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Rotation of rigid bodies

Rotation of rigid bodies - Chapter 10 Rotation of Rigid...

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1 Chapter 10: Rotation of Rigid Bodies Rotation of Rigid bodies circle6 A rigid body is an object that is non-deformable. barb4left The separations between all pairs of particles remain constant. barb4left All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible

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2 Angular Position Angular Position circle6 Axis of rotation is the center of the disc circle6 Every particle on the disc undergoes circular motion about a fixed axis of origin O , perpendicular to the plane circle6 Consider a point P located at ( r , θ ) where r is the distance from the origin to P and θ is the measured counterclockwise from the reference line circle6 As the particle moves through θ , it moves though an arc length s . circle6 The arc length and r are related: barb4left s = θ r θ s P r circle6 The angular coordinate r s = θ ө is in radian 2π rad = 360 0 θ r s = 1 rad = = 57.3° Angular Displacement, Velocity circle6 The angular displacement is defined as the angle the object rotates through during some time interval f i θ θ θ Δ = f i f i t t t θ θ θ ω Δ = = Δ circle6 The average angular speed, ω , of a rotating rigid object is the ratio of the angular displacement to the time interval circle6 The instantaneous angular speed lim 0 t d t dt θ θ ω Δ → Δ = Δ Unit = rad/s dt dx v x = (analogous to ) ω Δθ ω arrowrightnosp circle6 Direction: Right hand rule
3 Angular Acceleration Angular Acceleration circle6 The average angular acceleration, α , of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change: f i f i t t t ω ω ω α Δ = = Δ circle6 The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0 lim 0 t d t dt ω ω α Δ → Δ = Δ Unit = rad/s 2 ω arrowrightnosp ω arrowrightnosp α arrowrightnosp α arrowrightnosp Speeding up Speeding down circle6 Direction: circle6 Now suppose ω can change as a function of time Rotations: Quiz 1 Rotations: Quiz 1 circle6 Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go- round makes one complete revolution every two seconds. barb4left Klyde’s angular velocity is: (a) the same as Bonnie’s (b) (b) twice Bonnie’s (c) half Bonnie’s ω

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4 Rotations: Quiz 1 answer Rotations: Quiz 1 answer circle6 The angular velocity ω of any point on a solid object rotating about a fixed axis is the same . barb4left Both Bonnie & Klyde go around once (2 π radians) every two seconds. barb4left (a) is the correct answer barb4left What will be different? ω circle6 When a rigid object rotates about a fixed axis in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration barb4left So θ , ω , α all characterize the motion of the entire rigid object
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Rotation of rigid bodies - Chapter 10 Rotation of Rigid...

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