Download A grindstone with a radius of 0.610 m is being used to sharpen an ax

1
A grindstone with a radius of 0.610 m is being used to sharpen
an ax. What is the kinetic energy of the grindstone in if it
completes one revolution every 4.20 s?
4
A lawn mower has a flat, rod-shaped steel blade that rotates
about its center. The mass of the blade is 0.58 kg and its length
is 0.56 m.
(a) What is the rotational energy of the blade at its operating
angular speed of 3500 rpm?
(b) If all of the rotational kinetic energy of the blade could be
converted to gravitational potential energy, to what height
would the blade rise?
Answers
(a) 1.1 kJ
(b) 180 m
5
Answers
4.81 J
A diver tucks her body in midflight, decreasing her moment of
inertia by a factor of two.
(a) Does the diver's kinetic energy increase, decrease, or stay
the same?
(b) Calculate the ratio of the final kinetic energy to the initial
kinetic energy for the diver.
Answers
2
An electric fan spinning with an angular speed of 12 rad/s has a
kinetic energy of 4.1 J. What is the moment of inertia of the
fan?
Answers
(a) increases
(b) 2
6
0.057 kg.m2
3
When a pitcher throws a curve ball, the ball is given a fairly
rapid spin. If a 15-kg baseball with a radius of 3.7 cm is thrown
with a linear speed of 46 m/s and an angular speed of 41 rad/s,
how much of its kinetic energy is translational and how much
is rotational? Assume the ball is a uniform, solid sphere.
Calculate the kinetic energy of rotation of the earth about its
axis, and compare it with the kinetic energy of the orbital
motion of the earth's center of mass about the sun. Assume the
earth to be a homogeneous sphere of mass 6.0 x 1024 kg and
radius 6.4 x 106 m. The radius of the earth's orbit is 1.5x 1011
m.
Answer
1.03 x 104
7
The figure shows a uniform disk, with mass M = 2.5 kg and
radius R =- 20 cm, mounted on a fixed horizontal axle. A block
with mass m = 1.2 kg hangs from a massless cord that is
wrapped around the rim of the disk. What is its rotational
kinetic energy K at t = 2.5 s?
Answers
Kt = 160 J
Kr = 0.069 J
Answer
90 J
8
Answer the following questions.
(a) If R = 12 cm, M = 400 g, and m = 50 g in the figure, find
the speed of the block after it has descended 50 cm starting
from rest. Solve the problem using energy conservation
principles.
(b) Repeat (a) with R = 5.0 cm.
10
Answer
Answer
(a) 1.4 m/s
(b) 1.4 m/s
9
Answer the following questions
(a) A uniform solid disk of radius R and mass M is free to
rotate on a frictionless pivot through a point on its rim (see
the figure). If the disk is released from rest in the position
shown by the blue circle, what is the speed of its center of
mass when the disk reaches the position indicated by the
dashed circle?
(b) What is the speed of the lowest point on the disk in the
dashed position?
(c) What If? Repeat part (a) using a uniform hoop.
A uniform, hollow, cylindrical spool has inside radius R/2,
outside radius R, and mass M (see figure). 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 given by the equation below.
11
A bicycle is turned upside down while its owner repairs a flat
tire. A friend spins the other wheel, of radius 0.381 m, and
observes that drops of water fly off tangentially. She measures
the height reached by drops moving vertically. A drop that
breaks loose from the tire on one turn rises h = 54.0 cm above
the tangent point. A drop that breaks loose on the next turn
rises 51.0 cm above the tangent point. The height to which the
drops rise decreases because the angular speed of the wheel
decreases. From this information, determine the magnitude of
the average angular acceleration of the wheel.
Answer
(a) 2[(Rg)/3]1/2
(b) 4[((Rg)/3]1/2
(c) (Rg)1/2
Answer
-3.22 rad/s2
12
A uniform cylinder of radius 10 cm and mass 20 kg is mounted
so as to rotate freely about a horizontal axis that is parallel to
and 5.0 cm from the central longitudinal axis of the cylinder.
(a) What is the rotational inertia of the cylinder about the axis
of rotation?
(b) If the cylinder is released from rest with its central
longitudinal axis at the same height as the axis about which the
cylinder rotates, what is the angular speed of the cylinder as it
passes through its lowest position?
16
A block of mass m is attached to a string that is wrapped
around the circumference of a wheel of radius R and moment
of inertia I. The wheel rotates freely about its axis and the
string wraps around its circumference without slipping.
Initially the wheel rotates with an angular speed ω, causing the
block to rise with a linear speed v. Suppose the block has a
mass of 2.1 kg and an initial upward speed of 0.33 m/s. Find
the moment of inertia of the wheel if its radius is 8.0 cm and
the block rises to a height of 7.4 cm before momentarily
coming to rest.
Answer
11 rad/s
Answers
13
-26
An oxygen molecule, O2, has a total mass of 5.30 x 10 kg and
a rotational inertia of 1.94 x 10-46 kg.m2 about an axis through
the center perpendicular to the line joining the atoms. Suppose
that such a molecule in a gas has a speed of 500 m/s and that its
rotational kinetic energy is two-thirds of its translational
kinetic energy. Find its angular velocity.
0.17 kg.m2
17
Answer
6.75 x 1012 rad/sec
14
Two spheres have identical radii and masses. How might you
tell which of these spheres is hollow and which is solid?
A 1.3-kg block is tied to a string that is wrapped around the rim
of a pulley of radius 7.2 cm. The block is released from rest.
(a) Assuming the pulley is a uniform disk with a mass of
0.31 kg, find the speed of the block after it has fallen through a
height of 0.50 m.
(b) If a small lead weight is attached near the rim of the pulley
and this experiment is repeated, will the speed of the block
increase, decrease, or stay the same? Explain.
Answers
(a) 3.0 m/s
(b) decrease
Answers
Spin the two spheres with equal angular speeds. The one
with the larger moment of inertia – the hollow sphere – has
the greater kinetic energy, and hence will spin for a longer
time before stopping.
15
A 13-g CD with a radius of 6.0 cm rotates with an angular
speed of 32 rad/s.
(a) What is its kinetic energy?
(b) What angular speed must the CD have if its kinetic energy
is to be doubled?
Answers
(a) 1.2 x 10-2 J
(b) 45 rad/s
18
A 2.85 kg bucket is attached to a disk-shaped pulley of radius
0.121 m and mass 0.742 kg. The bucket is allowed to fall.
(a) what is its linear acceleration?
(b) What is the angular acceleration of the pulley?
(c) How far does the bucket drop in 1.50 s?
(d) What is the kinetic energy of the disk at 1.50 s?
(e) What is the kinetic energy of the bucket at 1.50?
Answers
(a) 8.68 m/s2
(b) 71.7 rad/s2
(c) 9.77 m
(d)
(e)
19
A uniform disk of radius 0.12 m and mass 5 kg is pivoted so
that it rotates freely about its central axis (see figure). A string
wrapped around the disk is pulled with a force of 20 N.
(a) What is the torque exerted on the disk?
(b) What is the angular acceleration of the disk?
(c) If the disk starts from rest, what is its angular velocity after
5 s?
(d) What is its kinetic energy after 5 s?
(e) What is the total angle θ that the disk turns through in 5 s?
(f) Show that the work done by the torque, τ∆θ, equals the
kinetic energy.
22
The two masses (m1 = 5.0 kg and m2 = 3.0 kg) in the Atwood's
machine shown in the figure are released from rest, with m1 at
a height of 0.75 m above the floor. When m1 hits the ground its
speed is 1.8 m/s. Assuming that the pulley is a uniform disk
with a radius of 12 cm,
(a) outline a strategy that allows you to find the mass of the
pulley.
(b) Implement the strategy given in part (a) and determine the
pulley’s mass.
Answers
(a) Equate the initial and final energies, then solve for the
mass of the pulley
(b) 1200 kg
Answers
(a) 2.4 n.m
(b) 66.7 rad/s2
(c) 333 rad/s
(d) 2000 j
(e) 834 rad
(f) τΔθ
20
A thin ring of mass 2.7 kg and radius 8 cm rotates about an
axis through its center and perpendicular to the plane of the
ring at 1.5 rev/so Calculate the kinetic energy of the ring.
Answers
0.763 J
21
A 15.0 kg object and a 10.0 kg object are suspended, joined by
a cord that passes over a pulley with a radius of 10.0 cm and a
mass of 3.00 kg (see the figure). The cord has a negligible
mass and does not slip on the pulley. The pulley rotates on its
axis without friction. The objects start from rest 3.00 m apart.
Treat the pulley as a uniform disk, and determine the speeds of
the two objects as they pass each other.
Answer
2.36 m/s
23
You pull downward with a force of 25 N on a rope that passes
over a disk-shaped pulley of mass 1.3 kg and radius 0.075 m.
The other end of the rope is attached to a 0.67 kg mass. Find
the linear acceleration of the 0.67 kg mass.
Answers
14 m/s2
24
The figure shows a ball with mass m = 0.341 kg attached to the
end of a thin rod with length L = 0.452 m and negligible mass.
The other end of the rod is pivoted so that the ball can move in
a vertical circle. The rod is held horizontally, as shown and
then given enough of a downward push to cause the ball to
swing down and around and just reach the vertically up
position, with zero speed there.
(a) How much work is done on the ball by the gravitational
force from the initial point to the lowest point?
(b) How much work is done on the ball by the gravitational
force from the initial point to the highest point?
(c) How much work is done on the ball by the gravitational
force from the initial point to the point on the right level with
the initial point?
(d) If the gravitational potential energy of the ball-Earth system
is taken to be zero at the initial point, what is it when the ball
reaches the lowest point?
(e) If the gravitational potential energy of the ball-Earth system
is taken to be zero at the initial point, what is it when the ball
reaches the highest point?
(f) If the gravitational potential energy of the ball-Earth system
is taken to be zero at the initial point, what is it when the ball
reaches the point on the right level with the initial point?
(g) Suppose the rod were pushed harder so that the ball passed
through the highest point with a nonzero speed. Would Ug
from the lowest point to the highest point then be greater than,
less than, or the same as it was when the ball stopped at the
highest point?
25
Answer
(a) 2.98 m/s
(b) 4.21m/s
(c) 2.98 m/s
(d) all the same
26
Answer
(a) 1.51 J
(b) -1.51 J
(c) 0
(d) -1.51 J
(e) 1.51 J
(f) 0
(h) same
The figure shows a ball with mass m = 0.341 kg attached to the
end of a thin rod with length L = 0.452 m and negligible mass.
The other end of the rod is pivoted so that the ball can move in
a vertical circle. The rod is held horizontally, as shown and
then given enough of a downward push to cause the ball to
swing down and around and just reach the vertically up
position, with zero speed there.
(a) What initial speed must be given the ball so that it reaches
the vertically upward position with zero speed?
(b) What then is its speed at the lowest point?
(c) What then is its speed at the point on the right at which the
ball is level with the initial point?
(d) If the ball's mass were doubled, would the answers to (a)
through (c) increase, decrease, or remain the same?
The figure shows a thin rod, of length L = 2.00 m and
negligible mass, that can pivot about one end to rotate in a
vertical circle. A ball of mass m = 5.00 kg is attached to the
other end. The rod is pulled aside to angle θ0 = 30.00 and
released with initial velocity v0 = 0.
(a) What is the speed of the ball at the lowest point?
(b) Does the speed increase, decrease, or remain the same if the
mass is increased?
Answer
(a) 2.29 m/s
(b) same
27
A rigid sculpture consists of a thin hoop (of mass m and radius
R = 0.15 m) and a thin radial rod (of mass m and length L =
2.0R), arranged as shown in the figure. The sculpture can pivot
around a horizontal axis in the plane of the hoop, passing
through its center.
(a) In terms of m and R, what is the sculpture's rotational
inertia I about the rotation axis?
(b) Starting from rest, the sculpture rotates around the rotation
axis from the initial upright orientation of of the figure. What is
its angular speed to about the axis when it is inverted?
30
The figure shows a rigid assembly of a thin hoop (of mass m
and radius R = 0.150 m) and a thin radial rod (of mass m and
length L = 2.00R). The assembly is upright, but if we give it a
slight nudge, it will rotate around a horizontal axis in the plane
of the rod and hoop, through the lower end of the rod.
Assuming that the energy given to the assembly in such a
nudge is negligible, what would be the assembly's angular
speed about the rotation axis when it passes through the
upside-down (inverted) orientation?
Answer
9.82 rad/s
Answer
31
(a) 4.83 mR2
(b) 10 rad/s
28
A rigid body is made of three identical thin rods, each with
length L = 0.600 m, fastened together in the form of a letter H
(see the figure). The body is free to rotate about a horizontal
axis that runs along the
length of one of the legs of the
H. The body is allowed to fall from rest from a position in
which the plane of the H is horizontal. What is the angular
speed of the body when the plane of the M is vertical?
A thin rod of length 0.75 m and mass 0.42 kg is suspended
freely from one end. It is pulled to one side and then allowed to
swing like a pendulum, passing through its lowest position
with angular speed 4.0 rad/s.
(a) Neglecting friction and air resistance, find the rod's kinetic
energy at its lowest position.
(b) Neglecting friction and air resistance, find how far above
that position the center of mass rises.
Answer
Answer
(a) .63 Nm
(b) .15 m
29
A meter stick is held vertically with one end on the floor and is
then allowed to fall. Find the speed of the other end just before
it hits the floor, assuming that the end on the floor does not
slip. (Hint: Consider the stick to be a thin rod and use the
conservation of energy principle.)
6.06 rad/s
32
The thin uniform rod in the figure has length 2.0 m and can
pivot about a horizontal, frictionless pin through one end. It is
released from rest at angle θ = 400 above the horizontal. Use
the principle of conservation of energy to determine the
angular speed of the rod as it passes through the horizontal
position.
Answer
5.42 m/s
Answer
3.1 rad/s
33
Four particles, each of mass 0.20 kg, are placed at the vertices
of a square with sides of length 0.50 m. The particles are
connected by rods of negligible mass. This rigid body can
rotate in a vertical plane about a horizontal axis A that passes
through one of the particles. The body is released from rest
with rod AB horizontal, as shown in in the figure.
(a) What is the rotational inertia of the body about axis A?
(b) What is the angular speed of the body about axis A at the
instant rod AB swings through the vertical position?
35
Answers
Answer
2
(a) 2 kgm
(b) 6.26 rad/s
34
A cylindrical rod 24.0 cm long with mass 1.20 kg and radius
1.50 cm has a ball of diameter 8.00 cm and mass 2.00 kg
attached to one end. The arrangement is originally vertical and
stationary, with the ball at the top. The system is free to pivot
about the bottom end of the rod after being given a slight
nudge.
(a) After rotates through ninety degrees, what is its rotational
kinetic energy?
(b) What is the angular speed of the rod and ball?
(c) What is the linear speed of the ball?
(d) How does this compare to the speed if the ball had fallen
freely through the same distance of 28 cm?
In the figure, a 2.0 kg uniform rod of length 3.0 m is mounted
to rotate freely about a horizontal axis that is perpendicular to
the rod at distance d = 1.0 m from one end. The rod is released
from rest when it is horizontal.
(a) What is its maximum angular speed?
(b) If its mass were increased, would the answer to (a) increase,
decrease, or stay the same?
(a)
(b)
(c)
(d)
36
6.90 nm
8.73 rad/s
2.44 m/s
1.0432 times
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 the figure below.
(a) At the instant the rod is horizontal, find its angular speed,
(b) At the instant the rod is horizontal, find the magnitude of its
angular acceleration,
(c) At the instant the rod is horizontal, find the x and y
components of the acceleration of its center of mass
(d) At the instant the rod is horizontal, find the components of
the reaction force at the pivot.
Answer
(a) 3.1 rad/s
(b) remains the same
Answer
(a) [(3g)/L]1/2
(b) (3g)/(2L)
(c) -(3g)/4
(d) Rx =-(3Mg)/2
Ry = (Mg)/4
37
In a microhematocrit centrifuge, small samples of blood are
placed in heparinized capillary tubes (heparin is an
anticoagulant). The tubes are rotated at 11,500 rpm, with the
bottom of the tubes 9.07 cm from the axis of rotation.
(a) Find the linear speed of the bottom of the tubes.
(b) What is the centripetal acceleration at the bottom of the
tubes?
(c) What is the kinetic energy of the system.
39
The figure shows a communications satellite that is a solid
cylinder with mass 1210 kg, diameter 1.21 m, and length 1.75
m. Prior to launch from the shuttle cargo bay, the satellite is set
spinning at 1.52 rev/s about its long axis. What is
(a) What is its rotational inertia about the rotation axis?
(b) What is its rotational kinetic energy?
Answer
(a) 221 kg.m2
(b) 1.10 x 104 J
Answers
131, 000 m/s
38
40
2
In the figure a thin uniform rod (mass 3.0 kg, length 4.0 m)
rotates freely about a horizontal axis A that is perpendicular to
the rod and passes through a point at distance d = 1.0 m from
the end of the rod. The kinetic energy of the rod as it passes
through the vertical position is 20 J.
(a) What is the rotational inertia of the rod about axis A?
(b) What is the (linear) speed of the end B of the rod as the rod
passes through the vertical position?
(c) At what angle θ will the rod momentarily stop in its upward
swing?
Two point particles with masses m1 and m2 are connected by a
massless rod of length L to form a dumbbell that rotates about
its center of mass with angular velocity w. Show that the ratio
of kinetic energies of the particles is K1/K2 = m1/m2
Answer
K1/K2 = m2/m1
41
A yo-yo has a rotational inertia of 950 g.cm2 and a mass of 120
g. Its axle radius is 3.2 mm, and its string is 120 cm long. The
yo-yo rolls from rest down to the end of the string.
(a) What is the magnitude of its linear acceleration?
(b) How long does it take to reach the end of the string?
(c) As it reaches the end of the string, what is its linear speed?
(d) As it reaches the end of the string, what is its translational
kinetic energy?
(e) As it reaches the end of the string, what is its rotational
kinetic energy?
(f) As it reaches the end of the string, what is its angular
speed?
Answer
Answer
(a) 7.0 kg.m2
(b) 7.2 m/s
(c) 710
(a) 13 cm/s2
(b) 4.4 s
(c) 55 cm/s
(d) 18 mJ
(e) 1.4 J
(f ) 27 rev/s
42
In 1980, over San Francisco Bay, a large yo-yo was released
from a crane. The 116 kg yo-yo consisted of two uniform disks
of radius 32 cm connected by an axle of radius 3.2 cm.
(a) What was the magnitude of the acceleration of the yoyo
during its fall?
(b) What was the magnitude of the acceleration of the yoyo
during its rise?
(c) What was the tension in the cord on which it rolled?
(d) Was that tension near the cord's limit of 52 kN?
(e) Suppose you build a scaled-up version of the yo-yo (same
shape and materials but larger). Will the magnitude of your yoyo's acceleration as it falls be greater than, less than, or the
same as that of the San Francisco yo-yo?
(f) How about the tension in the cord?
44
Yo-Yo man releases a yo-yo from rest and allows it to drop, as
he keeps the top end of the string stationary. The mass of the
yo-yo is 0.056 kg, its moment of inertia is 2.9 x 10-5 kg.m2, and
the radius of the axle the string wraps around is 0.0064 m.
Through what height must the yo-yo fall for its linear speed to
be 0.50 m/s?
Answer
(a) 0.19 m/s2
(b) 0.19 m/s2
(c) 1.1 kN
(d) no
(e) same
(f) greater
Answers
0.17 m
43
A yo-yo has a rotational inertia of 950 g.cm2 and a mass of 120
g. Its axle radius is 3.2 mm, and its string is 120 cm long. The
yo-yo is thrown so that its initial speed down the string is 1.3
m/s.
(a) How long does the yo-yo take to reach the end of the
string? As it reaches the end of the string, what is its
(b) As it reaches the end of the string, what is its total kinetic
energy?
(c) As it reaches the end of the string, what is its linear speed?
(d) As it reaches the end of the string, what is its translational
kinetic energy?
(e) As it reaches the end of the string, what is its angular
speed?
(f) As it reaches the end of the string, what is its rotational
kinetic energy
45
Yomega ("The yo-yo with a brain") is constructed with a
clever clutch mechanism in its axle that allows it to rotate
freely and "sleep" when its angular speed is greater than a
certain critical value. When the yo-yo's angular speed falls
below this value the clutch engages, causing the yo-yo to climb
the string to the user's hand. If the moment-5 kg.m2, its mass is
0.11 kg, and the string is 1.0 m long, what is the smallest
angular speed that will allow the yo-yo to return to the user's
hand?
Answer
(a) 0.89 s
(b) 9.4 J
(c) 1.4 m/s
(d) 0.12 J
(e) 4.4 x 102 rad/s
(f) 9.2 J
Answers
170 rad/s
46
Yo-Yo man releases a yo-yo from rest and allows it to drop, as
he keeps the top end of the string stationary. The mass of the
yo-yo is 0.056 kg, its moment of inertia is 2.9 x 10-5 kg.m2 and
the radius of the axle the string wraps around is 0.0064 m.
(a) What is the linear speed, v, of the yo-yo after it has
dropped through a height h = 0.50m?
(b) Assume the moment of inertia is increased to 3.9 x 10-5 kg.
m2.. What is the linear speed, v, of the yo-yo after it has
dropped through a height h = 0.50m?
(c) If the yo-yo’s moment of inertial is increased, does its final
speed increase, decrease or stay the same.
50
A car with tires of radius 32 cm drives on the highway at 55
mph.
(a) What is the angular speed of the tires?
(b) What is the linear speed of the top of the tires?
Answers
(a) 77 rad/s
(b) 110 mi/hr
Answers
0.85 m/s
47
51
As you drive down the highway, the top of your tires are
moving with a speed v. What is the reading on your
speedometer?
A thin-walled pipe rolls along the floor. What is the ratio of its
translational kinetic energy to its rotational kinetic energy
about the central axis parallel to its length?
Answer
1.00
48
A 1000 kg car has four 10 kg wheels. What fraction of the
total kinetic energy of the car is due to rotation of the wheels
about their axles? Assume that the wheels have the same
rotational inertia as disks of the same mass and size. Explain
why you do not need to know the radius of the wheels.
Answer
0.0196
49
An automobile has a total mass of 1700 kg. It accelerates from
rest to 40 km/h in 10 s. Each wheel has a mass of 32 kg and
radius of gyration of 30 cm.
(a). Find the rotational kinetic energy of each wheel about its
axle for the end of the 10 s interval.
(b) Find the total kinetic energy of each wheel for the end of
the 10 s interval.
(c) Find the total kinetic energy of the automobile for the end
of the 10 s interval .
Answer
(a) 990 J
(b) 3000 J
(c) 1.1 x 105 j
Answers
The reading on the speedometer gives the speed of the
axles of your car. This is the same as the speed of the
occupants inside the car. If the top of the tires have a speed
v, the axles have a speed v/2.
52
A 1.20 kg hoop with a radius of 10.0 cm rolls without slipping.
The linear speed of the disk is 1.41 m/s.
(a) Find the translational kinetic energy of the hoop.
(b) Find the rotational kinetic energy of the hoop.
(c) Find the total kinetic energy of the hoop.
55
A 1.20 kg disk with a radius of 10.0 cm rolls without slipping.
The linear speed of the disk is 1.41 m/s, find·
(a) Find the translational kinetic energy,
(b) Find the rotational kinetic energy, and
(c) Find the total kinetic energy of the disk.
Answers
(a) 1.19 J
(b) 0.595 J
(c) 1.79 J
Answers
(a) 1.19 J
(b) 1.19 J
(c) 2.39 J
53
A basketball rolls along the floor with a constant linear speed
v.
(a) Find the fraction of its total kinetic energy that is in the
form of rotational kinetic energy about the center of the ball.
(b) If the linear speed of the ball is doubled to 2v, does your
answer to part (a) increase, decrease, or stay the same?
Explain.
56
A solid sphere and a hollow sphere of the same mass and
radius roll without slipping at the same speed. Is the kinetic
energy of the solid sphere (a) more than, (b) less than, or (c)
the same as the kinetic energy of the hollow sphere
Answers
(b) less than
57
Two spheres of equal mass and radius are rolling across the
floor with the same speed. One sphere is solid, the other is
hollow. Which sphere is harder to stop? Why?
Answers
Answers
(a) 2/5
(b) It stays the same since the fraction doesn’t depend on
v.
54
What linear speed must a 0.050 kg hula hoop have if its total
kinetic energy is to be 0.10 J? Assume the hoop rolls on the
ground without slipping.
The hollow sphere is harder to stop because it – with its
greater moment of inertia – has more kinetic energy for a
given speed. The more the kinetic energy, the more work
that must be done to bring it to rest.
58
Answers
A solid ball of mass 1.4 kg and diameter 15 cm is rotating
about its diameter at 70 rev/min.
(a) What is its kinetic energy?
(b) If an additional 2 J of energy are supplied to the rotational
energy, what is the new angular speed of the ball?
1.4 m/s
Answer
(a) 84.6 mJ
(b) 347 rev/min
59
A homogeneous solid cylinder rolls without slipping on a
horizontal surface. The total kinetic energy is K. The kinetic
energy due to rotation about its center of mass is (a) 1/2 K, (b)
1/3 K, (c) 4/7 K, (d) none of these.
Answer
(b) 1/3 K,
60
Answer the following questions:
(a) Find the percentages of the total kinetic energy associated
with rotation and translation for an object that is rolling
without slipping if the object is a uniform sphere,
(b) Find the percentages of the total kinetic energy associated
with rotation and translation for an object that is rolling
without slipping if the object is a uniform cylinder
(c) Find the percentages of the total kinetic energy associated
with rotation and translation for an object that is rolling
without slipping if the object is a hoop.
64
A uniform ball, of mass M = 6.00 kg and radius R, rolls
smoothly from rest down a ramp at angle θ = 30.00
(a) The ball descends a vertical height h = 1.20 m to reach the
bottom of the ramp. What is its speed at the bottom?
(b) What are the magnitude and direction of the frictional force
on the ball as it rolls down the ramp?
Answer
(a) 28.6 %, 71.4%
(b) 33.3 %, 66.7 %
(c) 50%, 50%
Answer
61
A wheel has a thin 3.0 kg rim and four spokes, each of mass
1.2 kg. Find the kinetic energy of the wheel when it rolls at 6
m/s on a horizontal surface.
65
Answer
223 J
62
A 1.20 kg hollow sphere with a radius of 10.0 cm rolls without
slipping. The linear speed of the disk is 1.41 m/s, find·
(a) Find the translational kinetic energy,
(b) Find the rotational kinetic energy, and
(c) Find the total kinetic energy of the disk.
Answer
(a) 61.7 J
(b) 3.43 m
(c) no
66
Answers
(a) 1.19 J
(b) 0.793 J
(c) 1.98 J
63
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.
Assuming mechanical energy conservation, calculate the
moment of inertia of the can if it takes 1.50 s to reach the
bottom of the incline. Which pieces of data, if any, are
unnecessary for calculating the solution?
A solid sphere of weight 36.0 N rolls up an incline at an angle
of 30.00. At the bottom of the incline the center of mass of the
sphere has a translational speed of 4.90 m/s.
(a) What is the kinetic energy of the sphere at the bottom of the
incline?
(b) How far does the sphere travel up along the incline?
(c) Does the answer to (b) depend on the sphere's mass?
A uniform wheel of mass 10.0 kg and radius 0.400 in is
mounted rigidly on an axle through its center (see the figure).
The radius of the axle is 0.200 m, and the rotational inertia of
the wheel-axle combination about its central axis is 0.600 kg.
m2. The wheel is initially at rest at the top of a surface that is
inclined at angle θ = 30.00 with the horizontal; the axle rests on
the surface while the wheel extends into a groove in the surface
without touching the surface. Once released, the axle rolls
down along the surface smoothly and without slipping. When
the wheel-axle combination has moved down the surface by
2.00 in, what is
(a) When the wheel-axle combination has moved down the
surface by 2.00 in, what is its rotational kinetic energy?
(b) When the wheel-axle combination has moved down the
surface by 2.00 in, what is its translational kinetic energy?
Answer
1.21 x 10-4 kg.m2
Height of the can
Answer
(a) 58.8 J;
(b) 29.2J
67
A ball is released from rest on a frictionless surface, opposite
what is as shown. After reaching its lowest point, the ball
begins to rise again, this time on a no-slip surface. When the
ball comes to rest on the no-slip surface, is its height greater
than, less than, or the same as the height from which it was
released? Explain.
69
After you pick up a spare, your bowling ball rolls without
slipping back toward the ball rack with a linear speed of 2.85
m/s. To reach the rack, the ball rolls up a ramp that rises
through a vertical distance of 0.53 m.
(a) What is the linear speed of the ball when it reaches the top
of the ramp?
(b) If the radius of the ball were increased, would the speed
found in part
Answers
Assuming the ball starts spinning immediately on
encountering the no-slip surface, with no loss of energy, it
will rise to the same height from which it was released.
However, some energy will be lost in a real system as the
ball begins to spin; therefore, the ball should reach a height
slightly less than
68
A ball is released from rest on a no-slip surface, as shown.
After reaching its lowest point, the ball begins to rise again,
this time on a frictionless surface. Assume the ball is a solid
sphere of radius 2.8 cm and mass 0.10 kg. The ball is released
from rest at a height of 0.75 m above the bottom of the track on
the no-slip side,
(a) What is its angular speed when it is on the frictionless side
of the track?
(b) How high does the ball rise on the frictionless side?
Answers
(a) 0.83 m/s
(b) The speed is independent of the ball’s radius, and stays
the same
70
A 2.0 kg cylinder (radius = 0.10 m, length = 0.50 m) is
released from rest at the top of a ramp and allowed to roll
without slipping. The ramp is 0.75 m high and 5.0 m long. The
cylinder reaches the bottom of the ramp.
(a) What is its total kinetic energy?
(b) What is its rotational kinetic energy?
(c) What is its translational kinetic energy?
Answers
(a) 15 J
(b) 4.9 J
(c) 9.8 J
71
A 2.0 kg solid sphere (radius = 0.10 m) is released from rest at
the top of a ramp and allowed to roll without slipping. The
ramp is 0.75 m high and 5.0 m long. The sphere reaches the
bottom of the ramp.
(a) What is its total kinetic energy?
(b) What is its rotational kinetic energy?
(c) What is its translational kinetic energy?
Answers
(a) 120 rad/s
(b) 0.54 m
Answers
(a) 15 J
(b) 4.2 J
(c) 11 J
72
A ball is released from rest on a no-slip surface, as shown.
After reaching its lowest point, the ball begins to rise again,
this time on a frictionless surface. When the ball reaches its
maximum height on the frictionless surface, is it (a) at a greater
height, (b) at a lesser height, or (c) at the same height as when
it was released?
75
As a solid disk rolls over the top of a hill on a track, its speed is
80 cm/s. If friction losses are negligible, how fast is the disk
moving when it is 18 cm below the top?
Answers
1.73 m/s
76
Disks A and B are identical and roll across a floor with equal
speeds. Then disk A rolls up an incline, reaching a maximum
height h, and disk B moves up an incline that is identical
except that it is frictionless. Is the maximum height reached by
disk B greater than, less than, or equal to h?
Answer
less
Answers
(b) at a lesser height,
73
77
A marble of mass M and radius R rolls without slipping down
the track on the left from a height h1 as shown in the figure.
The marble then goes up the frictionless track on the right to a
height h2. Find h2.
Answer the following questions:
(a) Determine the acceleration of the center of mass of a
uniform solid disk rolling down an incline making angle θ with
the horizontal.
(b) Compare this acceleration with that of a uniform hoop.
What is the minimum coefficient of friction required to
maintain pure rolling motion for the disk
Answer
78
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.
Answer
(5/7) H1
74
As shown in the figure, a uniform solid sphere rolls on a
horizontal surface at 20 m/s. It then rolls up the incline shown.
If friction losses are negligible, what will be the value of h
where the ball stops?
Answer
Disk first
Answers
28.6 m
79
Suppose we change the race shown in the figure so that a disk
of radius R and mass M races a disk of radius 2R and mass M.
(a) Which disk wins, or do they finish at the same time?
Explain.
(b) How would your answer to part (a) change if the disks had
different masses? Explain.
82
The figure gives the potential energy function U(x) for a
system in which a particle is in one-dimensional motion.
(a) Rank regions AB, BC, and CD according to the magnitude
of the force on the particle, greatest first.
(b) What is the direction of the force when the particle is in
region AB?
Answer
(a) CD, AB, BC
(zero)
(b) positive x direction
Answers
(a) The disk of radius R has the smaller moment of inertia,
and therefore it wins the race.
(b) There would be no change. The speed of the disks
during the race is independent of their mass, just like
objects in free fall.
80
A disk and a hoop of the same mass and radius are released at
the same time at the top of a 1.1 m high inclined plane.
(a) Calculate the speeds of the disk bottom of the inclined
plane
(b) Calculate the speeds of the hoop at the bottom of the
inclined plane
Answers
(a) 3.8 m/s
(b) 3.3 m/s
81
A 2.0 kg solid sphere and a cylinder (radius = 0.10 m) are
released from rest at the top of a ramp and allowed to roll
without slipping. The ramp is 0.75 m high and 5.0 m long. The
objects reach the bottom of the ramp.
(a) Which object do you expect to have the greater speed at the
bottom of the ramp?
(b) Verify your answer to part (a) by calculating the speed of
the cylinder and the sphere when they reach the bottom of the
ramp.
Answers
3.1 m/s
3.2 m/s
83
The figure shows a plot of potential energy U versus position x
of a 0.90 kg particle that can travel only along an x axis. (Non
conservative forces are not involved.) The particle is released
at x = 4.5 m with an initial speed of 7.0 m/s, headed in the
negative x direction.
(a) If the particle can reach x = 1.0 m, what is its speed there,
and if it cannot, what is its turning point?
(b) What is the magnitude of the force on the particle as it
begins to move to the left of x = 4.0 m?
(c) What is the direction of the force on the particle as it begins
to move to the left of x = 4.0 m?
Suppose, instead, the particle is headed in the positive x
direction when it is released at x = 4.5 m at speed 7.0 m/s.
(d) If the particle can reach x = 7.0 m, what is its speed there,
and if it cannot, what is its turning point?
(e) What is the magnitude of the force on the particle as it
begins to move to the right of x = 5.0 m?
(f) What is the direction of the force on the particle as it begins
to move to the right of x = 5.0 m?
Answer
(a) 2.1 m/s
(b) 10 N
(c) positive x direction
(d) 5.7 m
(e) 30 N
(f) negative x direction
84
A conservative force F(x) acts on a 2.0 kg particle that moves
along an x axis. The potential energy U(x) associated with F(x)
is graphed in the figure. When the particle is at x = 2.0 m, its
velocity is -1.5 m/s.
(a) What is the magnitude?
(b) What is the direction of F(x) at this position?
(c) Between what positions on the left does the particle move?
(d) Between what positions on the right does the particle
move?
(e) What is its particle's speed at x = 7.0 m?
86
Figure a shows a molecule consisting of two atoms of masses
m and M (with m << M) and separation r. Figure b shows the
potential energy U(r) of the molecule as a function of r.
(a) Describe the motion of the atoms if the total mechanical
energy E of the two-atom system is greater than zero (as is E1).
(b) Describe the motion of the atoms if E is less than zero (as is
E2).
(c) For E1 = 1 x 10-19 J and r = 0.3 nm, find the potential energy
of the system.
(d) For E1 = 1 x 10-19 J and r = 0.3 nm, find the total kinetic
energy of the atoms.
(e) For E1 = 1 x 10-19 J and r = 0.3 nm, find the force
(magnitude and direction) acting on each atom.
(f) For what values of r is the force repulsive?
(g) For what values of r is the force attractive?
(h) For what values of r is the force zero?
Answer
(a) 4.8 N
(b) positive x direction
(c) 1.5 m
(d) 13.5 m
(e) 3.5 m/s
85
Answer
(a) turning point on left, none on right,
molecule breaks apart
(b) turning points
on both left and right, molecule does not
break apart
(c) -1.1 x 10-19 J
(d) 2.1 x 10-19 J
(e) ≈ 1 x 10-9 N on each, directed
toward the other
(f) r < 0.2 nm
(g) r > 0.2 nm
(h) r = 0.2 nm
A block with a kinetic energy of 30 J is about to collide with a
spring at its relaxed length. As the block compresses the
spring, a frictional force between the block and floor acts on
the block. The figure gives the kinetic energy K(x) of the block
and the potential energy U(x) of the spring as functions of
position x of the block, as the spring is compressed.
(a) What is the increase in thermal energy of the block and the
floor when the block reaches position x = 0.10 m and
(b) What is the increase in thermal energy of the block and the
floor when the spring reaches its maximum compression?
87
A conservative force F(x) acts on a particle that moves along
an x axis. The figure shows how the potential energy U(x)
associated with force F(x) varies with the position of the
particle.
(a) Plot F(x) for the range 0 < x < 6 m.
(b) The mechanical energy E of the system is 4.0 J. Plot the
kinetic energy K(x) of the particle directly on the figure.
Answer
(a) 7 J
(b) 16 J
Answer
88
The figure shows one direct path and four indirect paths from
point i to point f. Along the direct path and three of the
indirect paths, only a conservative force Fc acts on a certain
object. Along the fourth indirect path, both Fc and a non
conservative force Fnc act on the object. The change ∆Emec in
the object's mechanical energy (in joules) in going from i to f is
indicated along each straight-line segment of the indirect paths.
(a) What is ∆Emec from i to f along the direct path?
(b) What is ∆Emec due to Fnc along the one path where it acts?
90
The figure shows the potential energy U(x) of a solid ball that
can roll along an x axis. The ball is uniform, rolls smoothly,
and has a mass of 0.400 kg. It is released at x = 7.0 in headed
in the negative direction of the x axis with a mechanical
energy of 75 J.
(a) If the ball can reach x = 0 in, what is its speed there, and if
it cannot, what is its turning point?
(b) Suppose, instead, it is headed in the positive direction of
the x axis when it is released at x = 7.0 m with 75 J. If the ball
can reach x = 13 what is its speed there, and if it cannot, what
is its turning point?
Answer
no solution
(a) 12 J
(b) -2J
89
The figure gives the potential energy function of a particle.
(a) Rank regions AB, BC, CD, and DE according to the
magnitude of the force on the particle, greatest first.
(b) What value must the mechanical energy Emec of the particle
not exceed if the particle is to be trapped in the potential well at
the left?
(c) What value must the mechanical energy Emec of the particle
not exceed if the particle is to be trapped in the potential well at
the right?
(d) What value must the mechanical energy Emec of the particle
not exceed if the particle is to be able to move between the two
potential wells but not to the right of point H?
For the situation of (d), in which of regions BC, DE, and FG
will the particle have
(e) For the situation of (d), in which of regions BC, DE, and
FG will the particle have the greatest kinetic energy?
(f) For the situation of (d), in which of regions BC, DE, and
FG will the particle have the least speed?
Answer
(a) 2.0 m;
(b) 7.3 m/s
91
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.
Answer
(a) Tension 11.4 N
acc 7.57 m/s2
vel = 9.53 m/s
(b) 9.53 m/s
92
A torque of 0.97 N.m is applied to a bicycle wheel of radius 35
cm and mass 0.75 kg. Treating the wheel as a hoop, find its
angular acceleration.
Answers
Answer
no solution
(a) AB, CD, then BC and DE tie (zero force)
(b) 5 J
(c) 5 J
(d) 6 J
(e) FG
(f) DE
11 rad/s2
93
When the play button is pressed, a CD accelerates uniformly
from-rest to 450 rev/min in 3.0 revolutions. If the CD has a
radius of 6.0 cm and a mass of 17 g, what is the torque exerted
on it?
Answers
0.0072 Nm
94
The L-shaped object in the figure consists of three masses
connected by light rods. This object is given an angular
acceleration of 1.20 rad/s2.
(a) What torque must be applied to this object if it is rotated
about the x axis?
(b) What torque must be applied to this object if it is rotated
about the y axis?
(c) What torque must be applied to this object if it is rotated
about the z axis (which is through the origin and perpendicular
to the page)?
96
A fish takes the bait and pulls on the line with a force of 2.1 N.
The fishing reel, which rotates without friction, is a cylinder of
radius 0.055 m and mass 0.84 kg.
(a) What is the angular acceleration of the fishing reel?
(b) How much line does the fish pull from the reel in 0.25 s?
Answers
(a) 91 rad/s2
(b) 0.16 m
97
A fish takes the bait and pulls on the line with a force of 2.1 N.
The fishing reel, which rotates without friction, is a cylinder of
radius 0.055 m and mass 0.84 kg. Now assume the reel has a
friction clutch that exerts a restraining torque of 0.047 N.m.
(a) What is the angular acceleration of the fishing reel?
(b) How much line does the fish pull from the reel in 0.25 s?
Answers
Answers
(a) 11 Nm
(b) 12 Nm
(c) 23 Nm
95
A torque of 13 N.m is applied to the rectangular object shown
in the figure. The torque can act about the x axis, the y axis, or
the z axis, which passes through the origin and points out of
the page.
(a) In which case does the object experience the greatest
angular acceleration?
(b) In which case does the object experience the least angular
acceleration? Explain.
(c) Find the angular acceleration when the torque acts about the
x-axis.
(d) Find the angular acceleration when the torque acts about
the y-axis.
(e) Find the angular acceleration when the torque acts about the
z-axis.
(a) 54 rad/s2
(b) 0.092 m
98
To prepare homemade ice cream a crank must be turned with a
torque of 3.8 N.m. How much work is required for each
complete turn of the crank?
Answers
24 J
99
A hoop of radius 3.0 m has a mass of 150 kg. It rolls along a
horizontal floor so that its center of mass has a speed of 0.15
m/s. How much has to be done on the hoop to stop it? Show
the solution two ways.
Answer
3.375 J
Show two ways for credit
Answers
(a) greatest about the x axis, least about
the z axis
(b) 7.4 rad/s2
(c) 5.1 rad/s2
(d) 3.0 rad/s2
100
A grindstone with a radius of 0.610 m is being used to sharpen
an ax. The linear speed of the stone relative to the ax is 1.50
m/s and its rotational kinetic energy is 13.0 J. When the ax is
pressed firmly against the grindstone for sharpening, the
angular speed of the grindstone decreases. If the rotational
kinetic energy of the grindstone is cut in half to 6.50 J , what is
its angular speed?
105
After getting a drink of water, a hamster jumps onto an
exercise wheel for a run. A few seconds later the hamster is
running in place with a speed of 1.4 m/s. Find the work done
by the hamster to get the exercise wheel moving, assuming it is
a hoop of radius 0.13 m and mass 6.5 g.
Answers
6.4 x 10-3 J
106
The L-shaped object in the figure consists of three masses
connected by light rods.
(a) Find the work that must be done on this object to accelerate
it from rest to an angular speed of 2.75 rad/s about the x-axis,
(b) Find the work that must be done on this object to accelerate
it from rest to an angular speed of 2.75 rad/s about the y-axis.
(c) Find the work that must be done on this object to accelerate
it from rest to an angular speed of 2.75 rad/s about an axis
through the origin and perpendicular to the page.
Answers
4.30 kg.m2
101
It takes a good deal of effort to make homemade ice cream. If
the torque required to turn the handle on the ice cream maker is
5.00 N . m, how much work is expended on each complete
revolution of the handle?
Answers
31.4 J
102
When a ceiling fan rotating with an angular speed of 2.55 rad/s
is turned off, a frictional torque of 0.220 N.m slows it to a stop
in 5.75 s. What is the moment of inertia of the fan?
Answers
∆ω = .0496 kgm2
103
A wheel an a game show is given an initial angular speed of
1.22 rad/s. It comes to rest after rotating through 3/4 of a turn.
(a) Find the average torque exerted on the wheel given that it is
a disk of radius 0.71 m and mass 6.4 kg.
(b) If the mass of the wheel is doubled and its radius is halved,
will the angle through which it rotates before coming to rest
increase, decrease, or stay the same? Explain. (Assume that the
average torque is unchanged.)
Answers
(a) 0.25 Nm
(b) decrease
104
How much work must be done to accelerate a baton from rest
to an angular speed of 7.1 rad/s about its center. Consider the
baton to be a uniform rod of length 0.52 m and mass 0.46 kg.
Answers
0.26 J
Answers
(a) 34 J
(b) 38 J
(c) 72 J
107
The rectangular object in the figure consists of four small
masses connected by light rods.
(a) Find the work that must be done on this object to accelerate
it from rest to an angular speed of 2.75 rad/s about the x-axis.
(b) Find the work that must be done on this object to accelerate
it from rest to an angular speed of 2.75 rad/s about the y-axis.
(c) Find the work that must be done on this object to accelerate
it from rest to an angular speed of 2.75 rad/s about an axis
through the origin and perpendicular to the page.
111
When 100 J of work is done upon a flywheel, its angular speed
increases from 60 rev/min to 180 rev/min. What is its moment
of inertia?
Answers
0.63 kg.m2
112
A wheel with I = 20 kg· m2 is spinning at 3.0 rev/s on its axis.
How large is the frictional torque if the wheel coasts 40 rev
before stopping?
Answers
1.41 N.m
113
Answers
Answers
(a) 6.6 J
(b) 9.6 J
(c) 19 J
108
The rigid body shown in the figure consists of three particles
connected by massless rods. It is to be rotated about an axis
perpendicular to its plane through point P. If M = 0.40 kg, a =
30 cm, and b = 50 cm, how much work is required to take the
body from rest to an angular speed of 5.0 rad/s?
It is proposed to use a uniform disk 50 cm l in radius turning at
300 rev/s as an energy-storage device in a bus. How much
mass must the disk have if it is to be capable, while coasting to
rest, of furnishing the energy equivalent of a 100-hp motor
operating for 10 min? (Note: 1 hp = 746 watts)
202 kg
114
A metal tool is sharpened by being held against the rim of a
wheel on a grinding machine by a force of 180 N. The
frictional forces between the rim and the tool grind off small
pieces of the tool. The wheel has a radius of 20.0 cm and
rotates at 2.50 rev/s. The coefficient of kinetic friction between
the wheel and the tool is 0.320. At what rate is energy being
transferred from the motor driving the wheel to the thermal
energy of the wheel and tool and to the kinetic energy of the
material thrown from the tool?
Answer
2.6 J
109
Answer
A homogeneous cylinder of radius 18 cm and mass 60 kg is
rolling without slipping along a horizontal floor at 5 m/s. How
much work was required to give it this motion?
181 W
Answer
A flywheel having a moment of inertia of 900 kg.m2 rotates at
a speed of 120 rev/min. It is slowed down by a brake to a speed
of 90 rev/min. How much energy did the flywheel lose in
slowing down?
Resistance to the motion of an automobile consists of road
friction, which is almost independent of speed, and air drag,
which is proportional to speed-squared. For a certain car with a
weight of 12 000 N, the total resistant force F is given by F =
300 + 1.8v2, with F in Newtons and v in meters per second.
Calculate the power (in horsepower) required to accelerate the
car at 0.92 m/s2 when the speed is 80 km/h. (Note: 1 hp = 746
W)
Answers
Answer
31.0 x 103 J
69 hp
1.31 x 103 J
110
115
116
An engine develops 400 N.m of torque at 3700 rev/min. Find
the power developed by the engine.
122
Answer
1.511 x 105 W
117
An electric motor runs at 900 rev/min and delivers 2 hp. How
much torque does it deliver? (Note: 1 hp = 746 watts)
A skier weighing 600 N goes over a frictionless circular hill of
radius R = 20 m (see the figure). Assume that the effects of air
resistance on the skier are negligible. As she comes up the hill,
her speed is 8.0 m/s at point B, at angle θ = 200.
(a) What is her speed at the hilltop (point A) if she coasts
without using her poles?
(b) What minimum speed can she have at B and still coast to
the hilltop?
(c) Do the answers to these two questions increase, decrease, or
remain the same if the skier weighs 700 N?
Answers
15.8 N.m
118
119
A small motor delivers 0.20 hp when its shaft is turning at
1400 rev/min. How large a torque is it capable of providing?
(Note: 1 hp = 746 watts)
Answers
Answer
0.157 hp
(a) 6.4 m/s
(b) 4.9 m/s
(c) same
A certain motor is capable of producing an output torque of
0.80 N . m. It operates at a speed of 1400 rev/min. What is the
output horsepower of the motor under these conditions? (Note:
1 hp = 746 watts)
Answers
123
A boy is seated on the top of a hemispherical mound of ice.
He is given a very small push and starts sliding down the ice.
Show that he leaves the ice at a point whose height is 2R/3 if
the ice is frictionless. (Hint: The normal force vanishes as he
leaves the ice.)
79 N
120
A certain hp motor (based on output) normally operates at
1800 rev/min under load.
(a) How large a torque does the motor develop?
(b) If it drives a belt on a 5.0-cm-diameter pulley, what is the
difference in tensions in the two portions of the belt? (Note: 1
hp = 746 watts)
Answers
(a) 1.98 N.m
(b) 79 N
121
A boy is initially seated on the top of a hemispherical ice
mound of radius R = 13.8 m. He begins to slide down the
ice, with a negligible initial speed (see figure). Approximate
the ice as being frictionless. At what height does the boy
lose contact with the ice?
Answer
9.20 m
Answer
my solution
124
A uniform spherical shell of mass M = 4.5 kg and radius R =
8.5 cm can rotate about a vertical axis on frictionless bearings
(see the figure). A massless cord passes around the equator of
the shell, over a pulley of rotational inertia I = 3.0 x 10-3 kg.m2
and radius r = 5.0 cm, and is attached to a small object of mass
m = 0.60 kg. There is no friction on the pulley's axle; the cord
does not slip on the pulley. What is the speed of the object
when it has fallen 82 cm after being released from rest? Use
energy considerations.
126
Two blocks, as shown in the figure, 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.
Answer
Answer
1.4 m/s
125
The reel shown in the figure 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 if 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°.
(a) 156 N
118 N
(b) 1.17 kg.m2
127
In the figure, a solid cylinder of radius 10 cm and mass 12 kg
starts from rest and rolls without slipping a distance L = 6.0 m
down a roof that is inclined at angle θ = 300.
(a) What is the angular speed of the cylinder about its center as
it leaves the roof?
(b) The roof’s edge is at height H = 5.0 m. How far
horizontally from the roof's edge does the cylinder hit the level
ground?
Answer
Answer
(a)
(b) 1.74 rad/s
(a) 63 rad/s
(b) 4.0 m
128
In the figure, a solid brass ball of mass 0.280 g will roll
smoothly along a loop-the-loop track when released from rest
along the straight section. The circular loop has radius R = 14.0
cm, and the ball has radius r < < R.
(a) What is h if the ball is on the verge of leaving the track
when it reaches the top of the loop? The ball is released at
height h = 6.00R.
(b) What is the magnitude of the horizontal force component
acting on the ball at point Q?
(c) What is the direction of the horizontal force component
acting on the ball at point Q?
131
A homogeneous sphere starts from rest at the upper end of the
track shown in the figure to the left and rolls without slipping
until it rolls off the right-hand end. If H = 60 m and h =2 0 m
and the track is horizontal at the right-hand end, determine the
distance to the right of point A at which the ball strikes the
horizontal base line.
Answer
Answer
(a) 37.8 cm
(b) 1.96 x 102 N
(c) toward loop’s center
129
In the figure, a solid ball rolls smoothly from rest (starting at
height H = 6.0 m) until it leaves the horizontal section at the
end of the track, at height h = 2.0 m. How far horizontally from
point A does the ball hit the floor?
48 m
132
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 the figure. 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 that 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.
Answer
4.8 m
130
In the figure, a small, solid, uniform ball is to be shot from
point P so that it rolls smoothly along a horizontal path, up
along a ramp, and onto a plateau. Then it leaves the plateau
horizontally to land on a game board, at a horizontal distance d
from the right edge of the plateau. The vertical heights are h1 =
5.00 cm and h2 = 1.60 cm. With what speed must the ball be
shot at point P for it to land at d = 6.00 cm?
Answer
1.34 m/s
Answer
(a) 2.38 m/s
(b) 4.31 m/s
(c) never makes top
133
A large, cylindrical roll of tissue paper of initial radius R lies
on a long, horizontal surface with the outside end of the paper
nailed to the surface. The roll is given a slight shove (vi = 0)
and commences to unroll. Assume the roll has a uniform
density and that mechanical energy is conserved in the process.
(a) Determine the speed of the center of mass of the roll when
its radius has diminished to r.
(b) Calculate a numerical value for this speed at r = 1.00 mm,
assuming R = 6.00 m.
(c) What If? What happens to the energy of the system when
the paper is completely unrolled?
135
A solid sphere with a diameter of 0.17 m is released from rest;
it then rolls without slipping down a ramp, dropping through a
vertical height of 0.61 m. The ball leaves the bottom of the
ramp, which is 1.22 m above the floor, moving horizontally
(see the figure).
(a) Through what horizontal distance d does the ball move
before landing?
(b) How many revolutions does the ball make during its fall?
(c) If the ramp were to be made frictionless, would the distance
d increase, decrease, or stay the same? Explain.
Answer
(a) {[4g(R3 -r3)]/(3r2)}1/2
(b) 5.31 x 104 m/s
(c) the energy goes into internal energy.
134
A plank with a mass M = 6.00 kg rides on top of two
identical solid cylindrical rollers that have R = 5.00 cm and
m = 2.00 kg (see the figure below). The plank is pulled by a
constant horizontal force F of magnitude 6.00 N applied to
the end of the plank and perpendicular to the axes of the
cylinders (which are parallel). The cylinders roll without
slipping on a flat surface. There is also no slipping between
the cylinders and the plank.
(a) Find the acceleration of the plank and of the rollers.
(b) What friction forces are acting?
Answers
(a) 1.5 m
(b) 2.7 rev
(c) increase
136
A 0.105 kg yo-yo has an outer radius R that is 5.60 times
greater than the radius l' of its axle. The yo-yo is in equilibrium
if a mass m is suspended from its outer edge, as shown in the
figure. Find the tension in the two strings, Tl and T2, and the
mass m.
Answer
(a) 0.8 m/s
(b 0.6 N down
0.2 N forward
Answers
0.224 N
137
A 2000-kg block is lifted at a constant speed of 8 cm/s by a
steel cable that passes over a massless pulley to a motor-driven
winch as shown in the figure. The radius of the winch drum is
30 cm.
(a) What force must be exerted by the cable?
(b) What torque does the table exert on the winch drum?
(c) What is the angular velocity of the winch drum?
(d) What power must be developed by the motor to drive the
winch drum?
139
Answer
Answer
(a) 19.6 x 103 N
(b) 5.89 x 103 Nm
(c) 0.267 rad/s
(d) 1.57 x 103 W
(a) 230 N/m
(b) 160 J
140
138
A hollow sphere and uniform sphere of the same mass m1 and
radius R roll down an inclined plane from the same height H
without slipping (see figure). Each is moving horizontally as it
leaves the ramp. When the spheres hit the ground, the range of
the hollow sphere is L. Find the range L' of the uniform sphere
The figure shows a hollow cylinder of length 1.8 m, mass 0.8
kg, and radius 0.2 m. The cylinder is free to rotate about a
vertical axis that passes through its center and is perpendicular
to the cylinder's axis. Inside the cylinder are two masses of 0.2
kg each, attached to springs of spring constant k and
unstretched lengths 0.4 m. The inside walls of the cylinder are
frictionless.
(a) Determine the value of the spring constant if the masses are
located 0.8 m from the center of the cylinder when the cylinder
rotates at 24 rad/s.
(b) How much work was needed to bring the system from ω =
0 to ω = 24 rad/s?
When the system shown in the figure is released from rest, the
200-g mass slides down the incline against a frictional force of
0.50 N. If the moment of inertia of the wheel is 0.80 kg.m2,
how fast will the block be moving after it has slid 100 cm
along the incline?
Answers
Answer
1.09 L
0.86 m/s
141
The system shown in the figure is released from rest with the
spring in the unstretched position. If friction is negligible, how
far will the mass slide down the incline? k = 20 N/m.
144
A 34.0 kg child runs with a speed of 2.80 m/s tangential to the
rim of a stationary merry-go-round. The merry-go-round has a
moment of inertia of 510 kg.m2 and a radius of 2.31 m. When
the child jumps onto the merry-go-round, the entire system
begins to rotate. What initial speed does the child have if, after
landing on the merry-go-round, it takes her 22.5 s to complete
one revolution?
Answers
1.18 m
Answers
142
For a classroom demonstration, a student sits on a piano stool
holding a sizable mass in each hand. Initially, the student holds
his arms outstretched and spins about the axis of the stool with
an angular speed of 3.74 rad/s. The moment of inertia in this
case is 5.33 kg,m2. While still spinning, the student pulls his
arms in to his chest, reducing the moment of inertia to 1.60 kg,
m2. What is the student's angular speed now?
145
A beetle sits near the rim of a turntable that rotates without
friction about a vertical axis. What happens to the angular
speed of the turntable as the beetle walks toward the axis of
rotation?
Answers
Since there are no external torques acting on the system, its
angular momentum must remain constant. Thus, as the
beetle walks toward the axis of rotation, which reduces the
moment of inertia of the system, the angular speed of the
turntable increases.
146
Suppose a diver springs into the air with no initial angular
velocity. Can the diver begin to rotate by folding into a tucked
position? Explain.
Answers
12.5 rad/s
143
Answers
No. If the diver’s initial angular momentum is zero, it
must stay zero unless an external torque acts on her. A
diver needs to start off with at least a small angular speed,
which can then be increased by folding into a tucked
position.
A 34.0 kg child runs with a speed of 2.80 m/s tangential to the
rim of a stationary merry-go-round. The merry-go-round has a
moment of inertia of 510 kg.m2 and a radius of 2.31 m. When
the child jumps onto the merry-go-round, the entire system
begins to rotate. What is the angular speed of the system?
147
A 0.015 kg record with a radius of 15 cm rotates with an
angular speed of 33 1/3 rpm. A 1.1 g fly lands on the rim of
the record. What is the fly's angular momentum?
Answers
8.6 x 10-3 kgm2/s
Answers
0.318 rad/s
148
As an ice skater begins a spin, his angular speed is 3.28 rad/s.
After pulling in his arms, his angular speed increases to 5.72
rad/s. Find the ratio of the skater's final moment of inertia to
his initial moment of inertia.
152
Answers
0.573
A student sits at rest on a piano stool that can rotate without
friction. The moment of inertia of the student-stool system is
4.1 kg.m2. A second student tosses a 1.5 kg mass with a speed
of 2.7 m/s to the student on the stool, who catches it at a
distance of 0.40 m from the axis of rotation. What is the
resulting angular speed of the student and the stool?
Answers
0.37 rad/s
149
A 34.0 kg child runs with a speed of 2.80 m/s tangential to the
rim of a stationary merry-go-round. The merry-go-round has a
moment of inertia of 510 kg.m2 and a radius of 2.31 m. When
the child jumps onto the merry-go-round, the entire system
begins to rotate. Calculate both the initial and the final kinetic
energy of the system.
153
A student sits at rest on a piano stool that can rotate without
friction. The moment of inertia of the student-stool system is
4.1 kg.m2. A second student tosses a 1.5 kg mass with a speed
of 2.7 m/s to the student on the stool, who catches it at a
distance of 0.40 m from the axis of rotation.
(a) Does the kinetic energy of the mass-student-stooI system
increase, decrease, or stay the same as the mass is caught?
(b) Calculate the initial and final kinetic energy of the system.
Answers
(a) It decreases, because energy is dissipated in the
collision between the mass and the student’s hand.
(b) Initial = 5.5 J
Final = 0.30 J
Answers
154
Initial = 133 J
Final = 35.0 J
150
A disk-shaped merry-go-round of radius 2.63 m and mass 155
kg rotates freely with an angular speed of 0.641 rev/s. A 59.4
kg person running tangential to the rim of the merry-go-round
at 3.41 m/s jumps onto its rim and holds on. Before jumping on
the merry-go-round, the person was moving in the same
direction as the merry-go-round's rim. What is the final angular
speed of the merry-go-round?
A turntable with a moment of inertia of 5.4 x 10-3 kg.m rotates
freely with an angular speed of 33 1/3 rpm. Riding on the rim
of the turntable, 15 cm from the center, is a 1.3-g cricket.
(a) If the cricket walks to the center of the turntable, will the
turntable rotate faster, slower, or at the same rate? Explain.
(b) Calculate the angular speed of the turntable when the
cricket reaches the center.
Answers
(a) faster
(b) 3.5 rad/s
Answers
2.84 rad/s
151
A disk-shaped merry-go-round of radius 2.63 m and mass 155
kg rotates freely with an angular speed of 0.641 rev/s. A 59.4
kg person running tangential to the rim of the merry-go-round
at 3.41 m/s jumps onto its rim and holds on. Before jumping on
the merry-go-round, the person was moving in the same
direction as the merry-go-round's rim.
(a) does the kinetic energy of the system increase, decrease, or
stay the same when the person jumps on the merry-go-round?
(b) Calculate the initial and final kinetic energies for this
system.
Answers
(a) It must decrease, because some energy will be
dissipated in the “collision” between the person and the
merry-go-round.
(b) Initial = 4.69 J
Final = 4.31 J
155
A student on a piano stool rotates freely with an angular speed
of 2.95 rev/s. The student holds a 1.25 kg mass in each
outstretched arm, 0.759 m from the axis of rotation. The
combined moment of inertia of the student and the stool,
ignoring the two masses, is 5.43 kg.m2, a value that remains
constant.
(a) As the student pulls his arms inward, his angular speed
increases to 3.54 rev / s. How far are the masses from the axis
of rotation at this time, considering the masses to be points?
(b) Calculate the initial and final kinetic energy of the system.
Answers
(a) 0.0334 m
(b) Initial = 1.18 kJ
Final = 1.42 kJ