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Circular Motion

Circular Motion. Chin-Sung Lin. Rotation & Revolution. Axis. Axis. Rotation. Revolution. Period & Frequency. Period (T): seconds/cycle. Radius (r). Frequency (f): cycles/second (Hz). Period & Frequency. Radius (r). T = 1/f. f = 1/T. Period & Frequency Exercise. Radius (r).

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Circular Motion

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  1. Circular Motion Chin-Sung Lin

  2. Rotation & Revolution Axis Axis Rotation Revolution

  3. Period & Frequency Period (T): seconds/cycle Radius (r) Frequency (f): cycles/second (Hz)

  4. Period & Frequency Radius (r) T = 1/f f = 1/T

  5. Period & Frequency Exercise Radius (r) If the frequency is 40 Hz, what’s the period?

  6. Period & Frequency Exercise Radius (r) If the period is 0.05 s, what’s the frequency?

  7. Period & Frequency If the microprocessor clock of your computer is running at 2.5 GHz, what’s the period of the clock?

  8. Rotational & Linear Speed r A R 2πr A A 2πR B B B

  9. Rotational & Linear Speed r A R 2πr A A 2πR B B B ??? 2πR = 2πr ???

  10. Rotational & Linear Speed Linear speed: distance moved per unit of time v = Δd/ Δt The linear speed is greater on the outer edge of a rotational object than it is closer to the axis r R

  11. Rotational & Linear Speed v Tangential speed: The speed of an object moving along a circular path can be called tangential speed because the direction of motion is always tangent to the circle v v v

  12. Rotational & Linear Speed For circular motion, tangential speed = linear speed

  13. Rotational & Linear Speed Linear / Tangential Speed (v): Circumference = 2πr Period = T Radius (r) Linear/Tangential Speed = 2πr / T = 2πrf

  14. Rotational & Linear Speed Exercise Linear / Tangential Speed (v): Period = 2 s Tangential Speed ? 3 m

  15. Rotational & Linear Speed Exercise Linear / Tangential Speed (v): Frequency = 2 Hz Tangential Speed ? 4 m

  16. Rotational & Linear Speed Exercise Linear / Tangential Speed (v): Tangential Speed = 12π m/s Frequency = ? Period = ? 2 m

  17. Rotational & Linear Speed Rotational / Angular speed (): The number of rotations per unit of time All parts of a rotational object have the same rate of rotation, or same number of rotations per unit of time Unit of rotational speed: • Degrees/second or radians/second • Revolutions per minute (RPM)

  18. Rotational & Linear Speed Rotational / Angular speed (): 1 revolution = 2π Period = T Radius (r) Rotational Speed  = 2π/T = 2πf (rads/s)

  19. Rotational & Linear Speed Exercise Rotational / Angular speed (): Rotational Speed = ? Period = 2 s 5 m

  20. Rotational & Linear Speed Exercise Rotational / Angular speed (): Frequency = 2 Hz Rotational Speed = ? 5 m

  21. Rotational & Linear Speed Rotational / Angular speed (): Rotational Speed  = 2πf (rads/s) Tangential Speed v = 2πrf (m/s) v = r (Tangential speed) = (Radial distance) x (Rotational speed)

  22. Rotational & Linear Speed Rotational / Angular speed (): At the center (or axis) of the rotational platform, there is no tangential speed, but there is rotational speed

  23. Rotational & Linear Speed Exercise Rotational / Angular speed (): Rotational Speed = 4π Linear Speed = ? 3 m

  24. Rotational & Linear Speed Exercise Rotational / Angular speed (): Rotational Speed = ? Linear Speed = 6π m/s 2 m

  25. Rotational & Linear Speed Exercise Rotational / Angular speed (): Period = 3 s 4 m A B 2 m Rotational Speed = ? Linear Speed = ?

  26. Rotational & Linear Speed r A R B

  27. Rotational & Linear Speed r A R 2πR A A 2πR B B B 2πR

  28. Centripetal Force & Acceleration

  29. Centripetal Force & Acceleration Centripetal Force Inertia

  30. Centripetal Force & Acceleration

  31. Centripetal Force & Acceleration

  32. Centripetal Force & Acceleration Centripetal Force Inertia

  33. Centripetal Force & Acceleration Centripetal Acceleration Acceleration is a vector quantity a = Δv/ Δt Velocity can be changed by increasing/ decreasing the magnitude of v, or changing the direction

  34. Centripetal Force & Acceleration Centripetal Acceleration A A D B B D C C Change Speed Change Direction

  35. Centripetal Force & Acceleration Centripetal Acceleration An object moves around in a circle with constant speed has acceleration, because its direction is constantly changing This acceleration is called centripetal acceleration (Ac)

  36. Centripetal Force & Acceleration Centripetal Acceleration Centripetal acceleration is directed toward the center of the circle Ac Ac Ac Ac

  37. Centripetal Force & Acceleration Centripetal Acceleration An acceleration that is directed at a right angle to the path of a moving object and produces circular motion Centripetal acceleration (Ac) Ac = v 2 / r

  38. Centripetal Force & Acceleration Centripetal Acceleration Ac = v 2 / r = (r) 2 / r = r 2 Ac = v 2 / r = r 2

  39. Centripetal Acceleration Exercise Centripetal Acceleration (Ac): Linear speed = 6 m/s 3 m Centripetal Acceleration = ?

  40. Centripetal Acceleration Exercise Centripetal Acceleration (Ac): Rotational speed = 2 rad/s 3 m Centripetal Acceleration = ?

  41. Centripetal Acceleration Exercise Centripetal Acceleration (Ac): Period = 2 s Centripetal Acceleration = ? 5 m

  42. Centripetal Force & Acceleration Centripetal Force Centripetal force is a force directed toward the center of the circle Fc Fc Fc Fc

  43. Centripetal Force & Acceleration In linear motion Fnet = m a In circular motion Fc = m Ac

  44. Centripetal Force & Acceleration m v Fc Ac = v 2 / r Fc = m Ac Fc = m v 2 / r Fg

  45. Centripetal Force & Acceleration m v Fc Ac = v 2 / r Fc = m Ac Fc = m v 2 / r

  46. Centripetal Force & Acceleration Centripetal Force Centripetal force is a force directed toward the center of the circle Fc = m Ac = mv 2/r = mr 2

  47. Centripetal Force Exercise Centripetal Force (Fc): Linear speed = 4 m/s 2 kg 2 m Centripetal Force = ?

  48. Centripetal Force Exercise Centripetal Force (Fc): Angular speed = 3 rad/s 5 kg 2 m Centripetal Force = ?

  49. Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to mass (m) Fc ~ m (Fc = m Ac = mv 2/r = mr 2)

  50. Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to radius (r) Fc ~ r (Fc = m Ac = mv 2/r = mr 2)

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