7 Rotation 2009 - R v a rad 2 = Review: This is also...

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Unformatted text preview: R v a rad 2 = Review: This is also referred to as centripetal acceleration for uniform circular motion. Note the velocity (constant) is simply: T r v 2 = Where T is the period to complete one circle. 2 2 4 T R a rad = Rigid Bodies Real-world bodies can be complicated when forces act on them: They can deform They can twist They can stretch and be squeezed For now we will consider objects that do not change shape and label them rigid bodies. The first concepts to define are: Angular Velocity and Acceleration s = r Notice that is dimensionless. When r = s results in one radian: s/r = 1 = (rad) = (/180) (deg) Angular speed In the limit as t goes to zero the rate of change of angle becomes the derivative: is defined to be positive counterclockwise dt d = t Similarly the rate of change of angular velocity becomes angular acceleration As t goes to zero: When a rigid body rotates every particle rotating about a fixed axis has the same angle in a given time interval. The same angular speed and angular acceleration. and are the magnitudes of the angular velocity and angular acceleration. They are always positive. dt d = t Direction of the angular velocity vector is always along axis of rotation. You must specify the axis of rotation and keep it through out the problem. 1) f = i + t 2) f = i + i t + 1/2t 2 3) f 2 = i 2 + 2( f i ) 4) f- i = ( i + f )t Kinematics of Rotation For constant angular acceleration The angular kinematics are identical to the translation kinematics just replace position x by angle ; replace velocity v by angular velocity , and replace acceleration a by angular acceleration . This result is the tangential speed of a particle a distance r from the axis of rotation. Not every point has the same tangential speed because r to each point changes. However, every point in the rigid body has the same angular speed ( 29 r dt d r r dt d dt ds v T = = = = r v T = r v T = Recall that: The tangential acceleration a t = r Where the radial acceleration a r = v 2 /r = r 2 The total magnitude of acceleration is given by: a = (a t 2 + a r 2 ) = (r 2 2 + r 2 4 ) 1/2 = r( 2 + 4 ) 1/2 ( 29 r dt d r r dt d dt dv a T = = = = Question A bicycle wheel angular position is given by: (t) = (2.0 rad/s 3 )t 3 The diameter of the wheel is 0.36 meters. a) Find the angle , in radians and in degrees at times t 1 = 2.0s and t 2 = 5.0s b) Find the distance a particle moves on the rim during these two time intervals. c) Find the average angular velocity, in rad/s and rev/min during these two time intervals....
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This note was uploaded on 10/05/2011 for the course PHYS 4A taught by Professor Ernest during the Summer '10 term at Irvine Valley College.

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7 Rotation 2009 - R v a rad 2 = Review: This is also...

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