-round. What is the angular velocity of the merry-go-round now? What
if the people were on it initially and then jumped off in a radial direction
(relative to the merry-go-round)?
63.
(II) Suppose our Sun eventually collapses into a white dwarf, losing
about half its mass in the process, and winding up with a radius 1.0% of its
existing radius. Assuming the lost mass carries away no angular momentum,
what would the Sun’s new rotation rate be? (Take the Sun’s current period to
be about 30 days.) What would be its final
ke in terms of its initial
ke of today?
64.
(III) Hurricanes can involve winds in excess of
at the outer edge.
Make a crude estimate of (a) the
energy, and (b) the angular
momentum, of such a hurricane, approximating it as a rigidly rotating
uniform cylinder of air (density
) of radius 100 km and height 4.0 km.
65.
(III) An asteroid of mass
traveling at a speed
of
relative to the Earth,
hits the Earth at the equator tangentially, and in the direction of Earth’s
rotation. Use angular momentum to estimate the percent change in the angular
speed of the Earth as a result of the collision.