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Similar to instantaneous velocity, we can define the Instantaneous Angular Velocity A change in the Angular Velocity gives Average Angular Acceleration (rads/s 2 ) Example A fan takes 2.00 s to reach its operating angular speed of 10.0 rev/s. What is the average angular acceleration (rad/s 2 )?

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Solution: Given: t f =2.00 s, f =10.0 rev/s Recognize: t0=0, 0 =0, and that f needs to be converted to rads/s

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Equations of Rotational Kinematics Just as we have derived a set of equations to describe ``linear’’ or ``translational’’ kinematics, we can also obtain an analogous set of equations for rotational motion Consider correlation of variables Translational Rotational xdisplacement vvelocity aacceleration ttimet

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Replacing each of the translational variables in the translational kinematic equations by the rotational variables, gives the set of rotational kinematic equations We can use these equations in the same fashion we applied the translational kinematic equations

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Example: Problem 8.16 A figure skater is spinning with an angular velocity of +15 rad/s. She then comes to a stop over a brief period of time. During this time, her angular displacement is +5.1 rad. Determine (a) her average angular acceleration and (b) the time during which she comes to rest. Solution: Given: f =+5.1 rad, 0 =+15 rad/s Infer: 0 =0, f =0, t 0 =0 Find: , t f ?

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(a) Use last kinematic equation (b) Use first kinematic equation

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Or use the third kinematic equation Example: Problem 8.26 At the local swimming hole, a favorite trick is to run horizontally off a cliff that is 8.3 m above the water, tuck into a ``ball,’’ and rotate on the way down to the water. The average angular speed of rotation is 1.6 rev/s. Ignoring air resistance,

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Solution: Given: 0 = f = 1.6 rev/s, y 0 = 8.3 m Also, v 0y = 0, t 0 = 0, y f = 0 Recognize: two kinds of motion; 2D projectile motion and rotational motion with constant angular velocity. Method: #revolutions = = t. Therefore, need to find the time of the projectile motion, t f. determine the number of revolutions while on the way down. x y y0y0 yfyf v0v0

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Consider y-component of projectile motion since we have no information about the x- component.

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Tangential Velocity and Acceleration z r v For one complete revolution, the angular displacement is 2 rad From our studies of Uniform Circular Motion (Chap. 5), we know that the time for a complete revolution is a period T Therefore the angular frequency can be written

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Also from Chap. 5, we know that the speed for an object in a circular path is Tangential velocity (rad/s) The tangential velocity corresponds to the speed of a point on a rigid body, a distance r from its center, rotating at an angular velocity Each point on the rigid body rotates at the same angular velocity, but its tangential speed depends on its location r r vTvT r=0

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