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Uniform Circular Motion

Uniform Circular Motion. Position on a Circle. Motion in a circle is common. The most important measure is the radius ( r ). The position of a point on the circle is described by a radial vector . Origin is at the center. Magnitude is equal everywhere. r. r.

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Uniform Circular Motion

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  1. Uniform Circular Motion

  2. Position on a Circle • Motion in a circle is common. • The most important measure is the radius (r). • The position of a point on the circle is described by a radial vector . • Origin is at the center. • Magnitude is equal everywhere. r r

  3. We use degrees to measure position around the circle. There are 2pradians in the circle. This matches 360° The distance around a circle is s = rq, where q is in radians. Measuring a Circle Dq q r The angular displacement is Dq

  4. Period and Frequency • Movement around a circle takes time. • The period (T) is the time it takes to complete one revolution around the circle. • The frequency (f) is the number of cycles around completed in a time. • Cycles per second (cps or Hz) • Revolutions per minute (rpm) • Frequency is the inverse of period (f = 1/T).

  5. Frequency is measured in cycles per second. There is one cycle per period. Frequency is the inverse of the period, f =1/T. Angular velocity is measured in radians per second. There are 2p radians per period. Angular velocity, w = 2p/T. Angular velocity, w = 2pf. Cycles or Radians

  6. Angular Velocity • Displacement is related to the angle. • Displacement on the curve (s) • Angle around the circle (q) • Velocity has an angular equivalent. • Linear velocity (v) • Angular velocity (w) • Units (rad/s or 1/s = s-1)

  7. Speed on a Circle • The circumference of a circle is 2r. • The period is T. • The speed is related to the distance and the period or frequency. • v = 2r/T • v = 2rf • v = rw s = 2p r r

  8. Velocity on a Circle • Velocity is a vector change in position compared to time. • As the time gets shorter, the velocity gets closer to the tangent.

  9. Direction of Motion • In the limit of very small angular changes the velocity vector points along a tangent of the circle. • This is perpendicular to the position. • For constant w, the magnitude stays the same, but the direction always changes.

  10. No Slipping • A wheel can slide, but true rolling occurs without slipping. • As it moves through one rotation it moves forward 2pR. w v v = 2pR/T = wR R Dx = 2pR

  11. Point on the Edge • A point on the edge moves with a speed compared to the center, v = wr. • Rolling motion applies the same formula to the center of mass velocity, v = wR. • The total velocity of points varies by position. v =2vCM vCM v =0

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