Chapter 10 problems - Review & Summary Angular Position...

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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 θ 1 to θ 2 , undergoes an angular displacement (10-4) where Δ θ is positive for counterclockwise rotation and negative for clockwise rotation. Angular Velocity and Speed If a body rotates through an angular displacement Δ θ in a time interval Δ t , its average angular velocity ω avg is (10-5) The (instantaneous) angular velocity ω of the body is (10-6) Both ω avg 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 ω 1 to ω 2 in a time interval Δ t = t 2 - t 1 , the average angular acceleration α avg of the body is (10-7) The (instantaneous) angular acceleration α of the body is (10-8) Both α avg and α are vectors. The Kinematic Equations for Constant Angular Acceleration Constant angular acceleration ( α constant) is an important special case of rotational motion. The appropriate kinematic equations, given in Table 10-1, are (10-12)
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(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 (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)
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for a body with continuously distributed mass. The r and r i in these expressions represent the perpendicular distance from the axis of rotation to each mass element in the body, and the
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This note was uploaded on 01/12/2012 for the course PHYSICS phy 280 taught by Professor Griffo during the Fall '07 term at Bergen Community College.

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Chapter 10 problems - Review & Summary Angular Position...

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