Lect 20 - Rotation - Rotation of a Rigid Object about a...

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Unformatted text preview: Rotation of a Rigid Object about a Fixed Axis Chapter 10 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the object remain constant All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible Angular Position Axis of rotation is the center of the disc Choose a fixed reference line Point P is at a fixed distance r from the origin Angular Position Point P will rotate about the origin in a circle of radius r Every particle on the disc undergoes circular motion about the origin, O Polar coordinates are convenient to use to represent the position of P (or any other point) P is located at ( r , ) where r is the distance from the origin to P and is the measured counterclockwise from the reference line Angular Position As the particle moves, the only coordinate that changes is As the particle moves through , it moves though an arc length s . The arc length and r are related: s = r Radian This can also be expressed as is a pure number, but commonly is given the artificial unit, radian One radian is the angle subtended by an arc length equal to the radius of the arc = s r Conversions Comparing degrees and radians 1 rad = = 57.3 Converting from degrees to radians [rad] = [degrees] 360 o 2 180 o q Angular Position We can associate the angle with the entire rigid object as well as with an individual particle Remember every particle on the object rotates through the same angle The angular position of the rigid object is the angle between the reference line on the object and the fixed reference line in space The fixed reference line in space is often the x- axis Angular Displacement The angular displacement is defined as the angle the object rotates through during some time interval This is the angle that the reference line of length r sweeps out = f- q i Average Angular Speed The average angular speed, , of a rotating rigid object is the ratio of the angular displacement to the time interval = f- q i t f- t i = D q D t Instantaneous Angular Speed The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero lim D t 0 D q D t = dq dt Angular Speed Units of angular speed are radians/sec...
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Lect 20 - Rotation - Rotation of a Rigid Object about a...

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