Download CP7e: Ch. 7 Problems

Chapter 7 Problems
1, 2, 3 = straightforward, intermediate,
challenging
= full solution available in Student
Solutions Manual/Study Guide
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= biomedical application
Section 7.1 Angular Speed and Angular
Acceleration
1.
The tires on a new compact car have
a diameter of 2.0 ft and are warranted for 60
000 miles. (a) Determine the angle (in
radians) through which one of these tires
will rotate during the warranty period. (b)
How many revolutions of the tire are
equivalent to your answer in (a)?
2.
A wheel has a radius of 4.1 m. How
far (path length) does a point on the
circumference travel if the wheel is rotated
through angles of 30°, 30 rad, and 30 rev,
respectively?
3.
Find the angular speed of Earth
about the Sun in radians per second and
degrees per day.
4.
A potter’s wheel moves from rest to
an angular speed of 0.20 rev/s in 30 s. Find
its angular acceleration in radians per
second per second.
Section 7.2 Rotational Motion under
Constant Angular Acceleration
Section 7.3 Relations between Angular
and Linear Quantities
5.
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.
6.
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.
7.
A machine part rotates at an angular
speed of 0.60 rad/s; its speed is then
increased to 2.2 rad/s at an angular
acceleration of 0.70 rad/s2. Find the angle
through which the part rotates before
reaching this final speed.
8.
A tire placed on a balancing machine
in a service station starts from rest and
turns through 4.7 revolutions in 1.2 s before
reaching its final angular speed. Calculate
its angular acceleration.
9.
The diameters of the
main rotor and tail rotor of a single-engine
helicopter are 7.60 m and 1.02 m,
respectively. The 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.
10.
The tub of a washer goes into its
spin-dry cycle, starting from rest and
reaching an angular speed of 5.0 rev/s in 8.0
s. At this point, the person doing the
laundry opens the lid, and a safety switch
turns off the washer. The tub slows to rest
in 12.0 s. Through how many revolutions
does the tub turn during the entire 20-s
interval? Assume constant angular
acceleration while it is starting and
stopping.
11.
A standard cassette tape is placed in
a standard cassette player. Each side lasts
for 30 minutes. The two tape wheels of the
cassette fit onto two spindles in the player.
Suppose that a motor drives one spindle at
constant angular velocity of approximately
1 rad/s and the other spindle is free to
rotate at any angular speed. Find the order
of magnitude of the tape’s thickness.
Specify any other quantities you estimate
and the values you take for them.
12.
A coin with a diameter of 2.40 cm is
dropped on edge onto a horizontal surface.
The coin starts out with an initial angular
speed of 18.0 rad/s and rolls in a straight
line without slipping. If the rotation slows
with an angular acceleration of magnitude
1.90 rad/s2, how far does the coin roll before
coming to rest?
13.
A rotating wheel requires 3.00 s to
rotate 37.0 revolutions. Its angular velocity
at the end of the 3.00-s interval is 98.0 rad/s.
What is the constant angular acceleration of
the wheel?
Section 7.4
Centripetal Acceleration
14.
It has been suggested that rotating
cylinders about 10 mi long and 5.0 mi in
diameter be placed in space and used as
colonies. What angular speed must such a
cylinder have so that the centripetal
acceleration at its surface equals the free-fall
acceleration on Earth?
15.
Find the centripetal accelerations of
(a) a point on the equator of Earth and (b)
the North Pole, due to the rotation of Earth
about its axis.
16.
A tire 2.00 ft in diameter is placed on
a balancing machine, where it is spun so
that its tread is moving at a constant speed
of 60.0 mi/h. A small stone is stuck in the
tread of the tire. What is the acceleration of
the stone as the tire is being balanced?
17.
(a) What is the tangential
acceleration of a bug on the rim of a 10-in.diameter disk if the disk moves from rest to
an angular speed of 78 rev/min in 3.0 s? (b)
When the disk is at its final speed, what is
the tangential velocity of the bug? (c) One
second after the bug starts from rest, what
are its tangential acceleration, centripetal
acceleration, and total acceleration?
18.
A race car starts from rest on a
circular track of radius 400 m. The car’s
speed increases at the constant rate of 0.500
m/s2. At the point where the magnitudes of
the centripetal and tangential accelerations
are equal, determine (a) the speed of the
race car, (b) the distance traveled, and (c)
the elapsed time.
19.
A 55.0-kg ice-skater is moving at 4.00
m/s when she grabs the loose end of a rope,
the opposite end of which is tied to a pole.
She then moves in a circle of radius 0.800 m
around the pole. (a) Determine the force
exerted by the horizontal rope on her arms.
(b) Compare this force with her weight.
20.
A sample of blood is placed in a
centrifuge of radius 15.0 cm. The mass of a
red blood cell is 3.0 × 10–16 kg, and the
magnitude of the force acting on it as it
settles out of the plasma is 4.0 × 10–11 N. At
how many revolutions per second should
the centrifuge be operated?
21.
A certain light truck can go around a
flat curve having a radius of 150 m with a
maximum speed of 32.0 m/s. With what
maximum speed can it go around a curve
having a radius of 75.0 m?
22.
The cornering performance of an
automobile is evaluated on a skid pad,
where the maximum speed that a car can
maintain around a circular path on a dry,
flat surface is measured. Then the
centripetal acceleration, also called the
lateral acceleration, is calculated as a
multiple of the free-fall acceleration g. The
main factors affecting the performance of
the car are its tire characteristics and
suspension system. A Dodge Viper GTS can
negotiate a skid pad of radius 61.0 m at 86.5
km/h. Calculate its maximum lateral
acceleration.
23.
A 50.0-kg child
stands at the rim of a merry-go-round of
radius 2.00 m, rotating with an angular
speed of 3.00 rad/s. (a) What is the child’s
centripetal acceleration? (b) What is the
minimum force between her feet and the
floor of the carousel that is required to keep
her in the circular path? (c) What minimum
coefficient of static friction is required? Is
the answer you found reasonable? In other
words, is she likely to stay on the merry-goround?
24.
An engineer wishes to design a
curved exit ramp for a toll road in such a
way that a car will not have to rely on
friction to round the curve without
skidding. He does so by banking the road
in such a way that the force causing the
centripetal acceleration will be supplied by
the component of the normal force toward
the center of the circular path. (a) Show
that, for a given speed v and a radius r, the
curve must be banked at the angle θ such
that tan θ = v2/rg. (b) Find the angle at
which the curve should be banked if a
typical car rounds it at a 50.0-m radius and
a speed of 13.4 m/s.
25.
An air puck of mass 0.25 kg is tied to
a string and allowed to revolve in a circle of
radius 1.0 m on a frictionless horizontal
table. The other end of the string passes
through a hole in the center of the table,
and a mass of 1.0 kg is tied to it (Fig. P7.25).
The suspended mass remains in
equilibrium while the puck on the tabletop
revolves. (a) What is the tension in the
string? (b) What is the horizontal force
acting on the puck? (c) What is the speed of
the puck?
20.0 m/s at point , what is the force of the
track on the vehicle at this point? (b) What
is the maximum speed the vehicle can have
at point
in order for gravity to hold it on
the track?
Figure P7.25
Figure P7.28
26.
Tarzan (m = 85 kg) tries to cross a
river by swinging from a 10-m-long vine.
His speed at the bottom of the swing (as he
just clears the water) is 8.0 m/s. Tarzan
doesn’t know that the vine has a breaking
strength of 1 000 N. Does he make it safely
across the river? Justify your answer.
27.
A 40.0-kg child takes a ride on a
Ferris wheel that rotates four times each
minute and has a diameter of 18.0 m. (a)
What is the centripetal acceleration of the
child? (b) What force (magnitude and
direction) does the seat exert on the child at
the lowest point of the ride? (c) What force
does the seat exert on the child at the
highest point of the ride? (d) What force
does the seat exert on the child when the
child is halfway between the top and
bottom?
28.
A roller-coaster vehicle has a mass of
500 kg when fully loaded with passengers
(Fig. P7.28). (a) If the vehicle has a speed of
Section 7.5
Newtonian Gravitation
29.
The average distance separating
Earth and the Moon is 384 000 km. Use the
data in Table 7.3 to find the net
gravitational force exerted by Earth and the
Moon on a 3.00 × 104-kg spaceship located
halfway between them.
30.
During a solar eclipse, the Moon,
Earth, and Sun all lie on the same line, with
the Moon between Earth and the Sun. (a)
What force is exerted by the Sun on the
Moon? (b) What force is exerted by Earth
on the Moon? (c) What force is exerted by
the Sun on Earth? (See Table 7.3 and
Problem 29.)
31.
A coordinate system (in meters) is
constructed on the surface of a pool table,
and three objects are placed on the table as
follows: a 2.0-kg object at the origin of the
coordinate system, a 3.0-kg object at (0, 2.0),
and a 4.0-kg object at (4.0, 0). Find the
resultant gravitational force exerted by the
other two objects on the object at the origin.
32.
Use the data of Table 7.3 to find the
point between Earth and the Sun at which
an object can be placed so that the net
gravitational force exerted by Earth and the
Sun on that object is zero.
33.
Objects with masses of 200 kg and
500 kg are separated by 0.400 m. (a) Find
the net gravitational force exerted by these
objects on a 50.0-kg object placed midway
between them. (b) At what position (other
than infinitely remote ones) can the 50.0-kg
object be placed so as to experience a net
force of zero?
34.
Two objects attract each other with a
gravitational force of magnitude 1.00 × 10–8
N when separated by 20.0 cm. If the total
mass of the objects is 5.00 kg, what is the
mass of each?
Section 7.6
Kepler’s Laws
35.
A satellite moves in
a circular orbit around Earth at a speed of 5
000 m/s. Determine (a) the satellite’s
altitude above the surface of Earth and (b)
the period of the satellite’s orbit.
36.
A 600-kg satellite is in a circular orbit
about Earth at a height above Earth equal to
Earth’s mean radius. Find (a) the satellite’s
orbital speed, (b) the period of its
revolution, and (c) the gravitational force
acting on it.
37.
Io, a satellite of Jupiter, has an orbital
period of 1.77 days and an orbital radius of
4.22 × 105 km. From these data, determine
the mass of Jupiter.
38.
A satellite has a mass of 100 kg and
is located at 2.00 × 106 m above the surface
of Earth. (a) What is the potential energy
associated with the satellite at this location?
(b) What is the magnitude of the
gravitational force on the satellite?
39.
A satellite of mass 200 kg is launched
from a site on Earth’s equator into an orbit
200 km above the surface of Earth. (a)
Assuming a circular orbit, what is the
orbital period of this satellite? (b) What is
the satellite’s speed in it’s orbit? (c) What is
the minimum energy necessary to place the
satellite in orbit, assuming no air friction?
Additional Problems
40.
Neutron stars are extremely dense
objects that are formed from the remnants
of supernova explosions. Many rotate very
rapidly. Suppose that the mass of a certain
spherical neutron star is twice the mass of
the sun and its radius is 10.0 km. Determine
the greatest possible angular speed the
neutron star can have so that the matter at
its surface on the equator is just held in
orbit by the gravitational force.
41.
One method of pitching a softball is
called the “windmill” delivery method, in
which the pitcher’s arm rotates through
approximately 360° in a vertical plane
before the 198-gram ball is released at the
lowest point of the circular motion. An
experienced pitcher can throw a ball with a
speed of 98.0 mi/h. Assume that the angular
acceleration is uniform throughout the
pitching motion, and take the distance
between the softball and the shoulder joint
to be 74.2 cm. (a) Determine the angular
speed of the arm in rev/s at the instant of
release. (b) Find the value of the angular
acceleration in rev/s2 and the radial and
tangential acceleration of the ball just before
it is released. (c) Determine the force
exerted on the ball by the pitcher’s hand
(both radial and tangential components)
just before it is released.
42.
The Mars probe Pathfinder is
designed to drop the vehicle’s instrument
package from a height of 20 meters above
the Martian surface, after the speed of the
probe has been brought to zero by a
combination parachute–rocket system at
that height. To cushion the landing, giant
air bags surround the package. The mass of
Mars is 0.107 4 times that of Earth, and the
radius of Mars is 0.528 2 that of Earth. Find
(a) the acceleration due to gravity at the
surface of Mars and (b) how long it takes
for the instrument package to fall the last 20
meters.
43.
An athlete swings a 5.00-kg ball
horizontally on the end of a rope. The ball
moves in a circle of radius 0.800 m at an
angular speed of 0.500 rev/s. What are (a)
the tangential speed of the ball and (b) its
centripetal acceleration? (c) If the maximum
tension the rope can withstand before
breaking is 100 N, what is the maximum
tangential speed the ball can have?
44.
A digital audio compact disc carries
data along a continuous spiral track from
the inner circumference of the disc to the
outside edge. Each bit occupies 0.6 μm of
the track. 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.
45.
The Solar Maximum Mission
Satellite was placed in a circular orbit about
150 mi above Earth. Determine (a) the
orbital speed of the satellite and (b) the time
required for one complete revolution.
46.
A car rounds a banked curve where
the radius of curvature of the road is R, the
banking angle is θ, and the coefficient of
static friction is μ. (a) Determine the range
of speeds the car can have without slipping
up or down the road. (b) What is the range
of speeds possible if R = 100 m, θ = 10°, and
μ = 0.10 (slippery conditions)?
47.
A car moves at
speed v across a bridge made in the shape
of a circular arc of radius r. (a) Find an
expression for the normal force acting on
the car when it is at the top of the arc. (b) At
what minimum speed will the normal force
become zero (causing the occupants of the
car to seem weightless) if r = 30.0 m?
48.
A 0.400-kg pendulum bob passes
through the lowest part of its path at a
speed of 3.00 m/s. (a) What is the tension in
the pendulum cable at this point if the
pendulum is 80.0 cm long? (b) When the
pendulum reaches its highest point, what
angle does the cable make with the vertical?
(c) What is the tension in the pendulum
cable when the pendulum reaches its
highest point?
rotation at an angular speed of 5.00 rad/s, as
in Figure P7.51. The floor then drops away,
leaving the riders suspended against the
wall in a vertical position. What minimum
coefficient of friction between a rider’s
clothing and the wall is needed to keep the
rider from slipping? (Hint: Recall that the
magnitude of the maximum force of static
friction is equal to μn, where n is the
normal force—in this case, the force causing
the centripetal acceleration.)
49.
Because of Earth’s rotation about its
axis, a point on the equator has a centripetal
acceleration of 0.034 0 m/s2, while a point at
the poles has no centripetal acceleration. (a)
Show that, at the equator, the gravitational
force on an object (the object’s true weight)
must exceed the object’s apparent weight.
(b) What are the apparent weights of a 75.0kg person at the equator and at the poles?
(Assume Earth is a uniform sphere, and
take g = 9.800 m/s2.)
50.
A stuntman whose mass is 70 kg
swings from the end of a 4.0-m-long rope
along the arc of a vertical circle. Assuming
that he starts from rest when the rope is
horizontal, find the tensions in the rope that
are required to make him follow his circular
path (a) at the beginning of his motion, (b)
at a height of 1.5 m above the bottom of the
circular arc, and (c) at the bottom of the arc.
51.
In a popular amusement park ride, a
rotating cylinder of radius 3.00 m is set in
Figure P7.51
52.
A 0.50-kg ball that is tied to the end
of a 1.5-m light cord is revolved in a
horizontal plane, with the cord making a
30° angle with the vertical. (See Fig. P7.52.)
(a) Determine the ball’s speed. (b) If,
instead, the ball is revolved so that its speed
is 4.0 m/s, what angle does the cord make
with the vertical? (c) If the cord can
withstand a maximum tension of 9.8 N,
what is the highest speed at which the ball
can move?
Figure P7.52
53.
A skier starts at rest at the top of a
large hemispherical hill (Fig. P7.53).
Neglecting friction, show that the skier will
leave the hill and become airborne at a
distance h = R/3 below the top of the hill.
(Hint: At this point, the normal force goes
to zero.)
Figure P7.53
54.
After consuming all of its nuclear
fuel, a massive star can collapse to form a
black hole, which is an immensely dense
object whose escape speed is greater than
the speed of light. Newton’s law of
universal gravitation still describes the
force that a black hole exerts on objects
outside it. A spacecraft in the shape of a
long cylinder has a length of 100 m, and its
mass with occupants is 1 000 kg. It has
strayed too close to a 1.0-m-radius black
hole having a mass 100 times that of the
Sun (Figure P7.54). If the nose of the
spacecraft points toward the center of the
black hole, and if the distance between the
nose of the spacecraft and the black hole’s
center is 10 km, (a) determine the total force
on the spacecraft. (b) What is the difference
in the force per kilogram of mass felt by the
occupants in the nose of the ship versus
those in the rear of the ship farthest from
the black hole?
Figure P7.54
55.
In Robert Heinlein’s The Moon Is a
Harsh Mistress, the colonial inhabitants of
the Moon threaten to launch rocks down
onto Earth if they are not given
independence (or at least representation).
Assuming that a gun could launch a rock of
mass m at twice the lunar escape speed,
calculate the speed of the rock as it enters
Earth’s atmosphere.
56.
Show that the escape speed from the
surface of a planet of uniform density is
directly proportional to the radius of the
planet.
57.
A massless spring of constant k =
78.4 N/m is fixed on the left side of a level
track. A block of mass m = 0.50 kg is
pressed against the spring and compresses
it a distance d, as in Figure P7.57. The block
(initially at rest) is then released and travels
toward a circular loop-the-loop of radius R
= 1.5 m. The entire track and the loop-theloop are frictionless, except for the section
of track between points A and B. Given that
the coefficient of kinetic friction between
the block and the track along AB is μk = 0.30,
and that the length of AB is 2.5 m,
determine the minimum compression d of
the spring that enables the block to just
make it through the loop-the-loop at point
C. (Hint: The force exerted by the track on
the block will be zero if the block barely
makes it through the loop-the-loop.)
of the bottom track if the block just makes it
to point
on the first trip. (Hint: If the
block just makes it to point , the force of
contact exerted by the track on the block at
that point is zero.)
Figure P7.58
59.
A frictionless roller coaster is given
an initial speed v0 at height h, as in Figure
P7.59. The radius of curvature of the track
at point
is R. (a) Find the maximum
value of v0 so that the roller coaster stays on
the track at
solely because of gravity. (b)
Using the value of v0 calculated in (a),
determine the value of h’ that is necessary if
the roller coaster just makes it to point .
Figure P7.57
58.
A small block of mass m = 0.50 kg is
fired with an initial speed of v0 = 4.0 m/s
along a horizontal section of frictionless
track, as shown in the top portion of Figure
P7.58. The block then moves along the
frictionless, semicircular, vertical tracks of
radius R = 1.5 m. (a) Determine the force
exerted by the track on the block at points
and . (b) The bottom of the track
consists of a section (L = 0.40 m) with
friction. Determine the coefficient of kinetic
friction between the block and that portion
Figure P7.59
60.
A roller coaster travels in a circular
path. (a) Identify the forces on a passenger
at the top of the circular loop that cause
centripetal acceleration. Show the direction
of all forces in a sketch. (b) Identify the
forces on the passenger at the bottom of the
loop that produce centripetal acceleration.
Show these in a sketch. (c) Based on your
answers to (a) and (b), at what point, top or
bottom, should the seat exert the greatest
force on the passenger? (d) Assume the
speed of the roller coaster is 4.00 m/s at the
top of the loop of radius 8.00 m. Find the
force exerted by the seat on a 70.0-kg
passenger at the top of the loop. Then,
assume the speed remains the same at the
bottom of the loop, and find the force
exerted by the seat on the passenger at this
point. Are your answers consistent with
your choice of answers for (a) and (b)?
61.
Assume that you are agile enough to
run across a horizontal surface at 8.50 m/s,
independently of the value of the
gravitational field. What would be (a) the
radius and (b) the mass of an airless
spherical asteroid of uniform density 1.10 ×
103 kg/m3 on which you could launch
yourself into orbit by running? (c) What
would be your period?
62.
Figure P7.62 shows the elliptical
orbit of a spacecraft around Earth. Take the
origin of your coordinate system to be at
the center of Earth.
Figure P7.62
(a) On a copy of the figure (enlarged if
necessary), draw vectors representing
(i) the position of the spacecraft when it is
at
and ;
(ii) the velocity of the spacecraft when it is
at
and ;
(iii) the acceleration of the spacecraft when
it is at
and .
Make sure that each type of vector can be
distinguished. Provide a legend that shows
how each type is represented.
(b) Have you drawn the velocity vector at
longer than, shorter than, or the same
length as the one at ? Explain. Have you
drawn the acceleration vector at
longer
than, shorter than, or the same length as the
one at ? Explain. (Problem 62 is courtesy
of E. F. Redish. For more problems of this
type, visit www.physics.umd.edu/perg/)
63.
In a home laundry drier, a
cylindrical tub containing wet clothes
rotates steadily about a horizontal axis, as
in Figure P7.63. The clothes are made to
tumble so that they will dry uniformly. The
rate of rotation of the smooth-walled tub is
chosen so that a small piece of cloth loses
contact with the tub when the cloth is at an
angle of 68.0º above the horizontal. If the
radius of the tub is 0.330 m, what rate of
revolution is needed?
Figure P7.64
Figure P7.63
64.
Casting of molten metal is important
in many industrial processes. Centrifugal
casting is used for manufacturing pipes,
bearings, and many other structures. A
cylindrical enclosure is rotated rapidly and
steadily about a horizontal axis, as in Figure
P7.64. Molten metal is poured into the
rotating cylinder and then cooled, forming
the finished product. Turning the cylinder
at a high rotation rate forces the solidifying
metal strongly to the outside. Any bubbles
are displaced toward the axis so that
unwanted voids will not be present in the
casting.
Suppose that a copper sleeve of
inner radius 2.10 cm and outer radius 2.20
cm is to be cast. To eliminate bubbles and
give high structural integrity, the
centripetal acceleration of each bit of metal
should be 100g. What rate of rotation is
required? State the answer in revolutions
per minute.
65.
Suppose that a 1 800-kg car passes
over a bump in a roadway that follows the
arc of a circle of radius 20.4 m, as in Figure
P7.65. (a) What force does the road exert on
the car as the car passes the highest point of
the bump if the car travels at 8.94 m/s? (b)
What is the maximum speed the car can
have without losing contact with the road
as it passes this highest point?
Figure P7.65
66.
One popular design of a household
juice machine is a conical, perforated
stainless-steel basket 3.30 cm high, with a
closed bottom of diameter 8.00 cm and an
open top of diameter 13.70 cm, that spins at
20 000 revolutions per minute about a
vertical axis (Fig. P7.66). Solid pieces of
fruit or vegetables are chopped into
granules by cutters on the bottom of the
spinning cone. The dry pulp is ejected from
the top of the cone. After passing through
the perforations on the surface of the cone,
the juice is collected in an enclosure
immediately surrounding the cone. (a)
What centripetal acceleration does a bit of
fruit experience as it spins with the basket
at a point midway between the top and
bottom of the basket? Express the answer as
a multiple of g. (b) Observe that the weight
of the fruit is a negligible force. What is the
normal force on 2.00 g of fruit at the
midway point? (c) If the coefficient of
kinetic friction between the fruit and the
cone is 0.600, with what acceleration
relative to the cone will the bit of fruit start
to slide up the wall of the cone at that point,
after being temporarily stuck?
Figure P7.66
67.
(a) A luggage carousel at an airport
has the form of a section of a large cone,
steadily rotating about its vertical axis. Its
metallic surface slopes downward toward
the outside, making an angle of 20.0° with
the horizontal. A 30.0-kg piece of luggage is
placed on the carousel, 7.46 m from the axis
of rotation. The travel bag goes around
once in 38.0 s. Calculate the force of static
friction between the bag and the carousel.
(b) The drive motor is shifted to turn the
carousel at a higher constant rate of
rotation, and the piece of luggage is
bumped to a position 7.94 m from the axis
of rotation. The bag is on the verge of
slipping as it goes around once every 34.0 s.
Calculate the coefficient of static friction
between the bag and the carousel.
68.
A merry-go-round is stationary. A
dog is running on the ground just outside
its circumference, moving with a constant
angular speed of 0.750 rad/s. The dog does
not change his pace when he sees a bone
resting on the edge of the merry-go-round
one-third of a revolution in front of him. At
the instant the dog sees the bone, the
merry-go-round begins to move in the
direction the dog is running, with a
constant angular acceleration of 0.015 0
rad/s2. (a) After what time will the dog
reach the bone? (b) The confused dog keeps
running and passes the bone. How long
after the merry-go-round starts to turn do
the dog and the bone draw even with each
other for the second time?
69.
Figure P7.69 shows the drive train of
a bicycle that has wheels 67.3 cm in
diameter and pedal cranks 17.5 cm long.
The cyclist pedals at a steady angular rate
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?
banking angle that allows the bat to stay in
a horizontal plane? (c) What is the radius of
the circle of its flight when the bat flies at its
maximum speed? (d) Can the bat turn with
a smaller radius by flying more slowly?
Figure P7.69
70.
The maximum lift force on a bat is
proportional to the square of its flying
speed v. For the hoary bat (Lasiurus
cinereus), the magnitude of the lift force is
given by
FL ≤ (0.018 N ∙ s2/m2)v2
The bat can fly in a horizontal circle by
“banking” its wings at an angle θ, as shown
in Figure P7.70. In this situation, the
magnitude of the vertical component of the
lift force must equal the bat’s weight. The
horizontal component of the force provides
the centripetal acceleration. (a) What is the
minimum speed that the bat can have if its
mass is 0.031 kg? (b) If the maximum speed
of the bat is 10 m/s, what is the maximum
© Copyright 2004 Thomson. All rights reserved.
Figure P7.70
71.
(a) Find the acceleration due to
gravity at the surface of a neutron star of
mass 1.5 solar masses and having a radius
of 10.0 km. (b) Find the weight of a 0.120-kg
baseball on this star. (c) Assume the
equation PE = mgh applies, and calculate
the energy that a 70.0-kg person would
expend climbing a 1.00-cm-tall mountain
on the neutron star.