of the spacecraft around the same axis is I
5 5.00 3
105kg?m2. Neither the spacecraft nor the gyroscope
is originally rotating. The gyroscope can be powered
up in a negligible period of time to an angular speed
of 100 rad/s. If the orientation of the spacecraft is to
be changed by 30.08, for what time interval should the
gyroscope be operated?
43. The angular momentum vector of a precessing gyro-
scope sweeps out a cone as shown in Figure P11.43. The
angular speed of the tip of the angular momentum vec-
tor, called its precessional frequency, is given by v
t/L, where t is the magnitude of the torque on the gyro-
scope and L is the magnitude of its angular momen-
tum. In the motion called precession of the equinoxes, the
Earth’s axis of rotation precesses about the perpendicu-
lar to its orbital plane with a period of 2.58 3 104 yr.
Model the Earth as a uniform sphere and calculate the
torque on the Earth that is causing this precession.
angular momentum vector
sweeps out a cone in space.
44. A light rope passes over a light,
frictionless pulley. One end is fas-
tened to a bunch of bananas of
mass M, and a monkey of mass M
clings to the other end (Fig. P11.44).
The monkey climbs the rope in
an attempt to reach the bananas.
(a) Treating the system as consist-
ing of the monkey, bananas, rope,
and pulley, find the net torque on
the system about the pulley axis.
(b) Using the result of part (a),
determine the total angular momen-
tum about the pulley axis and describe the motion of
the system. (c)Will the monkey reach the bananas?
45. Comet Halley moves about the Sun in an elliptical
orbit, with its closest approach to the Sun being about
0.590 AU and its greatest distance 35.0 AU (1 AU 5 the
Earth–Sun distance). The angular momentum of the
comet about the Sun is constant, and the gravitational
force exerted by the Sun has zero moment arm. The
comet’s speed at closest approach is 54.0 km/s. What is
its speed when it is farthest from the Sun?
46. Review. Two boys are sliding toward each other on a
frictionless, ice-covered parking lot. Jacob, mass 45.0 kg,
is gliding to the right at 8.00 m/s, and Ethan, mass
31.0kg, is gliding to the left at 11.0 m/s along the same
der. (b) Is the mechanical energy of the clay–cylinder
system constant in this process? Explain your answer.
(c) Is the momentum of the clay–cylinder system con-
stant in this process? Explain your answer.
40. Why is the following situation impossible? A space station
shaped like a giant wheel has a radius of r 5 100 m and
a moment of inertia of 5.00 3 108 kg ? m2. A crew of
150 people of average mass 65.0 kg is living on the rim,
and the station’s rotation causes the crew to experience
an apparent free-fall acceleration of g (Fig. P11.29).
A research technician is assigned to perform an experi-
ment in which a ball is dropped at the rim of the station
every 15 minutes and the time interval for the ball to
drop a given distance is measured as a test to make sure
the apparent value of g is correctly maintained. One
evening, 100 average people move to the center of the
station for a union meeting. The research technician,
who has already been performing his experiment for an
hour before the meeting, is disappointed that he cannot
attend the meeting, and his mood sours even further by
his boring experiment in which every time interval for
the dropped ball is identical for the entire evening.
41. A 0.005 00-kg bullet traveling horizontally with speed
1.00 3 103 m/s strikes an 18.0-kg door, embedding itself
10.0 cm from the side opposite the hinges as shown in
Figure P11.41. The 1.00-m wide door is free to swing
on its frictionless hinges. (a) Before it hits the door,
does the bullet have angular momentum relative to the
door’s axis of rotation? (b) If so, evaluate this angu-
lar momentum. If not, explain why there is no angular
momentum. (c) Is the mechanical energy of the bullet–
door system constant during this collision? Answer
without doing a calculation. (d)At what angular speed
does the door swing open immediately after the colli-
sion? (e) Calculate the total energy of the bullet–door
system and determine whether it is less than or equal
to the kinetic energy of the bullet before the collision.
0.005 00 kg
An overhead view of a bullet striking a door.
Section 11.5 The Motion of Gyroscopes and Tops
42. A spacecraft is in empty space. It carries on board a
gyroscope with a moment of inertia of I
5 20.0 kg ? m2
about the axis of the gyroscope. The moment of inertia
RasterEdge Product Refund Policy
Refund Agreement that we will email to you. controls, PDF document, image to pdf files and for capturing, viewing, processing, converting, compressing and create html email from pdf; pdf to html converters
chapter 11 Angular Momentum
Assuming m and d are known, find (a) the moment
of inertia of the system of three particles about the
pivot, (b) the torque acting on the system at t 5 0,
(c) the angular acceleration of the system at t 5 0,
(d) the linear acceleration of the particle labeled 3 at
t 5 0, (e) the maximum kinetic energy of the system,
(f) the maximum angular speed reached by the rod,
(g) the maximum angular momentum of the system,
and (h) the maximum speed reached by the particle
50. Two children are playing on stools at a restaurant coun-
ter. Their feet do not reach the footrests, and the tops
of the stools are free to rotate without friction on ped-
estals fixed to the floor. One of the children catches a
tossed ball, in a process described by the equation
(a) Solve the equation for the unknown v
. (b) Com-
plete the statement of the problem to which this
equation applies. Your statement must include the
given numerical information and specification of the
unknown to be determined. (c) Could the equation
equally well describe the other child throwing the ball?
Explain your answer.
51. A projectile of mass m moves to the right with a speed v
(Fig. P11.51a). The projectile strikes and sticks to the end
of a stationary rod of mass M, length d, pivoted about
a frictionless axle perpendicular to the page through
O (Fig. P11.51b). We wish to find the fractional change
of kinetic energy in the system due to the collision.
(a) What is the appropriate analysis model to describe
the projectile and the rod? (b)What is the angular
momentum of the system before the collision about an
axis through O? (c)What is the moment of inertia of
the system about an axis through O after the projectile
sticks to the rod? (d)If the angular speed of the system
after the collision is v, what is the angular momentum
of the system after the collision? (e) Find the angular
speed v after the collision in terms of the given quanti-
line. When they meet, they grab each other and hang
on. (a) What is their velocity immediately thereafter?
(b) What fraction of their original kinetic energy is
still mechanical energy after their collision? That was
so much fun that the boys repeat the collision with the
same original velocities, this time moving along paral-
lel lines 1.20 m apart. At closest approach, they lock
arms and start rotating about their common center of
mass. Model the boys as particles and their arms as a
cord that does not stretch. (c) Find the velocity of their
center of mass. (d) Find their angular speed. (e) What
fraction of their original kinetic energy is still mechani-
cal energy after they link arms? (f) Why are the answers
to parts (b) and (e) so different?
47. We have all complained that there aren’t enough hours
in a day. In an attempt to fix that, suppose all the peo-
ple in the world line up at the equator and all start
running east at 2.50 m/s relative to the surface of the
Earth. By how much does the length of a day increase?
Assume the world population to be 7.00 3 109 people
with an average mass of 55.0kg each and the Earth to
be a solid homogeneous sphere. In addition, depend-
ing on the details of your solution, you may need to use
the approximation 1/(1 2 x) < 11 x for small x.
48. A skateboarder with his board can be modeled as a
particle of mass 76.0 kg, located at his center of mass,
0.500m above the ground. As shown in Figure P11.48,
the skateboarder starts from rest in a crouching posi-
tion at one lip of a half-pipe (point A). The half-pipe
forms one half of a cylinder of radius 6.80 m with its
axis horizontal. On his descent, the skateboarder moves
without friction and maintains his crouch so that his
center of mass moves through one quarter of a circle.
(a) Find his speed at the bottom of the half-pipe (point
B). (b) Find his angular momentum about the center
of curvature at this point. (c)Immediately after passing
point B, he stands up and raises his arms, lifting his
center of gravity to 0.950 m above the concrete (point
C). Explain why his angular momentum is constant in
this maneuver, whereas the kinetic energy of his body
is not constant. (d) Find his speed immediately after he
stands up. (e) How much chemical energy in the skate-
boarder’s legs was converted into mechanical energy in
the skateboarder–Earth system when he stood up?
49. A rigid, massless rod has three particles with equal
masses attached to it as shown in Figure P11.49. The rod
is free to rotate in a vertical plane about a frictionless
axle perpendicular to the rod through the point P and
is released from rest in the horizontal position at t 5 0.
XDoc.Converter for .NET Purchase information
Online Convert PDF to Html. SUPPORT: controls, PDF document, image to pdf files and components for capturing, viewing, processing, converting, compressing and best pdf to html converter online; batch convert pdf to html
XDoc.HTML5 Viewer for .NET Purchase information
Online Convert PDF to Html. SUPPORT: controls, PDF document, image to pdf files and components for capturing, viewing, processing, converting, compressing and convert pdf to html code c#; convert pdf to website html
of mass at speeds of 5.00 m/s. Treating the astronauts
as particles, calculate (a) the magnitude of the angu-
lar momentum of the two-astronaut system and (b) the
rotational energy of the system. By pulling on the rope,
one astronaut shortens the distance between them to
5.00 m. (c) What is the new angular momentum of
the system? (d) What are the astronauts’ new speeds?
(e) What is the new rotational energy of the system?
(f) How much chemical potential energy in the body
of the astronaut was converted to mechanical energy in
the system when he shortened the rope?
56. Two astronauts (Fig. P11.55), each having a mass M,
are connected by a rope of length d having negligible
mass. They are isolated in space, orbiting their center
of mass at speeds v. Treating the astronauts as particles,
calculate (a)the magnitude of the angular momen-
tum of the two-astronaut system and (b) the rotational
energy of the system. By pulling on the rope, one of the
astronauts shortens the distance between them to d/2.
(c) What is the new angular momentum of the system?
(d) What are the astronauts’ new speeds? (e) What is
the new rotational energy of the system? (f) How much
chemical potential energy in the body of the astronaut
was converted to mechanical energy in the system
when he shortened the rope?
57. Native people throughout North and South America
used a bola to hunt for birds and animals. A bola can
consist of three stones, each with mass m, at the ends
of three light cords, each with length ,. The other
ends of the cords are tied together to form a Y. The
hunter holds one stone and swings the other two above
his head (FigureP11.57a). Both these stones move
together in a horizontal circle of radius 2, with speed
. At a moment when the horizontal component of
their velocity is directed toward the quarry, the hunter
releases the stone in his hand. As the bola flies through
the air, the cords quickly take a stable arrangement
with constant 120-degree angles between them (Fig.
P11.57b). In the vertical direction, the bola is in free
fall. Gravitational forces exerted by the Earth make
the junction of the cords move with the downward
. You may ignore the vertical motion as
you proceed to describe the horizontal motion of the
bola. In terms of m, ,, and v
, calculate (a) the mag-
nitude of the momentum of the bola at the moment
of release and, after release, (b)the horizontal speed
of the center of mass of the bola and (c) the angu-
lar momentum of the bola about its center of mass.
(d) Find the angular speed of the bola about its center
of mass after it has settled into its Y shape. Calculate
ties. (f)What is the kinetic energy of the system before
the collision? (g)What is the kinetic energy of the sys-
tem after the collision? (h)Determine the fractional
change of kinetic energy due to the collision.
52.A puck of mass m 5 50.0 g is attached to a taut cord pass-
ing through a small hole in a frictionless, horizontal
surface (Fig. P11.52). The puck is initially orbiting with
5 1.50m/s in a circle of radius r
The cord is then slowly pulled from below, decreasing
the radius of the circle to r 5 0.100 m. (a)What is the
puck’s speed at the smaller radius? (b) Find the tension
in the cord at the smaller radius. (c) How much work is
done by the hand in pulling the cord so that the radius
of the puck’s motion changes from 0.300 m to 0.100 m?
Problems 52 and 53.
53. A puck of mass m is attached to a taut cord passing
through a small hole in a frictionless, horizontal sur-
face (Fig. P11.52). The puck is initially orbiting with
in a circle of radius r
. The cord is then slowly
pulled from below, decreasing the radius of the circle
to r. (a) What is the puck’s speed when the radius is r?
(b) Find the tension in the cord as a function of r.
(c) How much work is done by the hand in pulling the
cord so that the radius of the puck’s motion changes
54. Why is the following situation impossible? A meteoroid strikes
the Earth directly on the equator. At the time it lands,
it is traveling exactly vertical and downward. Due to the
impact, the time for the Earth to rotate once increases
by 0.5 s, so the day is 0.5 s longer, undetectable to layper-
sons. After the impact, people on the Earth ignore the
extra half-second each day and life goes on as normal.
(Assume the density of the Earth is uniform.)
55. Two astronauts (Fig. P11.55), each having a mass of
75.0kg, are connected by a 10.0-m rope of negligible
mass. They are isolated in space, orbiting their center
Problems 55 and 56.
chapter 11 Angular Momentum
rolling occurs. (c) Assume the coefficient of friction
between disk and surface is m. What is the time inter-
val after setting the disk down before pure rolling
motion begins? (d) How far does the disk travel before
pure rolling begins?
62. In Example 11.9, we investigated an elastic collision
between a disk and a stick lying on a frictionless sur-
face. Suppose everything is the same as in the example
except that the collision is perfectly inelastic so that
the disk adheres to the stick at the endpoint at which it
strikes. Find (a) the speed of the center of mass of the
system and (b)the angular speed of the system after
63. A solid cube of side 2a and mass M is sliding on a fric-
tionless surface with uniform velocity v
as shown in
FigureP11.63a. It hits a small obstacle at the end of
the table that causes the cube to tilt as shown in Fig-
ure P11.63b. Find the minimum value of the magni-
tude of v
such that the cube tips over and falls off the
table. Note: The cube undergoes an inelastic collision
at the edge.
64. A solid cube of wood of side 2a and mass M is resting
on a horizontal surface. The cube is constrained to
rotate about a fixed axis AB (Fig. P11.64). A bullet of
mass m and speed v is shot at the face opposite ABCD at
a height of 4a/3. The bullet becomes embedded in the
cube. Find the minimum value of v required to tip the
cube so that it falls on face ABCD. Assume m ,, M.
the kinetic energy of the bola (e) at the instant of
release and (f) in its stable Y shape. (g) Explain how
the conservation laws apply to the bola as its configu-
ration changes. Robert Beichner suggested the idea
for this problem.
58. A uniform rod of mass 300 g and length 50.0 cm
rotates in a horizontal plane about a fixed, frictionless,
vertical pin through its center. Two small, dense beads,
each of mass m, are mounted on the rod so that they
can slide without friction along its length. Initially,
the beads are held by catches at positions 10.0 cm on
each side of the center and the system is rotating at an
angular speed of 36.0 rad/s. The catches are released
simultaneously, and the beads slide outward along the
rod. (a) Find an expression for the angular speed v
the system at the instant the beads slide off the ends of
the rod as it depends on m. (b) What are the maximum
and the minimum possible values for v
and the values
of m to which they correspond?
59. Global warming is a cause for concern because even
small changes in the Earth’s temperature can have sig-
nificant consequences. For example, if the Earth’s polar
ice caps were to melt entirely, the resulting additional
water in the oceans would flood many coastal areas.
Model the polar ice as having mass 2.30 3 1019 kg and
forming two flat disks of radius 6.00 3 105 m. Assume
the water spreads into an unbroken thin, spherical shell
after it melts. Calculate the resulting change in the dura-
tion of one day both in seconds and as a percentage.
60. The puck in Figure P11.60 has a mass of 0.120 kg. The
distance of the puck from the center of rotation is
originally 40.0 cm, and the puck is sliding with a speed
of 80.0 cm/s. The string is pulled downward 15.0 cm
through the hole in the frictionless table. Determine
the work done on the puck. (Suggestion: Consider the
change of kinetic energy.)
61. A uniform solid disk of radius R is set into rotation
with an angular speed v
about an axis through its cen-
ter. While still rotating at this speed, the disk is placed
into contact with a horizontal surface and immedi-
ately released as shown in Figure P11.61. (a) What is
the angular speed of the disk once pure rolling takes
place? (b) Find the fractional change in kinetic energy
from the moment the disk is set down until pure
Balanced Rock in Arches National
Park, Utah, is a 3 000 000-kg
boulder that has been in stable
equilibrium for several millennia.
It had a smaller companion nearby,
called “Chip Off the Old Block,”
that fell during the winter of 1975.
Balanced Rock appeared in an
early scene of the movie Indiana
Jones and the Last Crusade. We will
study the conditions under which
an object is in equilibrium in this
(John W. Jewett, Jr.)
12.1 Analysis Model: Rigid
Object in Equilibrium
12.2 More on the Center of
12.3 Examples of Rigid Objects
in Static Equilibrium
12.4 Elastic Properties of Solids
c h a p p t t e r
In Chapters 10 and 11, we studied the dynamics of rigid objects. Part of this chapter
addresses the conditions under which a rigid object is in equilibrium. The term equilibrium
implies that the object moves with both constant velocity and constant angular velocity
relative to an observer in an inertial reference frame. We deal here only with the special
case in which both of these velocities are equal to zero. In this case, the object is in what
is called static equilibrium. Static equilibrium represents a common situation in engineering
practice, and the principles it involves are of special interest to civil engineers, architects,
and mechanical engineers. If you are an engineering student, you will undoubtedly take an
advanced course in statics in the near future.
The last section of this chapter deals with how objects deform under load conditions. An
elastic object returns to its original shape when the deforming forces are removed. Several
elastic constants are defined, each corresponding to a different type of deformation.
12.1 Analysis Model: Rigid Object in Equilibrium
In Chapter 5, we discussed the particle in equilibrium model, in which a particle
moves with constant velocity because the net force acting on it is zero. The situation
with real (extended) objects is more complex because these objects often cannot be
modeled as particles. For an extended object to be in equilibrium, a second condi-
tion must be satisfied. This second condition involves the rotational motion of the
chapter 12 Static equilibrium and elasticity
(Quick Quiz 12.2)
Three forces act on an object.
Notice that the lines of action of
all three forces pass through a
Consider a single force F
acting on a rigid object as shown in Figure 12.1. Recall
that the torque associated with the force F
about an axis through O is given by
The magnitude of t
is Fd (see Equation 10.14), where d is the moment arm shown
in Figure 12.1. According to Equation 10.18, the net torque on a rigid object causes
it to undergo an angular acceleration.
In this discussion, we investigate those rotational situations in which the angular
acceleration of a rigid object is zero. Such an object is in rotational equilibrium.
Because o t
5 Ia for rotation about a fixed axis, the necessary condition for rota-
tional equilibrium is that the net torque about any axis must be zero. We now have
two necessary conditions for equilibrium of a rigid object:
1. The net external force on the object must equal zero:
2. The net external torque on the object about any axis must be zero:
These conditions describe the rigid object in equilibrium analysis model. The first
condition is a statement of translational equilibrium; it states that the translational
acceleration of the object’s center of mass must be zero when viewed from an iner-
tial reference frame. The second condition is a statement of rotational equilibrium;
it states that the angular acceleration about any axis must be zero. In the special
case of static equilibrium, which is the main subject of this chapter, the object in
equilibrium is at rest relative to the observer and so has no translational or angular
speed (that is, v
5 0 and v 5 0).
Q uick Quiz 12.1 Consider the object subject to the two forces of equal magnitude
in Figure 12.2. Choose the correct statement with regard to this situation.
(a) The object is in force equilibrium but not torque equilibrium. (b)The object
is in torque equilibrium but not force equilibrium. (c) The object is in both
force equilibrium and torque equilibrium. (d) The object is in neither force
equilibrium nor torque equilibrium.
Q uick Quiz 12.2 Consider the object subject to the three forces in Figure 12.3.
Choose the correct statement with regard to this situation. (a)The object is in
force equilibrium but not torque equilibrium. (b) The object is in torque equi-
librium but not force equilibrium. (c) The object is in both force equilibrium
and torque equilibrium. (d) The object is in neither force equilibrium nor
The two vector expressions given by Equations 12.1 and 12.2 are equivalent,
in general, to six scalar equations: three from the first condition for equilibrium
and three from the second (corresponding to x, y, and z components). Hence, in a
complex system involving several forces acting in various directions, you could be
faced with solving a set of equations with many unknowns. Here, we restrict our
discussion to situations in which all the forces lie in the xy plane. (Forces whose
vector representations are in the same plane are said to be coplanar.) With this
restriction, we must deal with only three scalar equations. Two come from balanc-
ing the forces in the x and y directions. The third comes from the torque equa-
tion, namely that the net torque about a perpendicular axis through any point in
the xy plane must be zero. This perpendicular axis will necessarily be parallel to
A single force F
on a rigid object at the point P.
Pitfall Prevention 12.1
Zero Torque Zero net torque does
not mean an absence of rotational
motion. An object that is rotating
at a constant angular speed can
be under the influence of a net
torque of zero. This possibility
is analogous to the translational
situation: zero net force does not
mean an absence of translational
(Quick Quiz 12.1)
Two forces of equal magnitude are
applied at equal distances from
the center of mass of a rigid object.
Documents you may be interested
Documents you may be interested