1. Physics Lesson 11 - Circular Motion
Eleanor Roosevelt High School
Chin-Sung Lin

2. Rotation & Revolution
Axis
Axis
Rotation
Revolution

3. Period & Frequency Frequency (f): cycles/second (Hz)
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 SpeedB
2πR B B

9. Rotational & Linear SpeedB
2πR 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
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

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):
Frequency = ?
Tangential Speed = 12π m/s
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 ():
Period = 2 s
Rotational Speed = ?
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 ():
Linear Speed = 6π m/s
Rotational Speed = ?
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 SpeedB

27. Rotational & Linear SpeedB
2πR B B
2πR

28. Centripetal Force & Acceleration

29. Centripetal Force & Acceleration
Inertia

30. Centripetal Force & Acceleration

31. Centripetal Force & Acceleration

32. Centripetal Force & Acceleration
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 B C D
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

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 is a force directed toward the center of the circle Fc

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

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

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

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

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

51. Centripetal Force & Acceleration
Centripetal force is directly proportional to linear speed squared (v2)
Fc ~ v2
(Fc = m Ac = mv 2/r = mr 2)

52. Centripetal Force & Acceleration
Centripetal force is directly proportional to angular speed squared (2)
Fc ~ 2
(Fc = m Ac = mv 2/r = mr 2)

53. Centripetal Force Example
For a circular motion, what if mass is doubled? Fc will be …………
For a circular motion, what if radius is doubled? Fc will be …………
For a circular motion, what if linear speed is doubled? Fc will be …………
For a circular motion, what if angular speed is doubled? Fc will be …………

54. Centripetal Force Example
For a circular motion, what if mass is halved? Fc will be …………
For a circular motion, what if radius is halved? Fc will be …………
For a circular motion, what if linear speed is halved? Fc will be …………
For a circular motion, what if angular speed is halved? Fc will be …………

55. Centripetal Force Example
A 280 kg motorcycle traveling at 32 m/s enters a curve of radius = 130 m. What force of friction is required from the contact of the tires with the road to prevent a skid?

56. Centripetal Force Example
A 280 kg motorcycle traveling at 32 m/s enters a curve of radius = 130 m. What force of friction is required from the contact of the tires with the road to prevent a skid? Fc = 280kg x (32 m/s)2/130m = 2205 N

57. Centripetal Force Exercise
Astronauts are trained to tolerate greater acceleration than the gravity by using a spinning device whose radius is 10.0 m. With what linear speed and rotational speed would an astronaut have to spin in order to experience an acceleration of 3 g’s at the edge of the device?

58. Centripetal Force Exercise
To swing a pail of water around in a vertical circle fast enough so that the water doesn’t spill out when the pail is upside down. If Mr. Lin’s arm is 0.60 m long, what is the minimum speed with which he can swing the pail so that the water doesn’t spill out at the top of the path?

59. Centripetal Force Exercise
At the outer edge of a rotating space station, 1 km from its center, the rotational acceleration is 10.0 m/s2. What is the new weight of a 1000 N object being moved to a new storage room which is 500 m from the center of the space station?

60. Summary Rotation & revolution Period & frequency
Linear/tangential speed: v = Δd / Δt = 2πr / T = 2πrf (m/s)
Rotational/angular speed:  = 2π/T = 2πf (rads/s)
Tangential speed = Radius x Rotational speed: v = r

61. Summary Centripetal force & acceleration
Centripetal acceleration: Ac = v 2 / r = r 2
Centripetal force: Fc = m Ac = mv 2/r = mr 2
Centripetal force: Fc ~ m
Centripetal force: Fc ~ r
Centripetal force: Fc ~ v2
Centripetal force: Fc ~ 2

62. Centripetal Force Lab

63. Centripetal Force Lab

64. Centripetal Force Lab m v Fc Fg

65. Centripetal Force Lab m v Fc
Ac = v 2 / r
Fc = m Ac
Fc = m v 2 / r Fg

66. Centripetal Force Lab

67. Centripetal Force Lab

68. Centripetal Force Lab
Common Errors
The position of clip
The plane of circular motion
The washers are not identical