Chapter%208

Chapter%208 - COLLEGE PHYSICS, Part I Chapter 8: Rotational...

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COLLEGE PHYSICS, Part I Chapter 8: Rotational Kinematics Rotational Motion and Angular Displacement Angular Velocity and Angular Acceleration The Equations of Rotational Kinematics Angular Variables and Tangential Variables Centripetal and Tangential Acceleration Rolling Motion The Vector Nature of Angular Variables Kinetic Energy of Rotation and Moment of Inertia
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Rotational Motion and Angular Displacement We will consider rotational motion of rigid bodies , i.e. objects that do not undergo any kind of deformation and maintain their shape constant . Consider rotation about a fixed axis , i.e. an axis that does not change its location or direction . If there is a straight line within the object that passes through the axis of rotation, then the angle θ the line makes with the positive direction of the x -axis is a coordinate that describes the object’s rotational position . The angle is usually measured in radians rather than in degrees. A radian is an angle corresponding to an arc equal in length to the radius of the circle. When θ = 1 rad, s = r When θ = 2 rad, s = 2r When θ = n rad, s = nr rad r s θ = r s rad =
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180 o ( π ) y x + θ r = 1 0 o 90 o ( π /2) 270 o (3 π /2) 360 o (2 π ) x y deg deg 2 360 360 2 rad o o rad θ π = × = × Because the circumference of a circle is c = 2 π r , the angle in radians corresponding to a complete revolution is But we also know that the angle corresponding to a complete revolution is 360 o . From these two expressions we get: 2 2 = = r r rad O deg 360 θ = O O 57.3 360 1rad = =
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Angular Velocity When a rigid body rotates through an angular displacement ∆θ = θ 2 θ 1 in a time interval t = t 2 t 1 , the average angular velocity ϖ av is defined as: Unit: rad / s The instantaneous angular velocity is the limit of av as t approaches zero: All parts of a rigid body have the same angular velocity. Angular velocity can also be measured in revolutions per minute ( rpm ). 1 rpm = 2 π rad/min = 2 π /60 rad/s Example: 12,000 rpm = 1,257 rad/s = 72,000 deg/s t t t av θ = - - = 1 2 1 2 t t lim 0 =
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When the angular velocity of a rigid body changes from ϖ 1 to 2 during a time interval from t 1 to t 2 , then the object has an angular acceleration. The average angular acceleration is defined as: The instantaneous angular acceleration is: Angular Acceleration α av t t t = - - = 2 1 2 1 Unit: rad / s 2 = t t 0 lim The sign of angular velocity is determined the same way as the sign of an angle: Counterclockwise = positive Clockwise = negative
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Example: A compact disk spins at 7200 rpm. a)
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Chapter%208 - COLLEGE PHYSICS, Part I Chapter 8: Rotational...

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