Download POP4e: Ch. 10 Problems

Chapter 10 Problems
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Section 10.1 Angular Position, Speed, and
Acceleration
1.
During a certain period of time, the
angular position of a swinging door is
described by θ = 5.00 + 10.0t + 2.00t2, where
θ is in radians and t is in seconds.
Determine the angular position, angular
speed, and angular acceleration of the door
(a) at t = 0 and (b) at t = 3.00 s.
Section 10.2 Rotational Kinematics: The
Rigid Object Under Constant Angular
Acceleration
2.
A dentist’s drill starts from rest.
After 3.20 s of constant angular
acceleration, it turns at a rate of 2.51 × 104
rev/min. (a) Find the drill’s angular
acceleration. (b) Determine the angle (in
radians) through which the drill rotates
during this period.
3.
An electric motor
rotating a grinding wheel at 100 rev/min is
switched off. The wheel then moves with a
constant negative angular acceleration of
magnitude 2.00 rad/s2. (a) During what time
interval does the wheel come to rest? (b)
Through how many radians does it turn
while it is slowing down?
4.
A centrifuge in a medical laboratory
rotates at an angular speed of 3 600
rev/min. When switched off, it rotates
through 50.0 revolutions before coming to
rest. Find the constant angular acceleration
of the centrifuge.
5.
The tub of a washer goes into its spin
cycle, starting from rest and gaining
angular speed steadily for 8.00 s, at which
time it is turning at 5.00 rev/s. At this point,
the person doing the laundry opens the lid
and a safety switch turns off the washer.
The tub smoothly slows to rest in 12.0 s.
Through how many revolutions does the
tub turn while it is in motion?
6.
A rotating wheel requires 3.00 s to
rotate through 37.0 rev. Its angular speed at
the end of the 3.00-s interval is 98.0 rad/s.
What is the constant angular acceleration of
the wheel?
7.
(a) Find the angular speed of the
Earth’s rotation on its axis. As the Earth
turns toward the east, we see the sky
turning toward the west at this same rate.
(b)
The rainy Pleiads wester
And seek beyond the sea
The head that I shall dream of
That shall not dream of me.
—A. E. Housman (© Robert E. Symons)
Cambridge, England, is at longitude 0° and
Saskatoon, Saskatchewan, is at longitude
107° west. How much time elapses after the
Pleiades set in Cambridge until these stars
fall below the western horizon in
Saskatoon?
Section 10.3 Relations Between
Rotational and Translational Quantities
The cyclist pedals at a steady cadence of
76.0 rev/min. The chain engages with a
front sprocket 15.2 cm in diameter and a
rear sprocket 7.00 cm in diameter. (a)
Calculate the speed of a link of the chain
relative to the bicycle frame. (b) Calculate
the angular speed of the bicycle wheels. (c)
Calculate the speed of the bicycle relative to
the road. (d) What pieces of data, if any, are
not necessary for the calculations?
8.
Make an order-of-magnitude
estimate of the number of revolutions
through which a typical automobile tire
turns in 1 yr. State the quantities you
measure or estimate and their values.
9.
A disk 8.00 cm in
radius rotates at a constant rate of 1 200
rev/min about its central axis. Determine (a)
its angular speed, (b) the tangential speed
at a point 3.00 cm from its center, (c) the
radial acceleration of a point on the rim,
and (d) the total distance a point on the rim
moves in 2.00 s.
10.
A wheel 2.00 m in diameter lies in a
vertical plane and rotates with a constant
angular acceleration of 4.00 rad/s2. The
wheel starts at rest at t = 0, and the radius
vector of a certain point P on the rim makes
an angle of 57.3° with the horizontal at this
time. At t = 2.00 s, find (a) the angular speed
of the wheel, (b) the tangential speed and
the total acceleration of the point P, and (c)
the angular position of the point P.
11.
Figure P10.11 shows the drivetrain of
a bicycle that has wheels 67.3 cm in
diameter and pedal cranks 17.5 cm long.
Figure P10.11
12.
A digital audio compact disc carries
data, each bit of which occupies 0.6 μm
along a continuous spiral track from the
inner circumference of the disc to the
outside edge. A CD player turns the disc to
carry the track counterclockwise above a
lens at a constant speed of 1.30 m/s. Find
the required angular speed (a) at the
beginning of the recording, where the spiral
has a radius of 2.30 cm, and (b) at the end of
the recording, where the spiral has a radius
of 5.80 cm. (c) A full-length recording lasts
for 74 min 33 s. Find the average angular
acceleration of the disc. (d) Assuming that
the acceleration is constant, find the total
angular displacement of the disc as it plays.
(e) Find the total length of the track.
13.
A car traveling on a flat (unbanked)
circular track accelerates uniformly from
rest with a tangential acceleration of 1.70
m/s2. The car makes it one fourth of the way
around the circle before it skids off the
track. Determine the coefficient of static
friction between the car and track from
these data.
Section 10.4 Rotational Kinetic Energy
15.
This problem describes one
experimental method for determining the
moment of inertia of an irregularly shaped
object such as the payload for a satellite.
Figure P10.15 shows a counterweight of
mass m suspended by a cord wound
around a spool of radius r, forming part of
a turntable supporting the object. The
turntable can rotate without friction. When
the counterweight is released from rest, it
descends through a distance h, acquiring a
speed v. Show that the moment of inertia I
of the rotating apparatus (including the
turntable) is mr2(2gh/v2 – 1).
14.
Rigid rods of negligible mass lying
along the y axis connect three particles (Fig.
P10.14). The system rotates about the x axis
with an angular speed of 2.00 rad/s. Find (a)
the moment of inertia about the x axis and
the total rotational kinetic energy evaluated
from ½ Iω2 and (b) the tangential speed of
each particle and the total kinetic energy
evaluated from  1 2 mi vi 2 .
Figure P10.15
Figure P10.14
16.
Big Ben, the Parliament tower clock
in London, has an hour hand 2.70 m long
with a mass of 60.0 kg and a minute hand
4.50 m long with a mass of 100 kg (Fig.
P10.16). Calculate the total rotational kinetic
energy of the two hands about the axis of
rotation. (You may model the hands as
uniform long, thin rods.)
(John Lawrence/Stone/Getty Images)
Figure P10.16 Problems 10.16, 10.42, and
10.64.
17.
Consider two objects with m1 > m2
connected by a light string that passes over
a pulley having a moment of inertia of I
about its axis of rotation as shown in Figure
P10.17. The string does not slip on the
pulley or stretch. The pulley turns without
friction. The two objects are released from
rest separated by a vertical distance 2h. (a)
Use the principle of conservation of energy
to find the translational speeds of the
objects as they pass each other. (b) Find the
angular speed of the pulley at this time.
Figure P10.17
18.
As a gasoline engine operates, a
flywheel turning with the crankshaft stores
energy after each fuel explosion, providing
the energy required to compress the next
charge of fuel and air. For the engine of a
certain lawn tractor, suppose a flywheel
must be no more than 18.0 cm in diameter.
Its thickness, measured along its axis of
rotation, must be no larger than 8.00 cm.
The flywheel must release energy 60.0 J
when its angular speed drops from 800
rev/min to 600 rev/min. Design a sturdy,
steel flywheel to meet these requirements
with the smallest mass that you can
reasonably attain. Assume that the material
has the density listed for iron in Table 15.1.
Specify the shape and mass of the flywheel.
19.
A war-wolf or trebuchet is a device
used during the Middle Ages to throw
rocks at castles and sometimes now used to
fling pianos as a sport. A simple trebuchet
is shown in Figure P10.19. Model it as a stiff
rod of negligible mass, 3.00 m long, joining
particles of mass 60.0 kg and 0.120 kg at its
ends. It can turn on a frictionless horizontal
axle perpendicular to the rod and 14.0 cm
from the large-mass particle. The rod is
released from rest in a horizontal
orientation. Find the maximum speed that
the small-mass object attains.
Figure P10.20
21.
Find the net torque on
the wheel in Figure P10.21 about the axle
through O, taking a = 10.0 cm and b = 25.0
cm.
Figure P10.19
Section 10.5 Torque and the Vector
Product
20.
The fishing pole in Figure P10.20
makes an angle of 20.0° with the horizontal.
What is the torque exerted by the fish about
an axis perpendicular to the page and
passing through the angler’s hand?
Figure P10.21

22.
Given M  6iˆ  2ˆj  kˆ and

N  2iˆ  ˆj  3kˆ , calculate the vector product
 
M N.
23.

A force of F  ( 2.00iˆ  3.00ˆj) N is
applied to an object that is pivoted about a
fixed axle aligned along the z coordinate
axis. The force is applied at the point

r  ( 4.00iˆ  5.00ˆj) m. Find (a) the
magnitude of the net torque about the z axis

and (b) the direction of the torque vector  .
24.
Two vectors are given by


A  3iˆ  7 ˆj  4kˆ and B  6iˆ  10ˆj  9kˆ .
Evaluate the following quantities. (a)
 
 
cos 1 A  B / AB and (b) sin 1 A  B / AB .




(c) Which give(s) the angle between the
vectors?
25.
Use the definition of the vector
product and the definitions of the unit
vectors î , ĵ , and k̂ to prove Equations
10.23. You may assume that the x axis
points to the right, the y axis up, and the z
axis toward you (not away from you). This
choice is said to make the coordinate
system right-handed.
Figure P10.26
27.
A uniform beam of mass mb and
length ℓ supports blocks with masses m1
and m2 at two positions as shown in Figure
P10.27. The beam rests on two knife edges.
For what value of x will the beam be
balanced at P such that the normal force at
O is zero?
Section 10.6 The Rigid Object in
Equilibrium
26.
In exercise physiology studies, it is
sometimes important to determine the
location of a person’s center of mass, which
can be done with the arrangement shown in
Figure P10.26. A light plank rests on two
scales, which read Fg1 = 380 N and Fg2 = 320
N. A distance of 2.00 m separates the scales.
How far from the woman’s feet is her
center of mass?
Figure P10.27
28.
A uniform plank of length 6.00 m
and mass 30.0 kg rests horizontally across
two horizontal bars of a scaffold. The bars
are 4.50 m apart, and 1.50 m of the plank
hangs over one side of the scaffold. Draw a
free-body diagram of the plank. How far
can a painter of mass 70.0 kg walk on the
overhanging part of the plank before it tips?
29.
Figure P10.29 shows a claw hammer
as it is being used to pull a nail out of a
horizontal board. A force of 150 N is
exerted horizontally as shown. Find (a) the
force exerted by the hammer claws on the
nail and (b) the force exerted by the surface
on the point of contact with the hammer
head. Assume that the force the hammer
exerts on the nail is parallel to the nail.
31.
A uniform sign of
weight Fg and width 2L hangs from a light,
horizontal beam hinged at the wall and
supported by a cable (Fig. P10.31).
Determine (a) the tension in the cable and
(b) the components of the reaction force
exerted by the wall on the beam, in terms of
Fg, d, L, and θ.
Figure P10.31
Figure P10.29
30.
A uniform ladder of length L and
mass m1 rests against a frictionless wall. The
ladder makes an angle θ with the
horizontal. (a) Find the horizontal and
vertical forces the ground exerts on the base
of the ladder when a firefighter of mass m2
is a distance x from the bottom. (b) If the
ladder is just on the verge of slipping when
the fire-fighter is a distance d from the
bottom, what is the coefficient of static
friction between ladder and ground?
32.
A crane of mass 3 000 kg supports a
load of 10 000 kg as shown in Figure P10.32.
The crane is pivoted with a frictionless pin
at A and rests against a smooth support at
B. Find the reaction forces at A and B.
Figure P10.32
35.
An electric motor turns a flywheel
through a drive belt that joins a pulley on
the motor and a pulley that is rigidly
attached to the flywheel, as shown in
Figure P10.35. The flywheel is a solid disk
with a mass of 80.0 kg and a diameter of
1.25 m. It turns on a frictionless axle. Its
pulley has much smaller mass and a radius
of 0.230 m. The tension in the upper (taut)
segment of the belt is 135 N, and the
flywheel has a clockwise angular
acceleration of 1.67 rad/s2. Find the tension
in the lower (slack) segment of the belt.
Section 10.7 The Rigid Object Under a
Net Torque
33.
The combination of an applied force
and a friction force produces a constant
total torque of 36.0 N · m on a wheel
rotating about a fixed axis. The applied
force acts for 6.00 s. During this time the
angular speed of the wheel increases from 0
to 10.0 rad/s. The applied force is then
removed, and the wheel comes to rest in
60.0 s. Find (a) the moment of inertia of the
wheel, (b) the magnitude of the frictional
torque, and (c) the total number of
revolutions of the wheel.
34.
A potter’s wheel—a thick stone disk
of radius 0.500 m and mass 100 kg—is
freely rotating at 50.0 rev/min. The potter
can stop the wheel in 6.00 s by pressing a
wet rag against the rim and exerting a
radially inward force of 70.0 N. Find the
effective coefficient of kinetic friction
between wheel and rag.
Figure P10.35
36.
In Figure P10.36, the sliding block
has a mass of 0.850 kg, the counterweight
has a mass of 0.420 kg, and the pulley is a
hollow cylinder with a mass of 0.350 kg, an
inner radius of 0.020 0 m, and an outer
radius of 0.030 0 m. The coefficient of
kinetic friction between the block and the
horizontal surface is 0.250. The pulley turns
without friction on its axle. The light cord
does not stretch and does not slip on the
pulley. The block has a velocity of 0.820 m/s
toward the pulley when it passes through a
photogate. (a) Use energy methods to
predict its speed after it has moved to a
second photogate, 0.700 m away. (b) Find
the angular speed of the pulley at the same
moment.
Figure P10.36
37.
Two blocks, as shown in Figure
P10.37, are connected by a string of
negligible mass passing over a pulley of
radius 0.250 m and moment of inertia I. The
block on the frictionless incline is moving
up with a constant acceleration of 2.00 m/s2.
(a) Determine T1 and T2, the tensions in the
two parts of the string. (b) Find the moment
of inertia of the pulley.
at one end as shown in Figure 10.9. The rod
is released from rest in the horizontal
position. What are the initial angular
acceleration of the rod and the initial
translational acceleration of the right end of
the rod?
39.
An object with a weight of 50.0 N is
attached to the free end of a light string
wrapped around a reel of radius 0.250 m
and mass 3.00 kg. The reel is a solid disk,
free to rotate in a vertical plane about the
horizontal axis passing through its center.
The suspended object is released 6.00 m
above the floor. (a) Determine the tension in
the string, the acceleration of the object, and
the speed with which the object hits the
floor. (b) Verify your last answer by using
the principle of conservation of energy to
find the speed with which the object hits
the floor.
Section 10.8 Angular Momentum
40.
Heading straight toward the summit
of Pikes Peak, an airplane of mass 12 000 kg
flies over the plains of Kansas at nearly
constant altitude 4.30 km with constant
velocity 175 m/s west. (a) What is the
airplane’s vector angular momentum
relative to a wheat farmer on the ground
directly below the airplane? (b) Does this
value change as the airplane continues its
motion along a straight line? (c) What is its
angular momentum relative to the summit
of Pikes Peak?
Figure P10.37
38.
A uniform rod of length L and mass
M is free to rotate about a frictionless pivot
41.
The position vector of a
particle of mass 2.00 kg is given as a

function of time by r  (6.00iˆ  5.00tˆj) m.
Determine the angular momentum of the
particle about the origin as a function of
time.
42.
Big Ben (Fig. P10.16), the Parliament
tower clock in London, has hour and
minute hands with lengths of 2.70 m and
4.50 m and masses of 60.0 kg and 100 kg,
respectively. Calculate the total angular
momentum of these hands about the center
point. Treat the hands as long, thin,
uniform rods.
43.
A particle of mass 0.400 kg is
attached to the 100-cm mark of a meter
stick of mass 0.100 kg. The meter stick
rotates on a horizontal, frictionless table
with an angular speed of 4.00 rad/s.
Calculate the angular momentum of the
system when the stick is pivoted about an
axis (a) perpendicular to the table through
the 50.0-cm mark and (b) perpendicular to
the table through the 0-cm mark.
44.
A space station is constructed in the
shape of a hollow ring of mass 5.00 × 104 kg.
Members of the crew walk on a deck
formed by the inner surface of the outer
cylindrical wall of the ring, with radius 100
m. At rest when constructed, the ring is set
rotating about its axis so that the people
inside experience an effective free-fall
acceleration equal to g. (Fig. P10.44 shows
the ring together with some other parts that
make a negligible contribution to the total
moment of inertia.) The rotation is achieved
by firing two small rockets attached
tangentially to opposite points on the
outside of the ring. (a) What angular
momentum does the space station acquire?
(b) How long must the rockets be fired if
each exerts a thrust of 125 N? (c) Prove that
the total torque on the ring, multiplied by
the time interval found in part (b), is equal
to the change in angular momentum, found
in part (a). This equality represents the
angular impulse–angular momentum theorem.
Figure P10.44 Problems 10.44 and 10.50.
Section 10.9 Conservation of Angular
Momentum
45.
A cylinder with moment of inertia I1
rotates about a vertical, frictionless axle
with angular speed ωi. A second cylinder,
this one having moment of inertia I2 and
initially not rotating, drops onto the first
cylinder (Fig. P10.45). Because of friction
between the surfaces, the two eventually
reach the same angular speed ωf. (a)
Calculate ωf. (b) Show that the kinetic
energy of the system decreases in this
interaction and calculate the ratio of the
final to the initial rotational energy.
with an angular speed of 0.750 rad/s. The
moment of inertia of the student plus stool
is 3.00 kg · m2 and is assumed to be
constant. The student pulls the weights
inward horizontally to a position 0.300 m
from the rotation axis. (a) Find the new
angular speed of the student. (b) Find the
kinetic energy of the rotating system before
and after he pulls the weights inward.
Figure P10.45
46.
A playground merry-go-round of
radius R = 2.00 m has a moment of inertia I
= 250 kg · m2 and is rotating at 10.0 rev/min
about a frictionless vertical axle. Facing the
axle, a 25.0-kg child hops onto the merrygo-round and manages to sit down on the
edge. What is the new angular speed of the
merry-go-round?
47.
A 60.0-kg woman stands at the rim
of a horizontal turntable having a moment
of inertia of 500 kg · m2 and a radius of 2.00
m. The turntable is initially at rest and is
free to rotate about a frictionless, vertical
axle through its center. The woman then
starts walking around the rim clockwise (as
viewed from above the system) at a
constant speed of 1.50 m/s relative to the
Earth. (a) In what direction and with what
angular speed does the turntable rotate? (b)
How much work does the woman do to set
herself and the turntable into motion?
48.
A student sits on a freely rotating
stool holding two weights, each of mass
3.00 kg (Fig. P10.48). When his arms are
extended horizontally, the weights are 1.00
m from the axis of rotation and he rotates
Figure P10.48
49.
A puck of mass 80.0 g and radius
4.00 cm slides along an air table at a speed
of 1.50 m/s as shown in Figure P10.49a. It
makes a glancing collision with a second
puck of radius 6.00 cm and mass 120 g
(initially at rest) such that their rims just
touch. Because their rims are coated with
instant-acting glue, the pucks stick together
and spin after the collision (Fig. P10.49b).
(a) What is the angular momentum of the
system relative to the center of mass? (b)
What is the angular speed about the center
of mass?
Determine the work done on the puck.
(Suggestion: Consider the change of kinetic
energy.)
Section 10.10 Precessional Motion of
Gyroscopes
(a)
(b)
Figure P10.49
50.
A space station shaped like a giant
wheel has a radius of 100 m and a moment
of inertia of 5.00 × 108 kg · m2. A crew of 150
is living on the rim, and the station’s
rotation causes the crew to experience an
apparent free-fall acceleration of g (Fig.
P10.44). When 100 people move to the
center of the station for a union meeting,
the angular speed changes. What apparent
free-fall acceleration is experienced by the
managers remaining at the rim? Assume
that the average mass for each inhabitant is
65.0 kg.
51.
The puck in Figure 10.24 has a mass
of 0.120 kg. The distance of the puck from
the center of rotation is originally 40.0 cm,
and the puck is sliding with a speed of 80.0
cm/s. The string is pulled downward 15.0
cm through the hole in the frictionless table.
52.
The angular momentum vector of a
precessing gyroscope sweeps out a cone as
shown in Figure 10.25b. Its angular speed,
called its precessional frequency, is given
by ωp = τ/L, where τ is the magnitude of the
torque on the gyroscope and L is the
magnitude of its angular momentum. In the
motion called precession of the equinoxes,
represented in Figure P10.52, the Earth’s
axis of rotation precesses about the
perpendicular to its orbital plane with a
period of 2.58 × 104 yr. Model the Earth as a
uniform sphere and calculate the torque on
the Earth that is causing this precession.
NASA
Figure P10.52 (a) At present, the spin axis
of the Earth points toward the North Star.
(b) Torque on the spinning Earth will cause
it to precess, so the spin axis will no longer
be pointing in this direction in the future.
Section 10.11 Rolling Motion of Rigid
Objects
53.
A cylinder of mass 10.0 kg rolls
without slipping on a horizontal surface. At
a certain instant its center of mass has a
speed of 10.0 m/s. Determine (a) the
translational kinetic energy of its center of
mass, (b) the rotational kinetic energy about
its center of mass, and (c) its total energy.
54.
A uniform solid disk and a uniform
hoop are placed side by side at the top of an
incline of height h. If they are released from
rest and roll without slipping, which object
reaches the bottom first? Verify your
answer by calculating their speeds when
they reach the bottom in terms of h.
55.
A tennis ball is a hollow sphere with
a thin wall. It is set rolling without slipping
at 4.03 m/s on a horizontal section of a track
as shown in Figure P10.55. It rolls around
the inside of a vertical circular loop 90.0 cm
in diameter and finally leaves the track at a
point 20.0 cm below the horizontal section.
(a) Find the speed of the ball at the top of
the loop. Demonstrate that it will not fall
from the track. (b) Find its speed as it leaves
the track. (c) Suppose static friction between
ball and track were negligible so that the
ball slid instead of rolling. Would its speed
then be higher, lower, or the same at the top
of the loop? Explain.
Figure P10.55
56.
A metal can containing condensed
mushroom soup has mass 215 g, height 10.8
cm, and diameter 6.38 cm. It is placed at
rest on its side at the top of a 3.00-m-long
incline that is at 25.0° to the horizontal and
is then released to roll straight down. It
takes 1.50 s to reach the bottom of the
incline. Assuming mechanical energy
conservation, calculate the moment of
inertia of the can. Which pieces of data, if
any, are unnecessary in calculating the
solution?
Section 10.12 Context Connection—
Turning the Spacecraft
57.
A spacecraft is in empty space. It
carries on board a gyroscope with a
moment of inertia of Ig = 20.0 kg · m2 about
the axis of the gyroscope. The moment of
inertia of the spacecraft around the same
axis is Is = 5.00 × 105 kg · m2. Neither the
spacecraft nor the gyroscope is originally
rotating. The gyroscope can be powered up
in a negligible period of time to an angular
speed of 100 s–1. If the orientation of the
spacecraft is to be changed by 30.0°, for
how long should the gyroscope be
operated?
Additional Problems
58.
Review problem. A mixing beater
consists of three thin rods, each 10.0 cm
long. The rods diverge from a central hub,
separated from each other by 120°, and all
turn in the same plane. A ball is attached to
the end of each rod. Each ball has crosssectional area 4.00 cm2 and is so shaped that
it has a drag coefficient of 0.600. Calculate
the power input required to spin the beater
at 1 000 rev/min (a) in air and (b) in water.
59.
A long uniform rod of length L and
mass M is pivoted about a horizontal,
frictionless pin through one end. The rod is
released from rest in a vertical position as
shown in Figure P10.59. At the instant the
rod is horizontal, find (a) its angular speed,
(b) the magnitude of its angular
acceleration, (c) the x and y components of
the acceleration of its center of mass, and
(d) the components of the reaction force at
the pivot.
Figure P10.59
60.
A uniform, hollow, cylindrical spool
has inside radius R/2, outside radius R, and
mass M (Fig. P10.60). It is mounted so that
it rotates on a fixed, horizontal axle. A
counterweight of mass m is connected to
the end of a string wound around the spool.
The counterweight falls from rest at t = 0 to
a position y at time t. Show that the torque
due to the friction forces between spool and
axle is
 
2y 
5y 
 f  Rm g  2   M 2 
t 
4t 
 
Figure P10.60
61.
The reel shown in Figure P10.61 has
radius R and moment of inertia I. One end
of the block of mass m is connected to a
spring of force constant k, and the other end
is fastened to a cord wrapped around the
reel. The reel axle and the incline are
frictionless. The reel is wound
counterclockwise so that the spring
stretches a distance d from its unstretched
position and is then released from rest. (a)
Find the angular speed of the reel when the
spring is again unstretched. (b) Evaluate the
angular speed numerically at this point
taking I = 1.00 kg · m2, R = 0.300 m, k = 50.0
N/m, m = 0.500 kg, d = 0.200 m, and θ =
37.0°.
63.
A common demonstration,
illustrated in Figure P10.63, consists of a
ball resting at one end of a uniform board
of length ℓ, hinged at the other end, and
elevated at an angle θ. A light cup is
attached to the board at rc so that it will
catch the ball when the support stick is
suddenly removed. (a) Show that the ball
will lag behind the falling board when θ is
less than 35.3°. (b) Assume that the board is
1.00 m long and is supported at this
limiting angle. Show that the cup must be
18.4 cm from the moving end.
Figure P10.61
62.
A block of mass m1 = 2.00 kg and a
block of mass m2 = 6.00 kg are connected by
a massless string over a pulley in the shape
of a solid disk having radius R = 0.250 m
and mass M = 10.0 kg. These blocks are
allowed to move on a fixed block-wedge of
angle θ = 30.0° as shown in Figure P10.62.
The coefficient of kinetic friction is 0.360 for
both blocks. Draw free-body diagrams of
both blocks and of the pulley. Determine (a)
the acceleration of the two blocks and (b)
the tensions in the string on both sides of
the pulley.
Figure P10.62
Figure P10.63
64.
The hour hand and the minute hand
of Big Ben, the Parliament tower clock in
London, are 2.70 m and 4.50 m long and
have masses of 60.0 kg and 100 kg,
respectively (see Fig. P10.16). (a) Determine
the total torque due to the weight of these
hands about the axis of rotation when the
time reads (i) 3:00, (ii) 5:15, (iii) 6:00, (iv)
8:20, and (v) 9:45. (You may model the
hands as long, thin, uniform rods.) (b)
Determine all times when the total torque
about the axis of rotation is zero. Determine
the times to the nearest second, solving a
transcendental equation numerically.
child? List the assumptions you make in
solving this problem. The stove is supplied
with a wall bracket to prevent the accident.
65.
A string is wound around a uniform
disk of radius R and mass M. The disk is
released from rest with the string vertical
and its top end tied to a fixed bar (Fig.
P10.65). Show that (a) the tension in the
string is one-third the weight of the disk,
(b) the magnitude of the acceleration of the
center of mass is 2g/3, and (c) the speed of
the center of mass is (4gh/3)1/2 after the disk
has descended through distance h. Verify
your answer to (c) using the energy
approach.
Figure P10.66
Figure P10.65
66.
A new General Electric stove has a
mass of 68.0 kg and the dimensions shown
in Figure P10.66. The stove comes with a
warning that it can tip forward if a person
stands or sits on the oven door when it is
open. What can you conclude about the
weight of such a person? Could it be a
67.
(a) Without the wheels, a bicycle
frame has a mass of 8.44 kg. Each of the
wheels can be roughly modeled as a
uniform solid disk with a mass of 0.820 kg
and a radius of 0.343 m. Find the kinetic
energy of the whole bicycle when it is
moving forward at 3.35 m/s. (b) Before the
invention of a wheel turning on an axle,
ancient people moved heavy loads by
placing rollers under them. (Modern people
use rollers, too. Any hardware store will
sell you a roller bearing for a lazy Susan.) A
stone block of mass 844 kg moves forward
at 0.335 m/s, supported by two uniform
cylindrical tree trunks each of mass 82.0 kg
and radius 0.343 m. No slipping occurs
between the block and the rollers or
between the rollers and the ground. Find
the total kinetic energy of the moving
objects.
68.
A skateboarder with his board can
be modeled as a particle of mass 76.0 kg,
located at his center of mass. As shown in
Figure P7.59 on page 219, the skateboarder
starts from rest in a crouching position at
one lip of a half-pipe (point ). The halfpipe forms one half of a cylinder of radius
6.80 m with its axis horizontal. On his
descent, the skateboarder moves without
friction and maintains his crouch so that his
center of mass moves through one quarter
of a circle of radius 6.30 m. (a) Find his
speed at the bottom of the half-pipe (point
). (b) Find his angular momentum about
the center of curvature. (c) Immediately
after passing point , he stands up and
raises his arms, lifting his center of gravity
from 0.500 m to 0.950 m above the concrete
(point ). Explain why his angular
momentum is constant in this maneuver,
whereas his linear momentum and his
mechanical energy are not constant. (d)
Find his speed immediately after he stands
up, when his center of mass is moving in a
quarter circle of radius 5.85 m. (e) What
work did the skateboarder’s legs do on his
body as he stood up? Next, the
skateboarder glides upward with his center
of mass moving in a quarter circle of radius
5.85 m. His body is horizontal when he
passes point , the far lip of the half-pipe.
(f) Find his speed at this location. At last he
goes ballistic, twisting around while his
center of mass moves vertically. (g) How
high above point
does he rise? (h) Over
what time interval is he airborne before he
touches down, facing downward and again
in a crouch, 2.34 m below the level of point
? (i) Compare the solution to this
problem with the solution to Problem 7.59.
Which is more accurate? Why? (Caution: Do
not try this maneuver yourself without the
required skill and protective equipment, or
in a drainage channel to which you do not
have legal access.)
69.
Two astronauts (Fig. P10.69), each
having a mass M, are connected by a rope
of length d having negligible mass. They are
isolated in space, orbiting their center of
mass at speeds v. Treating the astronauts as
particles, calculate (a) the magnitude of the
angular momentum of the system and (b)
the rotational energy of the system. By
pulling on the rope, one of the astronauts
shortens the distance between them to d/2.
(c) What is the new angular momentum of
the system? (d) What are the astronauts’
new speeds? (e) What is the new rotational
energy of the system? (f) How much work
does the astronaut do in shortening the
rope?
Figure P10.69
70.
When a person stands on tiptoe (a
strenuous position), the position of the foot
is as shown in Figure P10.70a. The total

gravitational force on the body Fg is

supported by the force n exerted by the
floor on the toes of one foot. A mechanical
model for the situation is shown in Figure

P10.70b, where T is the force exerted by

the Achilles tendon on the foot and R is the
force exerted by the tibia on the foot. Find
the values of T, R, and θ when Fg = 700 N.
large forces exerted on the muscles and
vertebrae. The spine pivots mainly at the
fifth lumbar vertebra, with the principal
supporting force provided by the erector
spinalis muscle in the back. To see the
magnitude of the forces involved and to
understand why back problems are
common among humans, consider the
model shown in Fig. P10.71b for a person
bending forward to lift a 200-N object. The
spine and upper body are represented as a
uniform horizontal rod of weight 350 N,
pivoted at the base of the spine. The erector
spinalis muscle, attached at a point two
thirds of the way up the spine, maintains
the position of the back. The angle between
the spine and this muscle is 12.0°. Find the
tension in the back muscle and the
compressional force in the spine.
(a)
(b)
Figure P10.70
71.
A person bending forward to lift a
load “with his back” (Fig. P10.71a) rather
than “with his knees” can be injured by
(a)
friction and the position of the resultant
normal force. (b) Taking F = 300 N, find the
value of h for which the cabinet just begins
to tip.
(b)
Figure P10.71
72.
A wad of sticky clay with mass m

and velocity v i is fired at a solid cylinder of
mass M and radius R (Fig. P10.72). The
cylinder is initially at rest and is mounted
on a fixed horizontal axle that runs through
its center of mass. The line of motion of the
projectile is perpendicular to the axle and at
a distance d < R from the center. (a) Find the
angular speed of the system just after the
clay strikes and sticks to the surface of the
cylinder. (b) Is mechanical energy of the
clay–cylinder system conserved in this
process? Explain your answer.
Figure P10.73
74.
The following equations are
obtained from a free-body diagram of a
rectangular farm gate, supported by two
hinges on the left-hand side. A bucket of
grain is hanging from the latch.
–A + C = 0
+B – 392 N – 50.0 N = 0
A(0) + B(0) + C(1.80 m) – 392 N(1.50 m) –
50.0 N(3.00 m) = 0
Figure P10.72
73.
A force acts on a rectangular cabinet
weighing 400 N as shown in Figure P10.73.
(a) Assuming that the cabinet slides with
constant speed when F = 200 N and h =
0.400 m, find the coefficient of kinetic
(a) Draw the free-body diagram and
complete the statement of the problem,
specifying the unknowns. (b) Determine the
values of the unknowns and state the
physical meaning of each.
75.
A stepladder of negligible weight is
constructed as shown in Figure P10.75. A
painter of mass 70.0 kg stands on the ladder
3.00 m from the bottom. Assuming that the
floor is frictionless, find (a) the tension in
the horizontal bar connecting the two
halves of the ladder, (b) the normal forces at
A and B, and (c) the components of the
reaction force at the single hinge C that the
left half of the ladder exerts on the right
half. (Suggestion: Treat the ladder as a single
object, but also treat each half of the ladder
separately.)
What are the force components on the
sphere at the point P if h = 3R?
Figure P10.76
77.
Figure P10.77 shows a vertical force
applied tangentially to a uniform cylinder
of weight Fg. The coefficient of static friction
between the cylinder and all surfaces is
0.500. In terms of Fg, find the maximum
force P that can be applied that does not
cause the cylinder to rotate. (Suggestion:
When the cylinder is on the verge of
slipping, both friction forces are at their
maximum values. Why?)
Figure P10.75
76.
A solid sphere of mass m and radius
r rolls without slipping along the track
shown in Figure P10.76. It starts from rest
with the lowest point of the sphere at
height h above the bottom of the loop of
radius R, much larger than r. (a) What is the
minimum value of h (in terms of R) such
that the sphere completes the loop? (b)
Figure P10.77
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