Rotation
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Review & Summary
Angular Position
To describe the rotation of a rigid body about a fixed axis, called the rotation axis, we assume a reference line is fixed in
the body, perpendicular to that axis and rotating with the body. We measure the angular position θ of this line relative to a
fixed direction. When θ is measured in radians,
(10-1)
where s is the arc length of a circular path of radius r and angle θ. Radian measure is related to angle measure in revolutions
and degrees by
(10-2)
Angular Displacement
A body that rotates about a rotation axis, changing its angular position from θ1 to θ2, undergoes an angular displacement
(10-4)
where
is positive for counterclockwise rotation and negative for clockwise rotation.
Angular Velocity and Speed
If a body rotates through an angular displacement
in a time interval
, its average angular velocity ωavg is
(10-5)
The (instantaneous) angular velocity ω of the body is
(10-6)
Both ωavg and ω are vectors, with directions given by the right-hand rule of Fig. 10-6. They are positive for
counterclockwise rotation and negative for clockwise rotation. The magnitude of the body's angular velocity is the angular
speed.
Angular Acceleration
If the angular velocity of a body changes from ω1 to ω2 in a time interval
αavg of the body is
, the average angular acceleration
(10-7)
The (instantaneous) angular acceleration α of the body is
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Rotation
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(10-8)
Both αavg and α are vectors.
The Kinematic Equations for Constant Angular Acceleration
Constant angular acceleration
is an important special case of rotational motion. The appropriate kinematic
equations, given in Table 10-1, are
(10-12)
(10-13)
(10-14)
(10-15)
(10-16)
Linear and Angular Variables Related
A point in a rigid rotating body, at a perpendicular distance r from the rotation axis, moves in a circle with radius r. If the
body rotates through an angle θ, the point moves along an arc with length s given by
(10-17)
where θ is in radians.
The linear velocity
of the point is tangent to the circle; the point's linear speed v is given by
(10-18)
where ω is the angular speed (in radians per second) of the body.
The linear acceleration
of the point has both tangential and radial components. The tangential component is
(10-22)
where α is the magnitude of the angular acceleration (in radians per second-squared) of the body. The radial component of
is
(10-23)
If the point moves in uniform circular motion, the period T of the motion for the point and the body is
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Rotation
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(10-19, 10-20)
Rotational Kinetic Energy and Rotational Inertia
The kinetic energy K of a rigid body rotating about a fixed axis is given by
(10-34)
in which I is the rotational inertia of the body, defined as
(10-33)
for a system of discrete particles and defined as
(10-35)
for a body with continuously distributed mass. The r and ri in these expressions represent the perpendicular distance from
the axis of rotation to each mass element in the body, and the integration is carried out over the entire body so as to include
every mass element.
The Parallel-Axis Theorem
The parallel-axis theorem relates the rotational inertia I of a body about any axis to that of the same body about a parallel
axis through the center of mass:
(10-36)
Here h is the perpendicular distance between the two axes, and
is the rotational inertia of the body about the axis
through the com. We can describe h as being the distance the actual rotation axis has been shifted from the rotation axis
through the com.
Torque
Torque is a turning or twisting action on a body about a rotation axis due to a force
the position vector
relative to the axis, then the magnitude of the torque is
. If
is exerted at a point given by
(10-40, 10-41, 10-39)
where Ft is the component of
perpendicular to
and
is the angle between
and
perpendicular distance between the rotation axis and an extended line running through the
line of action of
, and
is called the moment arm of
The SI unit of torque is the newton-meter
. The quantity
is the
vector. This line is called the
. Similarly, r is the moment arm of Ft.
. A torque τ is positive if it tends to rotate a body at rest counterclockwise
and negative if it tends to rotate the body clockwise.
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Newton's Second Law in Angular Form
The rotational analog of Newton's second law is
(10-45)
where
is the net torque acting on a particle or rigid body, I is the rotational inertia of the particle or body about the
rotation axis, and α is the resulting angular acceleration about that axis.
Work and Rotational Kinetic Energy
The equations used for calculating work and power in rotational motion correspond to equations used for translational
motion and are
(10-53)
and
(10-55)
When τ is constant, Eq. 10-53 reduces to
(10-54)
The form of the work-kinetic energy theorem used for rotating bodies is
(10-52)
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
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