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# Uniform Circular Motion

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### Uniform Circular Motion

1. 1. Uniform Circular Motion Section 1 Topic 2
2. 2. Linear Motion <ul><li>All objects previously have been point objects. </li></ul><ul><li>Easy to do calculations but inaccurate. </li></ul><ul><li>If no dimensions it can’t rotate. </li></ul><ul><li>All objects rotate to some extent. </li></ul>
3. 3. Axis of Rotation <ul><li>A body rotates about an axis. </li></ul>
4. 4. Axis of Rotation <ul><li>This can be internal </li></ul>Such as a CD
5. 5. Axis of Rotation <ul><li>Or external </li></ul>such as a boomerang
6. 6. Axis of Rotation <ul><li>Or internal while revolving about an external axis </li></ul>
7. 7. Rigid Bodies <ul><li>All component particles are fixed in relation to each other as they rotate . </li></ul><ul><li>Not all objects are rigid bodies. </li></ul>
8. 8. Non Rigid Bodies <ul><li>Paint being stirred . </li></ul><ul><li>A diver who curls while somersaulting then straightens before hitting the water . </li></ul>
9. 9. Circular Motion <ul><li>An object moving in a circular path will have a constant speed . </li></ul><ul><li>I t is continually changing direction . </li></ul><ul><li>Therefore it’s velocity is continually changing. </li></ul><ul><li>A relationship can be determined for the speed of the object. </li></ul><ul><li>To do this some terms must be defined first. </li></ul>
10. 10. Circular Motion Terms <ul><li>Period </li></ul><ul><li>Is the time needed to complete one cycle/rev (in secs). The symbol T is used. </li></ul><ul><li>Frequency </li></ul><ul><li>Number of cycles/revs completed per unit time. </li></ul><ul><li>Units are Hertz (Hz) </li></ul>f =
11. 11. Circular Motion Terms <ul><li>In uniform circular motion, the object in one revolution moves 2  r in T seconds. </li></ul>
12. 12. Try Example 1 <ul><li>The Earth has a diameter of 1.276 x 10 7 m. Find the average linear speed of a point on the Earth’s equator. </li></ul>
13. 13. Solution <ul><li>r = 1.276 x 10 7 /2 = 6.38 x 10 6 m </li></ul><ul><li>T = 24 x 60 x 60 = 8.64 x 10 4 s </li></ul>v = 464 ms -1 (over 1600 km h -1 )
14. 14. Centripetal Acceleration <ul><li>A particle undergoing uniform circular motion is continually changing velocity. </li></ul><ul><li> acceleration is changing. </li></ul>
15. 15. Centripetal Acceleration <ul><li> v 1 = v b - v a. </li></ul><ul><li> v 2 = v c - v b and so on. </li></ul><ul><li>The magnitude of  v 1 =  v 2. </li></ul><ul><li>The direction is always to the centre of the circle . </li></ul>
16. 16. Centripetal Acceleration <ul><li>The acceleration which produces these velocity changes in a direction ….. </li></ul><ul><li>i s called centripetal (centre seeking) acceleration. </li></ul><ul><li>The direction is always towards the centre of the circular motion . </li></ul>
17. 17. Centripetal Acceleration
18. 18. Average Acceleration <ul><li>Defined as : </li></ul><ul><li>where  v = v f - v i . </li></ul><ul><li>  The instantaneous acceleration a at any instant ca n be obtained by allowing the time interval to become infinitesimal . </li></ul>
19. 19. Direction of Acceleration <ul><li>S tone attached to a string and whirled above the head . </li></ul><ul><ul><li>What type of motion has it? </li></ul></ul><ul><li>Circular . </li></ul><ul><ul><li>If string break s , what happens? </li></ul></ul>
20. 20. Direction of Acceleration <ul><li>S tone fl ies off in a direction that is tangential to the point at which the string breaks. </li></ul><ul><li>At any point, the tangent to the point gives the direction of the velocity. </li></ul>
21. 21. Relationship Between a and v in Circular Motion <ul><li>The magnitude of this acceleration is constant for a given speed and radius. </li></ul><ul><li>Circular Motion </li></ul>
22. 22. Force Causing the Centripetal Acceleration <ul><li>Newton’s 2 nd law tells us that a centripetal acceleration can only happen if there is an unbalanced force. </li></ul>
23. 23. Force Causing the Centripetal Acceleration <ul><li>Any particle undergoing uniform circular motion is acted upon by an unbalanced force which is…. </li></ul><ul><ul><li>C onstant in magnitude . </li></ul></ul><ul><ul><li>D irected towards the centre of the circle. </li></ul></ul><ul><ul><li>C auses the Centripetal Acceleration. </li></ul></ul>
24. 24. Force Causing the Centripetal Acceleration <ul><li>When an object undergoes uniform circular motion there is a net force </li></ul><ul><ul><li>Directed towards the centre of the circle, </li></ul></ul><ul><ul><li>Force has physical origin </li></ul></ul><ul><ul><ul><li>Gravity </li></ul></ul></ul><ul><ul><ul><li>Normal Force </li></ul></ul></ul><ul><ul><ul><li>Tension </li></ul></ul></ul><ul><ul><ul><li>Friction </li></ul></ul></ul>
25. 25. Force Causing the Centripetal Acceleration
26. 26. Force Causing the Centripetal Acceleration <ul><li>With a centripetal force, the object moves in a circular path. </li></ul>
27. 27. Force Causing the Centripetal Acceleration <ul><li>When the unbalanced force is released : </li></ul><ul><ul><li>the object moves along a tangential path , </li></ul></ul><ul><ul><li>at a constant velocity . </li></ul></ul>
28. 28. Force Causing the Centripetal Acceleration <ul><li>The force that causes this acceleration is not a new force . </li></ul><ul><li>Other objects must apply the force. </li></ul><ul><li>E xamples of centripetal forces include : </li></ul>
29. 29. Force Causing the Centripetal Acceleration <ul><li>Moon revolving around the Earth: </li></ul><ul><ul><li>Gravitational Force, </li></ul></ul><ul><ul><li>Directed towards the centre of the Earth, </li></ul></ul><ul><ul><li>Holds the moon in a near circular orbit. </li></ul></ul>
30. 30. Force Causing the Centripetal Acceleration <ul><li>Electrons revolve around the nucleus: </li></ul><ul><ul><li>Electric Force, </li></ul></ul><ul><ul><li>Directed to centre of the nucleus, </li></ul></ul><ul><ul><li>Holds electrons in circular orbit. </li></ul></ul>
31. 31. Force Causing the Centripetal Acceleration <ul><li>Car rounding a corner: </li></ul><ul><ul><li>Sideways frictional force, </li></ul></ul><ul><ul><li>Directed towards centre of turn, </li></ul></ul><ul><ul><li>Force between car tyre and road. </li></ul></ul><ul><li>If force not great enough: </li></ul><ul><ul><li>Car skids. </li></ul></ul>
32. 32. Force Causing the Centripetal Acceleration <ul><li>Billy can being swung. </li></ul><ul><ul><li>Vertically or horizontally </li></ul></ul><ul><ul><li>The tension force between arm and can </li></ul></ul><ul><ul><li>causes the can to move in circular motion. </li></ul></ul>
33. 33. Force Causing the Centripetal Acceleration <ul><li>Washing Machine tub on spin cycle: </li></ul><ul><ul><li>Tub rotates at high speed, </li></ul></ul><ul><ul><li>Inner wall exerts inwards force on clothes. </li></ul></ul><ul><ul><li>Holes in tub allow water to follow a straight line. </li></ul></ul><ul><ul><li>Water escapes. </li></ul></ul><ul><li>Force acts on clothes: </li></ul><ul><ul><li>not water. </li></ul></ul>
34. 34. Force Causing the Centripetal Acceleration <ul><li>The force can be found by combining Newton’s 2 nd Law and the equation for centripetal acceleration. </li></ul>
35. 35. Centripetal Acceleration and Tension Try Example 2
36. 36. Solution – Part (a) <ul><li>r = 1m </li></ul><ul><li>F = 196 N </li></ul><ul><li>m = 1 kg </li></ul>
37. 37. Solution – Part (a) <ul><li>v = 14 ms -1 tangential to the circle at the point of release. </li></ul>
38. 38. Solution – Part (b) <ul><li>F = 196 N </li></ul><ul><li>m = 1 kg </li></ul><ul><li>r = 1 m </li></ul><ul><li>g = 9.8 ms -2 </li></ul><ul><li>v = ? </li></ul>
39. 39. Solution – Part (b) <ul><li>M aximum tension occurs at the bottom of the path . </li></ul><ul><li>T ension must be sufficient both to provide the centripetal force and…… </li></ul><ul><li>balance the gravitational force . </li></ul>
40. 40. Solution – Part (b) <ul><li>v 2 =196 - 9.8 = 186.2 </li></ul><ul><li>v =13.6 ms -1 </li></ul>
41. 41. Centripetal Acceleration and Gravity <ul><li>Gravity is a force that always acts between bodies that have mass. </li></ul><ul><li>The force is so weak, it is only noticed when the mass is extremely large . </li></ul><ul><li>For example: </li></ul>
42. 42. Centripetal Acceleration and Gravity <ul><li>P lanets. </li></ul><ul><li>A lways a force of attraction and is directed towards the centre of the mass. </li></ul>
43. 43. Centripetal Acceleration and Gravity <ul><li>F orce of gravity causes the centripetal acceleration when a satellite moves in a circular orbit. </li></ul><ul><li>See T opic 3 for more details . </li></ul>
44. 44. Centripetal Acceleration and Friction <ul><li>There are three forces that act on a car which is turning a corner. </li></ul><ul><li>1. Gravity ( F G = m g ). Acting downwards. </li></ul>
45. 45. Centripetal Acceleration and Friction <ul><li>2. Normal Force ( F N ). Force that balances the force of the car on the road (by Newton’s III law). </li></ul><ul><ul><li>Sometimes, it is equal to F G , but acts in the opposite direction. </li></ul></ul>
46. 46. Centripetal Acceleration and Friction <ul><li>3. Frictional Force ( F fr ). Acting towards the centre of the turn. </li></ul>
47. 47. Centripetal Acceleration and Friction
48. 48. Centripetal Acceleration and Friction <ul><li>C ar turns a corner . </li></ul><ul><li>F eel as though you are pushed against the side of the car, away from the direction that the car is turning. </li></ul><ul><li>What is actually happening is : </li></ul><ul><ul><li>Y ou are trying to move in a straight line while the car is moving in a circular path. </li></ul></ul><ul><ul><li>The back of the seat (friction) or the door of the car exerts a force on you. </li></ul></ul>
49. 49. Centripetal Acceleration and Friction <ul><li>The force acts on the passenger in the car if they do not have their seat belt on. </li></ul><ul><li>Note: it is an European car. </li></ul>
50. 50. Centripetal Acceleration and Friction <ul><li>The car itself must also have a force acting on it to turn around the bend. </li></ul><ul><li>If the road is flat, the force is the friction between the tires and the road. </li></ul>
51. 51. Centripetal Acceleration and Friction <ul><li>Under some conditions </li></ul><ul><ul><li>water or ice on the road </li></ul></ul><ul><ul><li>excessive speed </li></ul></ul><ul><li>F rictional force is not enough . </li></ul><ul><li>C ar will skid in a near straight line path. </li></ul><ul><li>Cars & Ice </li></ul><ul><li>Cars and Ice 2 </li></ul>
52. 52. Centripetal Acceleration and Friction <ul><li>Press on brakes hard; </li></ul><ul><ul><li>Brakes lock. </li></ul></ul><ul><li>F rictional force is reduced. </li></ul><ul><li>C ar will skid in a near straight line path. </li></ul>
53. 53. Centripetal Acceleration and the Normal Force <ul><li>C ar turn s on a banked section of curved road : </li></ul><ul><ul><li>the chances of skidding is reduced. </li></ul></ul>
54. 54. <ul><li>Why? </li></ul><ul><ul><li>N ormal force does have a component acting towards the centre of the circle. </li></ul></ul>
55. 55. Centripetal Acceleration and the Normal Force
56. 56. Centripetal Acceleration and the Normal Force <ul><li>What does this mean? </li></ul><ul><ul><li>N ot only does friction supply the force to turn the car, so does some of the normal force. </li></ul></ul><ul><li>Can the entire force be supplied by horizontal component of normal force? </li></ul><ul><ul><li>Yes; at one specific angle </li></ul></ul><ul><li>This angle is given by: </li></ul>
57. 57. Centripetal Acceleration and the Normal Force <ul><li>In the vertical direction, there are 2 forces; </li></ul><ul><li>F N cos  acting upwards and mg acting downwards. </li></ul><ul><li>As there is no net vertical motion : </li></ul><ul><li>F N cos  = mg  </li></ul><ul><li>Now dividing  by  </li></ul>
58. 58. Centripetal Acceleration and the Normal Force <ul><li>For an y radius curve and ideal speed, the perfect banking angle can be found. </li></ul>
59. 59. Centrifugal Force <ul><li>Centripetal force holds objects in a circular motion. </li></ul><ul><li>Consider a stone whirling above a person’s head. </li></ul><ul><ul><li>By NIII, there is a reaction force which acts on the person’s hand. </li></ul></ul><ul><li>Th i s can be seen by an observer looking at the motion from the outside. </li></ul>
60. 60. Centrifugal Force <ul><li>Imagine yourself to be the stone </li></ul><ul><ul><li>new frame of reference </li></ul></ul><ul><ul><li>there appears to be a new force. </li></ul></ul><ul><li>This force has the same magnitude as the centripetal force </li></ul><ul><ul><li>Opposite in direction </li></ul></ul><ul><li>. Called the centrifugal force. </li></ul>
61. 61. Centrifugal Force <ul><li>As the forces are seen from two frames of reference, it cannot be said that the centrifugal force is a reaction to the centripetal force. </li></ul>
62. 62. Centrifugal Force <ul><li>Centripetal force is a real force as it is responsible for the circular motion . </li></ul>
63. 63. Centrifugal Force <ul><li>Centri fug al force is fictitious as it is only felt by the object in the rotating frame of reference. </li></ul>