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Unformatted text preview: 1 radians r r 2 2 = = A relationship exists between linear and angular quantities. To do this all angles must be measured in ___________. Instead of linear displacement we can now talk about ______ displacement o radian 3 . 57 1 = o radians 360 2 = ) degrees in ( 180 ) ( o radians in = i f = f i (in radians) Rotation of a Rigid Body Chapter 13 ____________ ____________ ____________ A complete circle has an angle of 360 o . In terms of radians, a complete circle has an arc length, s, equal to its circumference. s = C = __________ Arc Length This angle can be defined as the ratio of the arc length s to the radius r; that is, The angle has no dimensions but we call this unit of angular measure a radian. Similarly, we can define the arc length as s = r s r ____________ ____________ We have defined the angular displacement as = f I Where all angles are in radians. This is similar to a linear displacement. Average angular speed: Note: this looks like The instantaneous angular speed, , is defined as: t t t i f i f = = t x v = dt d t lim t = = Angular Velocity ____________ ____________ Angular Acceleration We can determine the average angular acceleration by taking the ratio of the angular velocity to the elapsed time. Note: this looks like The instantaneous angular acceleration, , is defined as: t t t i f i f = = t v a = dt d t lim t = = Interesting note: when a solid rotates about a fixed axis, every portion of the object has the same _________ velocity and _______________! ____________ 2 Rotational Kinematics Linear Motion Rotational Motion Position x Angular Position Velocity v Angular Velocity Acceleration a Angular acceleration Table 13.1 Relation of Angular and Linear Quantities Looking at the tangential velocity of a point on the circle, we can define the velocity of that point as We can further the relationship by remembering that s=r , such that Where the derivative of the angular position is the angular velocity, dt ds v = dt d r dt ds v = = = r v Every point on the object has the same _________ speed, but not the same ____________ speed! r v = r ____________ Angular Velocity and Acceleration Figure 13.5 We derived the relationship between the linear and angular velocities. Taking the time derivative of this function Where the derivative of the angular velocity is the angular acceleration. Relation of Angular and Linear Quantities = r v dt d r dt dv a t = = = r a t Only if the disk is speeding up or slowing down will there be a ______________ acceleration! r a t = r ____________ ____________ 3 Rotation about the Center of Mass ... m m ... x m x m x m M 1 x 2 1 2 2 1 1 i i i cm + + + + = = ... m m ... y m y m y m M 1 y 2 1 2 2 1 1 i i i cm + + + + = = The center of mass is the massweighted center of the object. = xdm M 1...
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This note was uploaded on 04/17/2008 for the course PHYSICS 122 taught by Professor Pope during the Spring '08 term at Clemson.
 Spring '08
 pope
 Power

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