Download About what axis is the rotational inertia of your body the least? 1

1
About what axis is the rotational inertia of your body the least?
5
Answer
Vertical Axis
2
The figure shows three small spheres that rotate about a
vertical axis. The perpendicular distance between the axis and
the center of each sphere is given. Rank the three spheres
according to their rotational inertia about that axis, greatest
first.
If two circular disks of the same weight and thickness are made
from metals having different densities, which disk, if either,
will have the larger rotational inertia about its central axis?
Answer
2
For both disks I = 1/2MR . The disk with the greater
density has the smaller radius and the smaller rotational
inertia.
Answer
All the same
3
Five solids are shown in cross section (shown in the figure).
The cross sections have equal heights and equal maximum
widths. The solids have equal masses.
(a) Which one has the largest rotational inertia about a
perpendicular axis through the center of mass? Explain your
logic
(b) Which the smallest rotational inertia about a perpendicular
axis through the center of mass? Explain your logic
6
The figure shows three 0.0100 kg particles that have been
glued to a rod of length L = 6.00 cm and negligible mass. The
assembly can rotate around a perpendicular axis through point
0 at the left end. We remove one particle (that is, 33% of the
mass).
(a) By what percentage does the rotational inertia of the
assembly around the rotation axis decrease when that removed
particle is the innermost one?
(b) By what percentage does the rotational inertia of the
assembly around the rotation axis decrease when that removed
particle is the outermost one?
Answer
The hoop has the largest rotational inertia since all of the
matter is at a large distance form the axis. On the other
hand, most of the matter comprising the prism is relatively
close to the axis so this object has the smallest rotational
inertia.
4
Why is it more difficult to do a sit-up with your hands behind
your head than when they are outstretched in front of you? A
diagram may help you to answer this.
Answer
Answer
(a) 7.14 %
(b) 64.28 %
7
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. What is the
rotational inertia of the body about axis A?
11
An oxygen molecule consists of two oxygen atoms whose total
-36
mass is 5.3 x 10 g and whose moment of inertia about an axis
perpendicular to the line joining the two atoms, midway
-46
2
between them, is 1.9 x 10 kg.m Estimate, from these data,
the effective distance between the atoms.
-6
6.69 x 10 m
if assume hollow sphere
-6
1.89 x 10 m
If assume point sources
-4
1.89 x 10 m
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Answer
.020 kg.m
Tons of dust and small particles rain down onto the Earth from
space everyday. As a result, does the Earth's moment of inertia
increase, decrease, or stay the same? Explain.
2
Answers
8
In principle, the Earth's moment of inertia increases
because both its mass and radius increase.
A tennis ball has a mass of 57 g and a diameter of 7 cm. Find
the moment of inertia about its diameter. Assume the ball is a
thin spherical shell.
13
Why should changing the axis of rotation of an object change
its moment of inertia, given that its shape and mass remain the
same?
Answer
-5
2
4.65 x 10 kg.m
9
Answers
Calculate the moment of inertia of a 14.0 kg solid sphere of
radius 0.623 m when the axis of rotation is through its center.
Answer
-5
2
2.17 x 10 kg.m
10
Calculate the moment of inertia of a 66.7 cm diameter bicycle
wheel. The rim and tire have a combined mass of 1.25 kg. The
mass of the hub can be ignored. (Why?)
Answer
0.139 kg.m
2
The moment of inertia of an object changes with the
position of the axis of rotation because the distance from
the axis to all the elements of mass have been changed. It
is not just the shape of an object that matters, but the
distribution of mass with respect to the axis of rotation.
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The L-shaped object in the figure can be rotated in one of the
following three ways: (i) about the x axis; (ii) about the y axis;
and (iii) about the z axis (which passes through the origin
perpendicular to the plane of the figure.)
(a) In which of these cases is the object's moment of inertia
greatest?
(b) In which case is it least? Explain.
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The L-shaped object in the figure can be rotated in one of the
following three ways: (i) about the x axis; (ii) about the y axis;
and (iii) about the z axis (which passes through the origin
perpendicular to the plane of the figure.)
(a) In which of these cases is the object's moment of inertia
greatest?
(b) In which case is it least? Explain.
Answers
Answers
The moment of inertia of the object is greatest when it is
rotated about the z axis. It is least when rotated about the x
axis.
The moment of inertia of the object is greatest when it is
rotated about the z axis. It is least when rotated about the x
axis.
The minute and hour hands of a clock have a common axis of
rotation and equal mass. The minute hand is long and thin; the
hour hand is short and thick. Which hand has the greatest
moment of inertia?
18
What is the moment of inertia of the wheel of mass of 1 kg and
an axle with a mass of 3 kg as seen in the figure . (You can
ignore the spokes in this problem)
Answers
The long, thin minute hand – with mass far from the axis
of rotation – has the greater moment of inertia.
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The moment of inertia of a 0.98 kg bicycle wheel rotating
2
about its center is 0.13 kg . m . What is the radius of this
wheel, assuming the weight of the spokes can be ignored?
Answer
-2
1.06 x 10 kgm
Answers
0.36 m
2
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The pulley shown in the figure is rotated by the falling masses.
The diameter of the wheel is 25 cm, has a thickness of 5 cm
and has a mass of 15 kg. The spokes of the wheel are each 5
cm wide, 5 cm thick. and has a mass of 2 kg. Determine the
moment of inertia of the pulley.
22
A bicycle wheel of radius 0.3 m has a rim of mass 1.0 kg and
50 spokes, each of mass 0.01 kg. What is its moment of inertia
about its axis of rotation?
Answer
2
0.105 kgm
23
In the figure the solid cylinder has a mass of 100 grams, a
radius of 3 cm and a length of 10 cm. The 20 gram metal plate
which bisects the rod is 1 cm thick, and 8 cm by 8 cm.
(a) Determine the moment of inertia of the object when it is
rotated around the x-axis
(b) Determine the moment of inertia of the object when it is
rotated around the y-axis.
Answer
.1674 kg,m
20
2
Small blocks, each of mass m, are clamped at the ends and at
the center of a light rigid rod of length L. Compute the
moment of inertia of the system about an axis perpendicular to
the rod and passing through a point one quarter of the length
from one end. Neglect the moment of inertia of the rod.
Answer
Answer
-5
11/16 mL
21
2
(a) 6.633 x 10 kgm
-4
2
(b) 1.167 x 10 kgm
2
A thin rectangular sheet of steel is 0.3 m by 0.4 m and has
mass 24 kg.
(a) Find the moment of inertia about an axis Through the
center, parallel to the long sides;
(b) Find the moment of inertia about an axis through the
center, parallel to the short sides;
(c) Find the moment of inertia about an axis through the
center, perpendicular to the plane.
(d) Do you notice anything strange about your results?
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A flywheel consists of a solid disk 0.5 m in diameter and 0.02
m thick, and two projecting hubs 0.1 m in diameter and 0.1 m
long. If the material of which it is constructed has a density of
3
6000 kg/m , find its moment of inertia about the axis of
rotation.
Answer
2
(a) .18 kg,m
2
(b) .32 kg.m
2
(c) .5 kg.m
(d) The answers to a + b add to c since the thickness is
zero
Answer
2
0.748 kgm
25
A solid sphere and a pulley are both rotated by a falling mass.
The solid cylinder has a diameter of 15 cm and a mass of 135
kg. The Solid metal disk has a diameter of 15 cm a thickness
of 10 cm and a mass of 125 kg. Determine the moment of
inertia of each object.
28
Calculate the moment of inertia of a 14.0 kg solid sphere of
radius 0.623 m when the axis of rotation is through its center.
Answer
2
2.17 kgm
29
Calculate the moment of inertia of a 66.7 cm diameter bicycle
wheel. The rim and tire have a combined mass of 1.25 kg. The
mass of the hub can be ignored. (Why?)
Answer
2
.139 kg.m
Answer
30
-1
2
Isphere = 3.038 x 10 kgm
-1
2
Idisk = 3.516 x 10 kgm
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In the figure, two particles, each with mass m = 0.85 kg, are
fastened to each other, and to a rotation axis at 0, by two thin
rods, each with length d = 5.6 cm and mass M = 1.2 kg. The
combination rotates around the rotation axis with angular speed
= 0.30 rad/s. What are the combination's rotational inertia
about O?
Find the moment of inertia of a disk of radius R and mass M
about an axis in the plane of the disk that passes through its
center.
Answer
2
0.023 kg.m
Answer
2
2
1/4 mR + 1/12 mR
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A uniform rectangular plate has mass m and sides a and b.
(a) What is the moment of inertia about an axis that is
perpendicular to the plate and passes through its center of
mass?
(b) Show by integration that the moment of inertia of the plate
about an axis that is perpendicular to the plate and passes
2
2
through one corner is (1/3)m(a + b ).
Answer
(a)
(b)
31
The figure shows an arrangement of 15 identical disks that
have been glued together in a rod-like shape of length L =
1.0000 m and (total) mass M = 100.0 mg. The arrangement can
rotate about a perpendicular axis through its central disk at
point 0.
(a) What is the rotational inertia of the arrangement about that
axis?
(b) If`we approximated the arrangement as being a uniform rod
of mass M and length L, what percentage error would we make
to calculate the rotational inertia?
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What is the moment of inertia of the wheel (mass of 1 kg) and
axle (mass of 3 kg) shown in the figure if it is rotated on an
axis parallel to the axle and 15 cm from the center of the axle.
You may ignore the spokes in this case.
Answer
Answer
-3
2
(a) 8.32 x 10 kg.m
-3
2
(b) 8.32 x 10 kg.m
32
An oxygen molecule consists of two oxygen atoms whose total
-36
mass is 5.3 x 10 kg and whose moment of inertia about an
axis perpendicular to the line joining the two atoms, midway
-46
2
between them, is 1.9 x 10 kg.m . Estimate, from these data,
the effective distance between the atoms.
-2
10.06 x 10 kgm
0
35
2
Compute the moment of inertia of a baton of dimensions
shown in the figure about a vertical axis that is parallel with
an axis that runs through the center of gravity (CG) and is
located 10 cm from the CG axis.
Answer
-5
1.19 x 10 m
33
Calculate the rotational inertia of a meter stick, with mass 0.56
kg, about an axis perpendicular to the stick and located at the
20 cm mark. (Treat the stick as a thin rod.)
Answer
-2
Answer
2
9.7 x 10 kg m
2
6.49 slug ft
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A rod 4 cm in diameter and 2 m long has a mass of 8 kg.
(a) Find the moment of inertia about an axis perpendicular to
the rod and passing through its center
(b) Find the moment of inertia about an axis perpendicular to
the rod and passing through one end,
(c) Find the moment of inertia about a longitudinal axis
passing through the center of the rod.
Answer
2
(a) 2.667 kgm
2
(b) 10.667 kgm
2
(c) .0016 kgm
37
A flywheel consists of a solid disk 0.5 m in diameter and 0.02
m thick, and two projecting hubs 0.1 m in diameter and 0.1 m
long. If the material of which it is constructed has a density of
3
4000 kg/m Find its moment of inertia if it is shifted 20 cm
from its normal axis of rotation.
39
In the figure the solid cylinder has a mass of 100 grams, a
radius of 3 cm and a length of 10 cm. The 20 gram metal plate
which bisects the rod is 1 cm thick, and 8 cm by 8 cm.
Determine the moment of inertia of the object when it is
rotated on an axis located parallel to the x axis and 20 cm from
it.
Answer
1.38 kgm
38
2
The four bodies shown in the figure have equal masses m.
Body A is a solid cylinder of radius R. Body B is a hollow thin
cylinder of radius R. Body C is a solid square with length of
side 2R. Body D is the same size as C, but hollow (i.e.) made
up of four thin sticks). The bodies have axes of rotation
perpendicular to the page and through the center of each body.
(a) Calculate the moment of inertia of the solid disk.
(b) Calculate the moment of inertia of the ring
(c) Calculate the moment of inertia of the solid cube
(d) Calculate the moment of inertia of the open box
(e) Show mathematically which body has the smallest moment
of inertia?
(f) Show mathematically which body has the largest moment
of inertia?
Answer
-3
4.866 x 10
40
The diagram shows 4 thin walled cylinders arranged
concentrically around the y-axis. The central cylinder has a
diameter of 20 cm and has a length of 36 cm. Each cylinder
increases in diameter by 10 cm. Each cylinder weighs 45 g.
Determine the moment inertia of the object if it is rotated on an
axis located 45 cm from the Y axis.
Answer
2
(a) (mR )/2
2
(b) mR
2
(c) 2/3 mR
2
(d) (16mR )/3
(e) Disk
(f) Open Cube
Answer
-2
4.25 x 10 kgm
2
41
The figure shown consists of a of metal rod, a thick walled
cylinder and a thin walled cylinder. Each object is 25 cm in
length. The rod has a diameter of 10 cm and weighs 300
grams. The thick walled cylinder has an inside diameter of 30
cm and an outside diameter of 45 cm . it weighs 420 grams.
The outer cylinder has a diameter of 66 cm and weighs 100
grams.
Determine the moment inertia of the object if it is rotated on an
axis parallel to the walls of the cylinder and located 120 cm
from the center of the central rod.
43
Answer
Answer
1.208 kgm
2
44
42
Answer the following questions:
(a) Find the moment of inertia I, for the four-particle system of
the figure above about the x-axis, which passes through m3 and
m4.
(b) Find Iy for the system about the y-axis, which passes
through m1 and m4
(c) Find the moment of inertia I,, about the z-axis, which
passes through m4 and is perpendicular to the plane of the
figure.
Use the parallel-axis theorem find the moment of inertia of the
four-particle system in the figure about an axis that is
perpendicular to the plane of the masses and passes through the
center of mass of the system.
Answer the following questions:
(a) Use the parallel-axis theorem to find the moment of inertia
about an axis that is parallel to the z axis and passes through
the center of mass of the system in the figure.
(b) Let x' and y' be axes in the plane of the figure that pass
through the center of mass and are parallel to the sides of the
rectangle. Compute Ix' and Iy' and use your results and Equation
8-30 to check your result for part
Answer
Answer
45
Use the parallel-axis theorem to find the moment of inertia of a
solid sphere of mass M and radius R about an axis that is
tangent to the sphere.
47
The figure shows a book-like object (one side is longer than
the other) and four choices of rotation axes, all perpendicular to
the face of the object. Rank the choices according to the
rotational inertia of the object about the axis, greatest first.
Answer
Answer
1, 2, 4, 3
46
The figure shows a pair of uniform spheres, each of mass 500
g and radius 5 cm. They are mounted on a uniform rod that has
a length L = 30 cm and a mass m = 60 g.
(a) Calculate the moment of inertia of this system about an
axis perpendicular to the rod through the center of the rod
using the approximation that the two spheres can be treated as
point particles that are 20 cm from the axis of rotation and that
the mass of the rod is negligible.
(b) Calculate the moment of inertia exactly and compare your
result with your answer for part a.
48
Figure a shows a rigid body consisting of two particles of mass
m connected by a rod of length L and negligible mass.
(a) What is the rotational inertia Icom about an axis through the
center of mass, perpendicular to the rod as shown?
(b) What is the rotational inertia I of the body about an axis
through the left end of the rod and parallel to the first axis (see
figure b)?
Answer
Answer
2
(a) 1/2 mL
2
(b) mL
49
The figure shows a thin, uniform rod of mass M and length L,
on an x axis with the origin at the rod's center.
(a) What is the rotational inertia of the rod about the
perpendicular rotation axis through the center?
(b) What is the rod's rotational inertia I about a new rotation
axis that is perpendicular to the rod and through the left end?
52
The uniform solid block in the figure has mass 0.172 kg and
edge lengths a = 3.5 cm, b = 8.4 cm, and c = 1.4 cm. Calculate
its rotational inertia about an axis through one corner and
perpendicular to the large faces.
Answer
Answer
-4
2
4.74 x 10 kg.m
50
The figure shows a uniform metal plate that had been square
before 25% of it was snipped off. Three lettered points are
indicated. Rank them according to the rotational inertia of the
plate around a perpendicular axis through them, greatest first.
Answer
c, a, b
51
Figure a shows a disk that can rotate about an axis that is
perpendicular to its face and at a radial distance h from the
center of the disk. figure b gives the rotational inertia I of the
disk about the axis as a function of that distance h, from the
center out to the edge of the disk. What is the mass of the
disk?
Answer
M = 2.5 kg
53
Four identical particles of mass 0.50 kg each are placed at the
vertices of a 2.0 m x 2.0 m square and held there by four
massless rods, which form the sides of the square. What is the
rotational inertia of this rigid body about an axis that
(a) What is the rotational inertia of this rigid body about an
axis that passes through the midpoints of opposite sides and
lies in the plane of the square?
(b) What is the rotational inertia of this rigid body about an
axis thatpasses through the midpoint of one of the sides and is
perpendicular to the plane of the square?
(c) What is the rotational inertia of this rigid body about an
axis thatlies in the plane of the square and passes through two
diagonally opposite particles?
Answer
54
The masses and coordinates of four particles are as follows: 50
g, x = 2.0 cm, y = 2.0 cm; 25 g, x = 0, y = 4.0 cm; 25 g, x =
-3.0 cm, y = -3.0 cm; 30 g, x = -2.0 cm, y = 4.0 cm.
(a) What is the rotational inertia of this collection about the x
axes?
(b) What is the rotational inertia of this collection about the y
axes?
(c) What is the rotational inertia of this collection about the z
axes?
(d) Suppose the answers to (a) and (b) are A and B,
respectively. Then what is the answer to (c) in terms of A and
B?
56
Answer
Answer
3
2
2
(a) 1.3 x 10 g cm
2
2
(b) 5.5 x 10 g cm
2
2
(c) 1.9 x 10 g cm
(d) A + B
(a) .27 kg m
2
(b) .22 kg m
2
(c) .10 kg m
57
55
Three 0.50 kg particles form an equilateral triangle with 0.60 m
sides. The particles are connected by rods of negligible mass.
What is the rotational inertia of this rigid body about
(a) What is the rotational inertia of this rigid body about an
axis that passes through one of the particles and is parallel to
the rod connecting the other two,
(b) What is the rotational inertia of this rigid body about an
axis that passes through the midpoint of one of the sides and is
perpendicular to the plane of the triangle, and
(c) What is the rotational inertia of this rigid body about an
axis that is parallel to one side of the triangle and passes
through the midpoints of the other two sides?
Answer the following questions.
(a) Show that the rotational inertia of a solid cylinder of mass
M and radius R about its central axis is equal to the rotational
inertia of a thin hoop of mass M and radius R/√2about its
central axis.
(b) Show that the rotational inertia I of any given body of mass
M about any given axis is equal to the rotational inertia of an
equivalent hoop about that axis, if the hoop has the same mass
M and a radius k given by the equation. The radius k of the
equivalent hoop is called the radius of gyration of the given
body.
In the figure , m = 1.5 kg and M = 3.0 kg. The array is
rectangular and it is split through the middle by the horizontal
axis. Assume the objects are wired together by very light rigid
pieces of wire.
(a) Calculate the moment of inertia of the array of point
objects shown in the figure about the vertical axis.
(b) Calculate the moment of inertia of the array of point
objects shown in the figure about the horizontal axis.
(c) About which axis would it be harder to accelerate this
array?
Answer
Answer
58
Many machines employ cams for various purposes, such as
opening and closing valves. In the figure, the cam is a circular
disk rotating on a shaft that does not pass through the center of
the disk. In the manufacture of the cam, a uniform solid
cylinder of radius R is first machined. Then an off-center hole
of radius R/2 is drilled, parallel to the axis of the cylinder, and
centered at a point a distance R/2 from the center of the
cylinder. The cam, of mass M, is then slipped onto the circular
shaft and welded into place. Assume R = 10 cm and M = 40
kg. The circular shaft is 20 kg. Calculate the Moment of
Inertia of the Cam Shaft.
60
Use the parallel-axis theorem find the moment of inertia of the
four-particle system in the figure about an axis that is
perpendicular to the plane of the masses and passes through the
center of mass of the system.
Answer
61
Answer
.326 kg m
59
2
Suppose a bicycle wheel is rotated about an axis through its
rim and parallel to its axle. Is its moment of inertia greater
than, less than, or the same as when it rotates about its axle?
Explain.
Answers
The moment of inertia is greater when the axis of rotation
is on the rim of the wheel. The reason is that much of the
wheel’s mass is now at a significantly greater distance
from the axis of rotation, compared with the case where the
axis is at the center of the wheel.
A uniform disk of radius 30 cm, thickness 3 cm, and mass 5 kg
rotates at omega ( ) = 10 rad/s about an axis parallel to the
symmetry axis but 0.5 cm from that axis.
(a) Find the net force on the bearings due to this imbalance.
(b) Where should a 100 g mass be placed on the disk to correct
this problem?
Answer
(a) .245 nm
(b) .25 m
62
A uniform, 100 kg cylinder of radius 0.60 m is placed flat on
some smooth ice. Two skaters wind ropes around the cylinder
in the same sense. The skaters then pull on their ropes as they
skate away, exerting constant forces of 40 N and 60 N,
respectively, for 5 s.
(a) Describe the motion of the cylinder.
(b) What is its angular acceleration?
(c) What is its angular velocity at the end of the 5 seconds?
Answer
(a)
2
(b) 3.33 rad/s
2
(c) 16.66 rad/s
63
A wheel mounted on an axis that is not frictionless is initially
at rest. A constant external torque of 50 N.m is applied to the
wheel for 20 s. At the end of the 20 s, the wheel has an angular
velocity of 600 rev/min. The external torque is then removed
and the wheel comes to rest after 120 s more.
(a) What is the moment of inertia of the wheel?
(b) What is the frictional torque, which is assumed to be
constant?
67
Answer
Answer
68
2
(a) 13.66 kg m
(b) 7.10 nm
64
A softball player swings a bat, accelerating it from rest to 3.0
rev/s in a time of 0.20 s. Approximate the bat as a 2.2 kg
uniform rod of length 0.95 m, and compute the torque the
player applies to one end of it.
A 2.0-kg stone is tied to a 0.50 m string and swung around a
circle at a constant angular velocity of 12 rad/s. The circle is
parallel to the xy-plane and is centered on the z-axis, 0.75 m
from the origin. The magnitude of the torque about the origin
is:
A day-care worker pushes tangentially on a small hand-driven
merry-go-round and is able to accelerate it from rest to a
spinning rate of 30 rpm in 10.0 s. Assume the merry-go-round
is a disk of radius 2.5 m and has a mass of 800 kg, and two
children (each with a mass of 25 kg) sit opposite each other on
the edge.
(a) Calculate the torque required to produce the acceleration,
neglecting frictional torque.
(b) What force is required?
Answer
Answer
.108 Nm
69
65
The bolts on the cylinder head of an engine require tightening
to a torque of 80 m-N. If a wrench is 30 cm long, what force
perpendicular to the wrench must the mechanic exert at its
end? If the six-sided bolt head is 15 mm in diameter, estimate
the force applied near each of the six points by a socket wrench
(see the figure).
A uniform rod of mass M and length L can pivot freely (i.e.,
we ignore friction) about a hinge attached to a wall, as in the
figure. The rod is held horizontally and then released. See the
figure .
(a) At the moment of release, determine the angular
acceleration of the rod.
(b) At the moment of release, determine the linear acceleration
of the tip of the rod. Assume that the force of gravity acts at the
center of mass of the rod, as shown.
Answer
Answer
66
A small 1.05-kg ball on the end of a light rod is rotated in a
horizontal circle of radius 0.900 m.
(a) Calculate the moment of inertia of the system about the
axis of rotation.
(b) Calculate the torque needed to keep the ball rotating at
constant angular velocity if air resistance exerts a force of
0.0800 N on the ball.
Answer
.01
70
A wheel of radius 0.250 m, which is moving initially at 43.0
m/s, rolls to a stop in 225 m.
(a) Calculate its linear acceleration.
(b) Calculate its angular acceleration.
2
(c) The wheel's rotational inertia is 0.155 kg.m about its
central axis. Calculate the torque exerted by the friction on the
wheel, about the central axis.
73
The figure is an overhead view of a horizontal bar that can
pivot; two horizontal forces act on the bar, but it is stationary.
0
If the angle between the bar and F2 is now decreased from 90
and the bar is still not to turn, should F2 be made larger, made
smaller, or left the same?
Answer
71
The figure shows an overhead view of a meter stick that can
pivot about the dot at the position marked 20 (for 20 cm). All
five forces on the stick are horizontal and have the same
magnitude. Rank the forces according to the magnitude of the
torque they produce, greatest first.
Answer
larger
74
The figure shows an overhead view of a horizontal bar that is
rotated about the pivot point by two horizontal forces, F1 and
F2, with F2 at angle to the bar, Rank the following values of
according to the magnitude of the angular acceleration of the
0
0
0
bar, greatest first: 90 , 70 , and 110 .
Answer
72
The figure shows an overhead view of a meter stick that can
pivot about the point indicated, which is to the left of the stick's
midpoint. Two horizontal forces, F1 and F2, are applied to the
stick. Only F1 is shown. Force F2 is perpendicular to the stick
and is applied at the right end.
(a) If the stick is not to turn, what should be the direction of F2?
(b) If the stick is not to turn, should F2 be greater than, less
than, or equal to F1?
Answer
(a) F2 down
(b) less
Answer
90•, then 70• and 110• tie
75
In the overhead view of the figure, five forces of the same
magnitude act on a strange merry-go-round; it is a square that
can rotate about point P, at midlength along one of the edges.
Rank the forces according to the magnitude of the torque they
create about point P, greatest first.
77
The figure shows three flat disks (of the same radius) that can
rotate about their centers like merry-go-rounds. Each disk
consists of the same two materials, one denser than the other
(density is mass per unit volume). In disks 1 and 3, the denser
material forms the outer half of the disk area. In disk 2, it forms
the inner half of the disk area. Forces with identical magnitudes
are applied tangentially to the disk, either at the outer edge or
at the interface of the two materials, as shown.
(a) Rank the disks according to the torque about the disk
center.
(b) Rank the disks according to the rotational inertia about the
disk center.
(c) Rank the disks according to the angular acceleration of the
disk, greatest first.
Answer
F5, F4, F2, F1, F3 (zero)
76
In the figure, two forces F1 and F2 act on a disk that turns about
its center like a merry-go-round. The forces maintain the
indicated angles during the rotation, which is counterclockwise
and at a constant rate, However, we are to decrease the angle
of F1 without changing the magnitude of F1.
(a) To keep the angular speed constant, should we increase,
decrease, or maintain the magnitude of F2?
(b) Does force F1 tend to rotate the disk clockwise or
counterclockwise?
(c) Does force F2 tend to rotate the disk clockwise or
counterclockwise?
Answer
(a) 1 and 2 tie, then 3;
(b) 1 and 3 tie, then 2;
(c) 2, 1, 3
78
The length of a bicycle pedal arm is 0.152 m, and a downward
force of 111 N is applied to the pedal by the rider.
(a) What is the magnitude of the torque about the pedal arm's
0
pivot when the arm is at angle 30 with the vertical?
(b) What is the magnitude of the torque about the pedal arm's
0
pivot when the arm is at angle 90 with the vertical?
(c) What is the magnitude of the torque about the pedal arm's
0
pivot when the arm is at angle 180 with the vertical?
Answer
(a) 8.4 N.m
(b) 17 N.m
(c) 0
Answer
(a) decrease;
(b) clockwise;
(c) counterclockwise
79
The body in the figure is pivoted at 0, and two forces act on it
as shown. If r1 = 1.30 m, r2 = 2.15 m, F1 = 4.20 N, F2 = 4.90
0
0
N, 1 = 75.0 , and 2 = 60.0 , what is the net torque about the
pivot?
82
If a 32.0 N.m torque on a wheel causes angular acceleration
2
25.0 rad/s , what is the wheel's rotational inertia?
Answer
2
1.28 kg m
83
Answer
The figure shows a uniform disk that can rotate around its
center like a merry-go-round. The disk has a radius of 2.00 cm
and a mass of 20.0 grams and is initially at rest. Starting at time
t = 0, two forces are to be applied tangentially to the rim as
indicated, so that at time t = 1.25 s the disk has an angular
velocity of 250 rad/s counterclockwise. Force F1 has a
magnitude of 0.100 N. What is magnitude F2?
-3.85 N.m
80
The body in the figure is pivoted at 0. Three forces act on it: FA
= 10 N at point A, 8.0 m from 0; FB = 16 N at B 4.0 m from 0;
and FC = 19 at C, 3.0 m from 0. What is the net torque about 0?
Answer
0.140 N
84
Answer
12 N.m
81
During the launch from a board, a diver's angular speed about
her center of mass changes from zero to 6.20 rad/s in 220 ms.
2
Her rotational inertia about her center of mass is 12.0 kg.m .
During the launch, what is the magnitude of
(a) During the launch, what is the magnitude of her average
angular acceleration?
(b) During the launch, what is the magnitude of the average
external torque on her from the board?
In the figure, a cylinder having a mass of 2.0 kg can rotate
about its central axis through point 0. Forces are applied as
shown: F1 = 6.0 N, F2 = 4.0 N, F3 = 2.0 N, and F4 = 5.0 N. Also,
r = 5.0 cm and R = 12 cm.
(a) Find the magnitude of the angular acceleration of the
cylinder.
(b) Find the direction of the angular acceleration of the
cylinder. (During the rotation, the forces maintain their same
angles relative to the cylinder.)
Answer
(a) 28.2 rad/s
(b) 338 N.m
2
Answer
2
(a) 9.7 rad/s
(b) The direction is counterclockwise (which is the positive
sense of rotation)
85
The figure shows particles 1 and 2, each of mass m, attached to
the ends of a rigid massless rod of length L1 + L2, with L1 = 20
cm and L2 = 80 cm. The rod is held horizontally on the fulcrum
and then released.
(a) What is the magnitude of the initial accelerations of particle
1?
(b) What is the magnitude of the initial accelerations of particle
2?
87
-3
2
A pulley, with a rotational inertia of 1.0 x 10 kg.m about its
axle and a radius of 10 cm, is acted on by a force applied
tangentially at its rim. The force magnitude varies in time as F
2
= 0.50t + 0.30t , with F in newtons and t in seconds. The pulley
is initially at rest.
(a) At t = 3.0 s what is its angular acceleration?
(b) At t = 3.0 s what is its angular speed?
Answer
2
2
(a) 4.2 x 10 rad/s
2
(b) 5.0 x 10 rad/s
88
Answer
2
(a) 1.7 m/s
2
(b) 6.9 m/s
86
In figure a, an irregularly shaped plastic plate with uniform
thickness and density (mass per unit volume) is to be rotated
around an axle that is perpendicular to the plate face ind
through point 0. The rotational inertia of the plate about that
axle is measured with the following method. A circular disk of
mass 0.500 kg and radius 2.00 cm is glued to the plate, with its
center aligned with point 0 (figure b). A string is wrapped
around the edge of the disk the way a string is wrapped around
a top. Then the string is pulled for 5.00 s. As a result, the disk
and plate are rotated by a constant force of 0.400 N that is
applied by the string tangentially to the edge of the disk. The
resulting angular speed is 114 rad/s. What is the rotational
inertia of the plate about the axle?
Beverage engineering. The pull tab was a major advance in the
engineering design of beverage containers. The tab pivots on a
central bolt in the can's top. When you pull upward on one end
of the tab, the other end presses downward on a portion of the
can's top that has been scored. If you pull upward with a 10 N
force, approximately what is the magnitude of the force applied
to the scored section? (You will need to examine a can with a
pull tab.)
Answer
25 N
89
Answer
--4
2
2.51 x 10 kg m
Two uniform solid spheres have the same mass of 1.65 kg, but
one has a radius of 0.226 m and the other has a radius of 0.854
m. Each can rotate about an axis through its center.
(a) What is the magnitude of the torque required to bring the
smaller sphere from rest to an angular speed of 317 rad/s in
15.5 s?
(b) What is the magnitude F of the force that must be applied
tangentially at the sphere's equator to give that torque?
(c) What is the corresponding values of for the larger sphere?
(d) What is the corresponding values of F for the larger sphere?
Answer
(a) .689 N.m
(b) 3.05 N
(c) 9.84 Nm
(c) 11.5 N
90
A small ball with mass 1.30 kg is mounted on one end of a rod
0.780 m long and of negligible mass. The system rotates in a
horizontal circle about the other end of the rod at 5010 rev/min.
(a) Calculate the rotational inertia of the system about the axis
of rotation.
-2
(b) There is an air drag of 2.30 x 10 N on the ball, directed
opposite its motion. What torque must be applied to the system
to keep it rotating at constant speed?
93
A uniform disk of radius 0.12 m and mass 5 kg is pivoted in
such that it rotates freely about its axis. 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
3 s?
(d) Find the total angle the disk turns in the 3 seconds.
Answer
2
(a) .791 kg m
-2
(b) 1.79 x 10 N.m
91
A bicyclist of mass 70 kg puts all his mass on each downwardmoving pedal as he pedals up a steep road. Take the diameter
of the circle in which the pedals rotate to be 0.40 m, and
determine the magnitude of the maximum torque he exerts.
Answer
Answer
2
1.4 x 10 N.m
94
92
A uniform, hollow, cylindrical spool has an outside radius R =
12 cm, inside radius R/2, and mass M = 5 kg. It is mounted so
that it rotates on a fixed horizontal axle of mass 10 kg
Calculate the moment of inertia of the object
A thin spherical shell has a radius of 1.90 m. An applied torque
of 960 N.m gives the shell an angular acceleration of 6.20
2
rad/s about an axis through the center of the shell.
(a) What is the rotational inertia of the shell about that axis?
(b) What is the mass of the shelf?
Answer
2
(a) 155 kg.m
(b) 64.4 kg
95
A potter's wheel—a thick stone disk of radius 0.500 m and
mass 100 kg—is freely rotating at 50.0 rev/min. The potter can
stop the wheel in 6.00 s by pressing a wet rag against the rim
and exerting a radially inward force of 70.0 N. Find the
effective coefficient of kinetic friction between wheel and rag.
Answer
Answer
96
Find the net torque on the wheel in the Figure about the axle
through O if a = 10.0 cm and b = 25.0 cm.
100
Force F = (2.0 N)i + (3.0 N)k acts on a pebble with postion
vector r = (0.50 m)j – (2.0 m)k, relative to the origin.
(a) What is the resulting torque acting on the the pebble about
the origin?
(b) What is the resulting torque acting on a point with
coordinates (2.0 m, 0, -3.0 m)
Answer
(a) (-1.5i – 4.0j – k) N.m
(b) (-1.5i – 4.0j – k) N.m
101
Answer
97
What is the net torque about the origin on a flea located at
coordinates (0, -4.0 m, 5.0 m) when forces F1 = (3.0 N)k and F2
= (-2.0 N)j act on the flea?
Answer
Given that r = ix + y j + zk and F = Fxi + Fyj + Fzk, show that
the torque = r X F is given by
-2.0i N.m
= (yFz –zFy)i + (zFx – x Fz)j + x(Fy – yFx)k
102
Answer
98
Show that, if r and F lie in a given plane, the torque
has no component in that plane
Answer
=rXF
(a) 50 k N.m
(b) 90 degrees
Answer
103
99
Answer the following questions:
(a) What are the magnitude and direction of torque about the
origin on a particle located at coordinates (0, -4.0 m, 3.0 m)
due to a force F1 with components Fx1 = 2.0 N and F1y = F1z =
0.
(b) What are the magnitude and direction of torque about the
origin on a particle located at coordinates (0, -4.0 m, 3.0 m)
due toa force F2 with components F2x = 0, F2y = 2.0 N, and F2z =
4.0 N?
Answer
(a) 10 N.m parallel to the yz plane at 53 degrees to the +y
axis.
(b) 22 N.m, -x
Force F = (-8.0 N)i + (6.0 N) j acts on a partice with position
vector r = (3.0 m)i + (4.0 m)j.
(a) What is the torque of the particle around the origin?
(b) What is the angle between the directions of r and F?
At t = 0, a 2.0 kg particle has position vector r = (4.0 m)i – (2.0
4
m)j relative to the origin. Its velocity is given by v = (-6t m/s)
i + (3.0 m/s)j. What is the torque acting on the particle about
the origin and for t > 0.
Answer
3
-96t k N.m
104
At t = 0, a 2.0 kg particle has position vector r = (4.0 m)i – (2.0
4
m)j relative to the origin. Its velocity is given by v = (-6t m/s)
i + (3.0 m/s)j. What is the torque acting on the particle at
coordinates (-2.0 m. –3.0 m, 0) and for t > 0.
Answer
3
48t k N.m
105
The position vector r of a particle relative to a certain point has
a magnitude of 3 m, and the force F on the particle has a
magnitude of 4 N.
(a) What is the angle between the directions of r and F if the
magnitude of the associated torque equals zero?
(b) What is the angle between the directions of r and F if the
magnitude of the associated torque equals 2 N.m?
108
Answer
106
Answer the following questions:
(a) What is the torque about the origin on a jar of jalapeno
peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to a
force F1 = (3.0 N)i - (4.0 Nj + (5.0 N)k,
(b) What is the torque about the origin on a jar of jalapeno
peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to a
force F2 = (-3.0 N)i - (4.0 N)j - (5.0 N)k
(c) What is the torque about the origin on a jar of jalapeno
peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to the
vector sum of F1 and F2?
(d) Repeat (c) for a point with coordinates (3.0 m, 2.0 m, 4.0
m) instead of the origin.
Answer the following questions.
(a) What is the torque about the origin on a jar of jalapeno
peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to a
force F1 = (3.0 N)i – (4.0 N)j – (5.0 N)k
(b) What is the torque about the origin on a jar of jalapeno
peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to a
force F2 = (-3.0 N)i – (4.0 N)j – (5.0 N)k
(c) What is the torque about the origin on a jar of jalapeno
peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to the
sum of force F1 = (3.0 N)i – (4.0 N)j – (5.0 N)k and F2 = (-3.0
N)i – (4.0 N)j – (5.0 N)k.
(d) What is the torque about the origin on a jar of jalapeno
peppers located at coordinates (3.0 m, 2.0 m, 4.0 m) due to the
sum of force F1 = (3.0 N)i – (4.0 N)j – (5.0 N)k and F2 = (-3.0
N)i – (4.0 N)j – (5.0 N)k.
Answer
(a)
(b)
(c)
(d)
109
Answer
(6.0i – 3.0j – 6.0k) N.m
(26i + 3.0j – 18k) N.m
(32i – 24k) N.m
0
At a certain time the position vector in meters of a 0.25 kg
object is r = 2.0i – 2.0k. At that instant, its velocity in meters
is v = -5.0i + 5.0k., and the force in newtons action on it is F =
4.0 j. What torque acts on it?
Answer
107
The figure shows plots of angular position versus time t for
three cases in which a disk is rotated like a merry-go- round. In
each ease, the rotation direction changes at a certain angular
position change.
(a) For each case, determine whether change, is clockwise or
counterclockwise from = 0, or whether it is at = 0.
(b) For each case, determine whether is zero before, after, or
at t = 0?
(c) For each case, determine whether is positive, negative, or
zero.
(8.1i + 8.0k) N.m
110
A 3.0 kg particle is x = 3.0 m, y = 8.0 m with a velocity of v =
(5.0 m/s)i + (6.0 m/s)j. It is acted on by a 7.0 N force in the
negative x direction. What torque about the origin acts on the
particle?
Answer
+56k N.m
111
Answer
A performer, seated on a trapeze, is swinging back and forth
with a period of 8.85 s. If she stands up, thus raising the center
of mass of the system trapeze + performer by 35.0 cm, what
will be the new period of the trapeze? Treat trapeze +
performer as a simple pendulum.
Answers
My solution
8.77 seconds
112
A disk whose radius R is 12.5 cm is suspended, as a physical
pendulum, from a point at distance h from its center C (See the
figure on the left). Its period T is 0.871 s when h = R/2. What
is the free-fall acceleration g at the location of the pendulum?
115
A pendulum is formed by pivoting a long thin rod of length L
and mass m about a point on the rod that is a distance d above
the center of the rod.
(a) Find the period of this pendulum in terms of d, L, m, and g,
assuming that it swings with small amplitude
(b) What happens to the period if d is decreased,
(c) What happens to the period if L is increased?
(d) What happens to the period if m is increased?
Answers
116
A meter stick swinging from one end oscillates with a
frequency f0. What would be the frequency, in terms of f0, if
the bottom half of the stick were cut off?
Answers
Answers
113
The figure at the left shows three physical pendulums
consisting of identical uniform spheres of the same mass that
are rigidly connected by identical rods of negligible mass. Each
pendulum is vertical and can pivot about suspension point 0.
Rank the pendulums according to period of oscillation, greatest
first.
117
A stick with length L oscillates as a physical pendulum,
pivoted about point 0 in the figure at the left.
(a) Derive an expression for the period of the pendulum in
terms of L and x, the distance from the point of support to the
center of mass of the pendulum.
(b) For what value of x/L is the period a minimum?
2
(c) Show that if L = 1.00 m and g = 9.80 m/s this minimum is
1.53 s.
Answers
114
A physical pendulum consists of a meter stick that is pivoted at
a small hole drilled through the stick a distance x from the 50
cm mark. The period of oscillation is observed to be 2.5 s. Find
the distance x.
Answers
Answers
118
The center of oscillation of a physical pendulum has this
interesting property: if an impulsive force (assumed horizontal
and in the plane of oscillation) acts at the center of oscillation,
no reaction is felt at the point of support. Baseball players (and
players of many other sports) know that unless the ball hits the
bat at this point (called the "sweet spot" by athletes), the
reaction due to the impact will sting their hands. To prove this
property, let the stick in the figure at the left simulate a
baseball bat. Suppose that a horizontal force F (due to impact
with the ball) acts toward the right at P, the center of
oscillation. The batter is assumed to hold the bat at 0, the point
of support of the stick.
(a) What acceleration does point 0 undergo as a result of F?
(b) What angular acceleration is produced by F about the
center of mass of the stick?
(c) As a result of the angular acceleration in (b), what linear
acceleration does point 0 undergo?
(d) Considering the magnitudes and directions of the
accelerations in (a) and (c), convince yourself that P is indeed
the "sweet spot."
Answers