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

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

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

30. The magnitude of the acceleration depends on

31. The magnitude of the acceleration depends on
i) the speed of the body

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

33. We might suggest that
a a

34. We might suggest that
a a v

35. We might suggest that
a a v
and that
a a

36. We might suggest that
a a v
and that 1
a a r
and therefore

37. We might suggest that
a a v
and that 1
a a r
and therefore v
a a r
Consideration of the units suggests that

38. We might suggest that
a a v
and that 1
a a r
and therefore v
a a r
Consideration of the units suggests that v2
a a r

39. It can be shown that the magnitude of the acceleration is given by

40. It can be shown that the magnitude of the acceleration is given by
or in terms of angular speed

41. It can be shown that the magnitude of the acceleration is given by
or in terms of angular speed
a = r w2
or in terms time period

42. It can be shown that the magnitude of the acceleration is given by
or in terms of angular speed
a = r w2
or in terms time period
4 p2 r
a = T2

43. The direction of this acceleration is

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

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

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 v2
Fc = m r
Fc = mrw2

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

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

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 ?

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

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 ?
Friction
Estimates needed:

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 ?
Friction
Estimates needed:
mass of car, m

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 ?
Friction
Estimates needed:
mass of car, m
radius of path, r

54. v2
Fc = m r

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

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

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

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

62. mg

63. Te cos q mg

64. Te cos q
Te sin q mg

65. Te cos q
Te sin q mg
The vertical forces acting on the mass are in equilibrium, therefore

66. Te cos q
Te sin q mg
The vertical forces acting on the mass are in equilibrium, therefore
Te cos q must have the same magnitude as mg

67. Te cos q
Te sin q mg
The mass is accelerating to the left, horizontally

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

69. Te cos q
Te sin q mg
Therefore v2
Te sin q
= m r

70. ...need I say more ?