Rotation 8/23/15, 2:04 PM

Print this page 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 , the average angular acceleration αavg of the body is

(10-7)

The (instantaneous) angular acceleration α of the body is

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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 http://edugen.wileyplus.com/edugen/courses/crs7165/halliday9781118230…k5NzgxMTE4MjMwNzI1YzEwLXNlYy0wMDUwLnhmb3Jt.enc?course=crs7165&id=ref Page 2 of 4 Rotation 8/23/15, 2:04 PM

(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 . If is exerted at a point given by the position vector relative to the axis, then the magnitude of the torque is

(10-40, 10-41, 10-39)

where Ft is the component of perpendicular to and is the angle between and . The quantity is the perpendicular distance between the rotation axis and an extended line running through the vector. This line is called the

line of action of , and is called the moment arm of . Similarly, r is the moment arm of Ft. The SI unit of torque is the newton-meter . 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)

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