Download Ch 8 Rotational Motion and Equilibrium

Chapter 8 Lecture
Pearson Physics
Rotational Motion
and Equilibrium
Prepared by
Chris Chiaverina
© 2014 Pearson Education, Inc.
Chapter Contents
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Describing Angular Motion
Rolling Motion and the Moment of Inertia
Torque
Static Equilibrium
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Describing Angular Motion
• Rotation is a part of everyday life. We live on a
planet that rotates and revolves around the Sun.
Automobiles have parts that rotate, as do
devices that play CDs and DVDs.
• To gain a better understanding of rotational
motion, we begin by considering the position,
speed, and acceleration of a rotating object.
• A coordinate system with an origin is used to
describe the motion of an object moving in a
straight line. The same thing is done to describe
the motion of a rotating object.
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Describing Angular Motion
• The bicycle wheel shown in the figure below is
free to rotate about its axle.
• The axle is the axis of rotation for the wheel. As
the wheel rotates, every point on the wheel
moves in a circular path centered about the axis
of rotation.
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Describing Angular Motion
• The location of a red spot on the bicycle tire in
the figure is described by its angular position,
that is, the angle θ that it makes with a given
reference line.
• The reference line defines where the angular
position is zero, θ = 0.
• The sign of angular position depends on its
orientation relative to the reference line.
– Counterclockwise rotation from the reference
line corresponds to positive angles, θ > 0.
– Clockwise rotation from the reference line
corresponds to negative angles, θ < 0.
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Describing Angular Motion
• While common units for measuring angles are the
degree () and revolution (rev), the most convenient unit
for angle measurements in scientific calculations is the
radian (rad).
• As the figure below indicates, the radian is the angle for
which the length of a circular arc is equal to the radius of
the circle.
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Describing Angular Motion
• A comparison between angles measured in
degrees and in radians is shown in the figure
below. Angles are indicated around the
circumference of the circle in both degrees and
radians.
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Describing Angular Motion
• As is illustrated in the following figure, the radian
is useful because it provides a simple
relationship between angles and arc lengths.
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Describing Angular Motion
• Multiplying the angle θ (measured in radians) by
the radius gives the arc length s: s = rθ.
• This relationship does not hold for degrees or
revolutions.
• Since the arc length corresponding to the
circumference of a circle equals 2πr, it follows
from the relationship s = rθ that a complete
revolution corresponds to θ = 2π radians.
Therefore,
1 rev = 360 = 2π rad
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Describing Angular Motion
• The units for angles—radians, degrees, and
revolutions—are all dimensionless. In the
relation s = rθ, for example, the arc length (s)
and the radius (r) both have the SI unit of meter;
hence the angle has no dimensions.
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Describing Angular Motion
• As the figure below indicates, as the bicycle
wheel rotates, the angular position of the spot of
red paint changes.
• The angular displacement of the spot, Δθ, is
defined as the difference between its final angle
and its initial angle:
Δθ = θf − θi
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Describing Angular Motion
• The angular displacement divided by the time
during which the displacement occurs is the
average angular velocity, ωav.
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© 2014 Pearson Education, Inc.
Describing Angular Motion
• Notice that ω is called the angular velocity, not
the angular speed. This is because ω can be
positive or negative, depending on the direction
of rotation.
– Counterclockwise rotation corresponds to
positive angular velocity, ω > 0.
– Clockwise rotation corresponds to negative
angular velocity, ω < 0.
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Describing Angular Motion
• The sign of ω indicates the direction of angular
velocity, as shown in the figure below. The
magnitude of the angular velocity is called the
angular speed.
• The following example illustrates how the
angular velocity is calculated.
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© 2014 Pearson Education, Inc.
Describing Angular Motion
• The angular and linear velocities of an object
moving in a circle are related.
• At any instant, an object moving in a circle is
also moving in a direction tangent to the circular
path.
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Describing Angular Motion
• The object's tangential speed is equal to the
circumference of the circular path, d = 2πr,
divided by the time required to complete one
circuit, t = T. This gives the following:
vt = d/t = 2πr/T = r(2π/T)
• Since 2π/T is the angular velocity ω, the
tangential speed of a rotating object may be
expressed as follows:
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© 2014 Pearson Education, Inc.
Describing Angular Motion
• If the angular velocity of a rotating object
increases or decreases with time, then we say
that the object has an angular acceleration.
• The average angular acceleration, αav, is the
change in angular velocity, Δω, in a given time
interval, Δt.
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© 2014 Pearson Education, Inc.
Describing Angular Motion
• The sign of the angular acceleration is determined by
whether the change in angular velocity is positive or
negative.
– If ω and α have the same sign, then the speed of
rotation is increasing.
– If ω and α have opposite signs, then the speed of
rotation is decreasing.
• These relationships are summarized in the following
figure.
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Describing Angular Motion
• The following example illustrates an application of the
definition of average angular acceleration.
• Just as linear speed is equal to the radius times the
angular speed, the linear acceleration is equal to the radius
times the angular acceleration:
tangential acceleration = radius x angular
acceleration
αt = rα
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© 2014 Pearson Education, Inc.
Rolling Motion and the Moment of Inertia
• As a bicycle wheel rolls freely, with no slipping
between the tire and the ground, the wheel
rotates about its axle, and at the same time the
axle moves in a straight line.
• Thus, the rolling motion of a wheel is a
combination of both rotational motion and linear
motion.
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Rolling Motion and the Moment of Inertia
• Because the forward motion of the axle exactly
cancels out the backward motion of the bottom
of the wheel, the instantaneous speed of the
bottom of the wheel is zero.
• The top of the wheel has twice the speed of the
axle. Thus, if a car's speed is v, as read on the
speedometer, then the tops of its wheels have
the speed 2v.
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Rolling Motion and the Moment of Inertia
• Due to its inertia, an object at rest or moving in a
straight line resists changes to that motion. A
similar relationship occurs with rotational motion.
• Each object has a moment of inertia, I, which
determines how easy or hard it is to change its
rotation. An object with a large moment of inertia
is difficult to start or stop rotating. For example, a
merry-go-round has a large moment of inertia,
whereas a baseball has a relatively small
moment of inertia.
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Rolling Motion and the Moment of Inertia
• Experiments show that an object's moment of
inertia depends linearly on the mass of the
object and on the square of the distance to the
object from the axis of rotation.
• For a mass m at a distance r from the axis of
rotation, the moment of inertia is I = mr2.
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© 2014 Pearson Education, Inc.
Rolling Motion and the Moment of Inertia
• In a system with several particles with different
masses and at different distances from the axis
of rotation, the moment of inertia is the sum of
the mr2 terms for each object.
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© 2014 Pearson Education, Inc.
Rolling Motion and the Moment of Inertia
• The preceding results are valid for individual
particles or groups of particles. What about a
solid object, like a disk or sphere?
• The table below contains the moment of inertia
for several uniform, rigid objects of various
shapes and total mass m.
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Rolling Motion and the Moment of Inertia
• For linear motion, the kinetic energy of a mass m
moving with a speed v is
.
• To find the kinetic energy of a rotating object, we
replace the linear speed v with the angular
speed ω and the mass m with the moment of
inertia I, which we can think of as rotational
mass.
• Thus the rotational kinetic energy of an object is
one-half the product of the moment of inertia
and the square of the angular speed.
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Rolling Motion and the Moment of Inertia
• Thus the rotational kinetic energy of an object is
one-half the product of the moment of inertia
and the square of the angular speed.
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© 2014 Pearson Education, Inc.
Rolling Motion and the Moment of Inertia
• Connections between linear and rotational quantities
such as these are summarized in the table below.
• We can apply these connections to momentum. By
replacing mass with the moment of inertia and linear
speed with angular speed, we obtain the angular
momentum, L.
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© 2014 Pearson Education, Inc.
© 2014 Pearson Education, Inc.
Torque
• A force can cause a rotation. The effectiveness of the
force depends on where the force is applied and in which
direction.
• In the figure below, we see that the force required far
from the nut is much less than the force required near
the nut. Less force is needed to open a revolving door if
the force is applied far from the door's axis of rotation.
r1
F1
r2
r2
F2
F2
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(a)
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(b)
r1
F1
Torque
• Thus the effectiveness of a force in causing a
rotation depends on both the magnitude of the
force and the distance from the axis of rotation
to the force. Taking this information into account,
we define a physical quantity called torque.
Torque, , is the product of force and distance.
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© 2014 Pearson Education, Inc.
Torque
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© 2014 Pearson Education, Inc.
Torque
• The figure on the left below shows that a force acting in
a direction that is directly toward or away from the axis of
rotation will cause no rotation. That is, radial forces
produce zero torque.
• The figure on the right below shows that if the force is at
an angle θ relative to the radial line, then only the
tangential component of the force will produce a torque.
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Torque
• Therefore, the torque in the general case is
distance times the tangential component of the
force, rF cosθ:
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© 2014 Pearson Education, Inc.
Torque
• The figure below shows how torque can be defined in
terms of a quantity known as a moment arm.
• The perpendicular distance from the axis of rotation to
the line of the force is defined as the moment arm. From
the figure, the length of the moment arm is r = r sinθ.
Referring to the general equation for torque,
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Torque
• The sign associated with a torque is determined
by the type of angular rotation it would produce.
– A torque that causes a counterclockwise
rotation is defined to be positive.
– A torque that causes a clockwise rotation is
defined to be negative.
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Torque
• The figure below illustrates a system with more
than one torque. The net torque acting on the
system is the sum of the individual torques,
taking into account the appropriate sign for
each.
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© 2014 Pearson Education, Inc.
Torque
• We can once again draw on the connections
between linear and rotational quantities when
considering the effect of torque on an object.
Replacing the force with torque, , and the linear
acceleration with angular acceleration, α, we
obtain the rotational version of Newton's second
law:
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© 2014 Pearson Education, Inc.
Torque
• Physical quantities that are conserved, such as
energy and linear momentum, play an important
role in physics. Angular momentum is another
quantity that is conserved.
• Angular momentum is conserved when the net
torque is zero.
• The motion of an ice skater may be used to
illustrate how angular momentum is conserved.
In the following figure, the skater pulls in her
arms, reducing her momentum of inertia.
However, since her angular momentum must
stay the same, her angular speed must increase.
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Torque
• In general, a large I and a small ω give the same
angular momentum as a small I and a large ω:
Iω = Iω
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Static Equilibrium
• When an object is at rest and remaining at rest, it
is said to be in a state of static equilibrium.
• The conditions for static equilibrium are the
following:
– The total force acting on the object must be
zero. This ensures that there is no linear
acceleration.
– The total torque acting on the object must also
be zero. This ensures that there is no angular
acceleration.
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Static Equilibrium
• An application of the these two conditions to the
plank that supports the child in the figure below
may be used to calculate the forces F1 and F2
acting on the plank.
3L/ 4
L/ 4
F2
F1
Axis of rotation
y
mg
x
• The total force acting on the plank must be zero;
therefore,
F1 + F2 – mg = 0
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© 2014 Pearson Education, Inc.
Static Equilibrium
• The total torque acting on the plank may be
found by choosing the left end of the plank to be
the axis of rotation. With this choice:
– Force F1 exerts zero torque.
– Force F2 produces a torque of +F2L since it
acts at the far end of the plank in the
counterclockwise direction.
– The weight of the child mg exerts a torque of
–mg
since it acts in the clockwise direction
at a distance from the axis.
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Static Equilibrium
• Setting the total torques equal to zero, we find
0 + F2L – mg(¾L) = 0
Therefore,
and
.
• In this example, the left end of the plank was
arbitrarily selected as the axis of rotation. In
general, you are free to chose an axis of rotation
that is most convenient for a given situation.
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Static Equilibrium
• The center of mass (CM) is the point at which an
object can be balanced. In many ways, an object
behaves as if all of its mass were concentrated
at its center of mass.
• The location of the center of mass for two
objects may be calculated using the following
simple equation.
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© 2014 Pearson Education, Inc.
Static Equilibrium
• The figure below shows the center of mass of
two objects. The center of mass is closer to the
larger mass, or equidistant between the masses
The center of mass
if they are equal.
(CM) is closer to the
m1
m1
m1
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more massive object.
CM
m2
m2
CM
CM
m2
• When an object is supported at its center of
mass, it is in static equilibrium since the torques
acting on the object are balanced.
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Static Equilibrium
• When the center of mass of an object is directly
below the suspension point, the torque due to
gravity is zero. This is because the force of
gravity acts through the axis of rotation.
• Therefore, if you allow an object with any shape
to hang freely, its center of mass will be directly
below its suspension point.
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Static Equilibrium
• The center of mass of an irregularly shaped
object, such as a piece of wood cut into the
shape of the continental United States, may be
located by suspending the model from two or
more points. The center of mass lies at the
intersection of vertical lines extending downward
from the suspension points.
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