1. Circular Motion © David Hoult 2009

27. Acceleration of a Body Moving in a Circular Path at Constant Speed The magnitude of the velocity of the body is constant but the direction is constantly changing, therefore, the body is © David Hoult 2009

28. Acceleration of a Body Moving in a Circular Path at Constant Speed The magnitude of the velocity of the body is constant but the direction is constantly changing, therefore, the body is accelerating At any instant, the direction of the velocity is a © David Hoult 2009

29. Acceleration of a Body Moving in a Circular Path at Constant Speed The magnitude of the velocity of the body is constant but the direction is constantly changing, therefore, the body is accelerating At any instant, the direction of the velocity is a tangent to the circular path © David Hoult 2009

30. The magnitude of the acceleration depends on © David Hoult 2009

31. The magnitude of the acceleration depends on i) the speed of the body © David Hoult 2009

32. The magnitude of the acceleration depends on i) the speed of the body ii) the radius of the circular path © David Hoult 2009

33. We might suggest that a  © David Hoult 2009

34. We might suggest that a  v © David Hoult 2009

35. We might suggest that a  v and that a  © David Hoult 2009

36. We might suggest that a  v and that a  1 r and therefore © David Hoult 2009

37. We might suggest that a  v and that a  1 r Consideration of the units suggests that and therefore a  v r © David Hoult 2009

38. We might suggest that a  v and that a  1 r Consideration of the units suggests that and therefore a  v r v2v2 r © David Hoult 2009

39. It can be shown that the magnitude of the acceleration is given by a = v2v2 r © David Hoult 2009

40. It can be shown that the magnitude of the acceleration is given by a = v2v2 r or in terms of angular speed © David Hoult 2009

41. It can be shown that the magnitude of the acceleration is given by a = v2v2 r or in terms of angular speed a = r  2 or in terms time period © David Hoult 2009

42. It can be shown that the magnitude of the acceleration is given by a = v2v2 r or in terms of angular speed a = r  2 or in terms time period a = 4 2 r4 2 r T2T2 © David Hoult 2009

43. The direction of this acceleration is © David Hoult 2009

44. The direction of this acceleration is towards the centre of the circle For this reason it is called a © David Hoult 2009

45. The direction of this acceleration is towards the centre of the circle For this reason it is called a centripetal acceleration and is said to be caused by a centripetal force © David Hoult 2009

46. The direction of this acceleration is towards the centre of the circle For this reason it is called a centripetal acceleration and is said to be caused by a centripetal force F c = m v2v2 r F c = mr  2 © David Hoult 2009

47. A centripetal force does not change the kinetic energy of the body on which it acts because it acts © David Hoult 2009

48. A centripetal force does not change the kinetic energy of the body on which it acts because it acts at 90° to the direction of the motion of the body © David Hoult 2009

49. Estimate the magnitude of the force needed to cause a car to move around a curve in a road at 50 km h -1. What force causes the centripetal acceleration in this situation ? © David Hoult 2009

50. Estimate the magnitude of the force needed to cause a car to move around a curve in a road at 50 km h -1. What force causes the centripetal acceleration in this situation ? Friction © David Hoult 2009

51. Estimate the magnitude of the force needed to cause a car to move around a curve in a road at 50 km h -1. What force causes the centripetal acceleration in this situation ? Estimates needed: Friction © David Hoult 2009

52. Estimate the magnitude of the force needed to cause a car to move around a curve in a road at 50 km h -1. What force causes the centripetal acceleration in this situation ? Estimates needed: mass of car, m Friction © David Hoult 2009

53. Estimate the magnitude of the force needed to cause a car to move around a curve in a road at 50 km h -1. What force causes the centripetal acceleration in this situation ? Estimates needed: mass of car, m radius of path,r Friction © David Hoult 2009

54. F c = m v2v2 r © David Hoult 2009

55. A small mass hangs on a string inside the car. It is observed by a passenger. © David Hoult 2009

56. If the car turns to the left: A small mass hangs on a string inside the car. It is observed by a passenger*. * the mass is in front of the passenger. © David Hoult 2009

57. If the car turns to the left: A small mass hangs on a string inside the car. It is observed by a passenger. © David Hoult 2009

58. Find the angle  during the time the car is moving round the curved path. A small mass hangs on a string inside the car. If the car turns to the left:  © David Hoult 2009

62. mg © David Hoult 2009

63. Te cos  mg © David Hoult 2009

64. Te cos  mg Te sin  © David Hoult 2009

65. Te cos  mg Te sin  The vertical forces acting on the mass are in equilibrium, therefore © David Hoult 2009

66. Te cos  mg Te sin  The vertical forces acting on the mass are in equilibrium, therefore Te cos  must have the same magnitude as mg © David Hoult 2009

67. Te cos  mg Te sin  The mass is accelerating to the left, horizontally © David Hoult 2009

68. Te cos  mg Te sin  The mass is accelerating to the left, horizontally The horizontal component of the tension provides the centripetal force needed for this acceleration. © David Hoult 2009

69. Te cos  mg Te sin  Therefore Te sin  = m v2v2 r © David Hoult 2009

70. ...need I say more ? © David Hoult 2009