P2F11-Week9-Barwick

# P2F11-Week9-Barwick - P2 Week 9 Rotational motion and...

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P2 Week 9 Rotational motion and kinematic equations Reading: Sec 3.4

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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

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Angular Position, 2 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 , θ ) where r is the distance from the origin to P and θ is the measured counterclockwise from the reference line
Angular Position, 3 As the particle moves, the only coordinate that changes is θ As the particle moves through θ , it moves though an arc length s . The arc length and r are related: s = θ r compare to calculation of s in cartesian coord.!

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Radian This can also be expressed as θ 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
Conversions Comparing degrees and radians 1 rad = = 57.3° Converting from degrees to radians θ [rad] = θ [degrees] " 180 !

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Angular Position, final We can associate the angle θ 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 θ 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

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Average Angular Speed The average angular speed, ω , of a rotating rigid object is the ratio of the angular displacement to the time interval " av # z = \$ 2 # \$ 1 t 2 # t 1 = % \$ % t Text Notation (Eqn 9.2) Alternate Notation
Instantaneous Angular Speed The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero

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Angular Speed Units of angular speed are radians/sec rad/s or s -1 Angular speed will be positive if θ is increasing (counterclockwise) Angular speed will be negative if θ is decreasing (clockwise)
Directions, details Strictly speaking, the speed and acceleration ( ω , α ) are the magnitudes of the velocity and acceleration vectors, which point along the rotation axis The directions are actually given by the right-hand rule

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Angular Speed, final Because a rotating object often returns to its initial orientation, the time to complete one revolution is a convenient measure.
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