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Chapter 10 problems

# Chapter 10 problems - Review Summary Angular Position To...

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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)
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 integration is carried out over the entire body so as to include every mass element.

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

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