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  1. 1. Topic 2 Mechanics
  2. 2. Contents <ul><li>Kinematic Concepts: Displacement Speed vs. Velocity Acceleration Changing Units Instantaneous vs. Average </li></ul><ul><li>Frames of Reference Graphical Representation of Position </li></ul><ul><li>Graphical Representation of Velocity </li></ul><ul><li>Graphical Representation of Acceleration </li></ul><ul><li>Equations for Constant Acceleration </li></ul><ul><li>Vertical Motion of Objects </li></ul><ul><li>Air Resistance </li></ul><ul><li>Force and Mass: Springs </li></ul>
  3. 3. Mechanics <ul><li>Mechanics is the study of : </li></ul><ul><ul><li>motion, </li></ul></ul><ul><ul><li>force and , </li></ul></ul><ul><ul><li>energy. </li></ul></ul>
  4. 4. Kinematic Concepts <ul><li>Kinematics is the part of mechanics that describe s how objects move. </li></ul><ul><li>The why objects move as they do is called dynamics . </li></ul>
  5. 5. Kinematic Concepts <ul><li>Displacement </li></ul><ul><li>When a person moves over a time interval, </li></ul><ul><ul><li>they change their position in space. </li></ul></ul><ul><li>This change in position is called the : </li></ul><ul><ul><li>displacement . </li></ul></ul><ul><li>It is a vector quantity . </li></ul>
  6. 6. Kinematic Concepts <ul><li>The further it moves from its starting position, </li></ul><ul><ul><li>the greater it’s displacement. </li></ul></ul>
  7. 7. Kinematic Concepts <ul><li>If the object moves in the opposite direction ; </li></ul><ul><li>as defined, </li></ul><ul><li>its displacement will be negative. </li></ul>
  8. 8. Kinematic Concepts <ul><li>Displacement is given the symbol s ; </li></ul><ul><ul><li>or sometimes  x , </li></ul></ul><ul><ul><li>it’s S.I. unit is the (m)etre. </li></ul></ul>
  9. 9. Kinematic Concepts <ul><li>The term distance is a scalar : </li></ul><ul><ul><li>the symbol is d . </li></ul></ul><ul><li>Distance is more useful when purchasing a car. </li></ul><ul><ul><li>The distance travelled is important ; </li></ul></ul><ul><ul><li>while the direction it travelled in is not. </li></ul></ul>
  10. 10. Kinematic Concepts <ul><li>A person who walks 100 m east and then ; </li></ul><ul><ul><li>100 m west has travelled a distance of , </li></ul></ul><ul><ul><li>200 m. </li></ul></ul><ul><li>Their displacement however is 0 m. </li></ul><ul><li>They have ended up at their starting point. </li></ul><ul><li>Distance vs Displacement </li></ul>
  11. 11. Kinematic Concepts <ul><li>Speed vs Velocity </li></ul><ul><li>Although speed and velocity are used interchangeably in everyday life, </li></ul><ul><ul><li>both terms have specific meanings. </li></ul></ul>
  12. 12. Kinematic Concepts <ul><li>Speed is defined as ; </li></ul><ul><ul><li>the distance travelled by an object in a given time interval. </li></ul></ul><ul><li>This will give us the average speed. </li></ul><ul><li>Mathematically, it can be represented by: </li></ul>
  13. 13. Kinematic Concepts <ul><li>As distance is one of the variables, </li></ul><ul><ul><li>speed must be a scalar. </li></ul></ul><ul><li>Velocity is used to signify both magnitude and direction hence ; </li></ul><ul><ul><li>it is a vector. </li></ul></ul>
  14. 14. Kinematic Concepts <ul><li>The average velocity is defined as : </li></ul><ul><ul><li>the change in position of the object ; </li></ul></ul><ul><ul><ul><li>in a given time interval. </li></ul></ul></ul><ul><li>Mathematically, it can be described as: </li></ul>
  15. 15. Kinematic Concepts <ul><li>The S.I. unit for both speed and velocity is m s -1 . </li></ul><ul><li>Always include a direction when giving the value of the velocity. </li></ul>
  16. 16. Kinematic Concepts <ul><li>Acceleration. </li></ul><ul><li>An object whose velocity is changing is : </li></ul><ul><ul><li>accelerating. </li></ul></ul><ul><li>Acceleration is defined as : </li></ul><ul><ul><li>the rate of change in velocity. </li></ul></ul>
  17. 17. Kinematic Concepts <ul><li>Which cars below are acceleration and which cars are traveling at constant velocity? </li></ul><ul><li>Which car accelerates fastest? </li></ul>
  18. 18. Kinematic Concepts <ul><li>The red car is traveling at constant velocity. </li></ul><ul><li>The blue car is accelerating fastest. </li></ul><ul><ul><li>Its rate of change of velocity is greatest. </li></ul></ul>
  19. 19. Kinematic Concepts <ul><li>Since an object always accelerates in a given direction, </li></ul><ul><ul><li>acceleration is a vector quantity. </li></ul></ul><ul><li>The S.I. units are m s -2 </li></ul>
  20. 20. Kinematic Concepts <ul><li>Changing Units </li></ul><ul><li>When using kinematic equations, </li></ul><ul><ul><li>the data given is not always given in S.I. units. </li></ul></ul><ul><li>The data needs to be converted to S.I. Units; </li></ul><ul><ul><li>before they can be substituted into an equation. </li></ul></ul>
  21. 21. Kinematic Concepts <ul><li>An example is speed. </li></ul><ul><li>Very often speed is given in km h -1 . </li></ul><ul><li>The kilometres need to be converted to metres and the hours into seconds. </li></ul><ul><li>There are 1000 m in 1 km </li></ul><ul><ul><li>3600 s in 1 hr (60 x 60). </li></ul></ul>
  22. 22. Kinematic Concepts <ul><li>There is doubt over whether the correct answer should be : </li></ul><ul><ul><li>30 m s -1 , </li></ul></ul><ul><ul><li>28 m s -1 or , </li></ul></ul><ul><ul><li>27.7 m s -1 </li></ul></ul><ul><ul><ul><li>due to the number of significant figures given. </li></ul></ul></ul>
  23. 23. Kinematic Concepts <ul><li>To avoid this problem, always give your data using scientific notation. </li></ul><ul><li>In the above example, </li></ul><ul><ul><li>1.00 x 10 2 m s -1 would eliminate the problem </li></ul></ul><ul><ul><li>hence the answer would be 27.7 m s -1 . </li></ul></ul>
  24. 24. Kinematic Concepts <ul><li>Instantaneous vs Average </li></ul><ul><li>If you drive a car for 240 km in 3 hrs, </li></ul><ul><ul><li>your average speed is 80 km h -1 . </li></ul></ul><ul><li>It is unlikely that for every part of the journey, </li></ul><ul><ul><li>you would be travelling at 80 km h -1 . </li></ul></ul><ul><li>At each instant, your speed would change. </li></ul>
  25. 25. Kinematic Concepts <ul><li>The speedometer in the car gives ; </li></ul><ul><ul><li>the instantaneous speed. </li></ul></ul><ul><li>The instantaneous speed is defined as : </li></ul><ul><ul><li>the average speed over an indefinitely short time interval . </li></ul></ul>
  26. 26. Kinematic Concepts <ul><li>The same definition can be used for velocity. </li></ul><ul><li>The formula for velocity needs to be changed to : </li></ul><ul><ul><li>accommodate the difference between average , </li></ul></ul><ul><ul><li>and instantaneous velocity. </li></ul></ul>
  27. 27. Kinematic Concepts <ul><li>Average velocity: </li></ul><ul><li>Instantaneous velocity: </li></ul>
  28. 28. Kinematic Concepts <ul><li>As  t becomes very small, </li></ul><ul><ul><li>approaching zero, </li></ul></ul><ul><ul><li> s approaches zero as well. </li></ul></ul><ul><li>However, the ratio approaches a definite value. </li></ul>
  29. 29. Kinematic Concepts <ul><li>This definite value is known as the instantaneous velocity. </li></ul><ul><li>The symbol for instantaneous velocity is : </li></ul><ul><ul><li>v no av or bar above the v . </li></ul></ul>
  30. 30. Kinematic Concepts <ul><li>The same procedure can be used for acceleration and so the equations become: </li></ul><ul><li>Average acceleration: </li></ul><ul><li>Instantaneous acceleration: </li></ul>
  31. 31. Kinematic Concepts <ul><li>Relationship between Velocity & Acceleration </li></ul>
  32. 32. Kinematic Concepts <ul><li>Frames of Reference </li></ul><ul><li>The simple question ‘how fast are you moving’ is quite complex. </li></ul><ul><li>In your seat now, you are moving at : </li></ul><ul><ul><li>0 m s -1 relative to the room. </li></ul></ul><ul><li>If the room were on the equator, </li></ul><ul><ul><li>you would be moving at 1700 km h -1 . </li></ul></ul>
  33. 33. Kinematic Concepts <ul><li>At the pole you would be spinning on the spot. </li></ul><ul><li>This however is not true relative to the Sun. </li></ul><ul><li>Two planes moving at 1000 km h -1 relative to the ground : </li></ul><ul><ul><li>may be stationary relative to each other. </li></ul></ul>
  34. 34. Kinematic Concepts <ul><li>All motion is relative . </li></ul><ul><li>We must choose our frame of reference in each description of motion we give. </li></ul><ul><li>Relative Motion (frame of reference) </li></ul>
  35. 35. Kinematic Concepts <ul><li>Usually the motion of objects is considered relative to the earth . </li></ul><ul><li>This is called: </li></ul><ul><ul><li>the inertial frame of reference . </li></ul></ul><ul><li>Unless otherwise stated, </li></ul><ul><ul><li>this will be assumed. </li></ul></ul><ul><li>Inertial Frame </li></ul>
  36. 36. Graphical Representation of Motion <ul><li>Graphical Representation of Position </li></ul><ul><li>Multiflash photographs of an object in motion ; </li></ul><ul><ul><li>can be taken and , </li></ul></ul><ul><ul><li>data collected from it. </li></ul></ul><ul><li>An example of a multiflash photo is the toy car on the next slide. </li></ul>
  37. 37. Graphical Representation of Motion <ul><li>To take this photo ; </li></ul><ul><ul><li>the film needs to be exposed for a period of time in darkness , </li></ul></ul><ul><ul><li>with a strobe flashing at a known rate. </li></ul></ul>
  38. 38. Graphical Representation of Motion <ul><li>From the data collected, </li></ul><ul><ul><li>graphs can be drawn. </li></ul></ul><ul><li>By plotting position on the vertical axis and ; </li></ul><ul><ul><li>time on the horizontal, </li></ul></ul><ul><ul><li>the graph will look like: </li></ul></ul>
  39. 39. Graphical Representation of Motion
  40. 40. Graphical Representation of Motion <ul><li>A number of facts can be ascertained from this graph. </li></ul><ul><li>As the line is straight, </li></ul><ul><ul><li>the change in position per unit time, is constant. </li></ul></ul><ul><ul><li>This means the velocity is constant. </li></ul></ul>
  41. 41. Graphical Representation of Motion <ul><li>2. The magnitude of the velocity can be obtained by the slope; </li></ul>
  42. 42. Graphical Representation of Motion <ul><li>If the graph is steep, it means : </li></ul><ul><ul><li>there is a greater change in position , </li></ul></ul><ul><ul><li>per unit time and the object , </li></ul></ul><ul><ul><li>is moving relatively fast. </li></ul></ul>
  43. 43. Graphical Representation of Motion
  44. 44. Graphical Representation of Motion <ul><li>What happens when two cars traveling at different speeds but with constant velocity? </li></ul>
  45. 45. Graphical Representation of Motion <ul><li>If the graph is horizontal : </li></ul><ul><ul><li>the object is stationary. </li></ul></ul><ul><li>If the slope is negative : </li></ul><ul><ul><li>the object is moving back towards its starting position with , </li></ul></ul><ul><ul><li>constant velocity . </li></ul></ul>
  46. 46. Graphical Representation of Motion
  47. 47. Graphical Representation of Motion <ul><li>In graph a); </li></ul><ul><li>The person has moved at constant velocity over the interval t 1 to point p </li></ul><ul><li>They then remained stationary for the period t 2 </li></ul><ul><li>The person then returned to the original starting position at a constant, but slower velocity over the interval t 3 . </li></ul>
  48. 48. Graphical Representation of Motion <ul><li>In graph b); </li></ul><ul><li>The person has moved at constant velocity over the interval t 1 to point p </li></ul><ul><li>They then remained stationary for the period t 2 </li></ul><ul><li>The person then continued in the original direction for the same distance p as in interval t 1 at a constant, but slower velocity over the interval t 3 . </li></ul>
  49. 49. Graphical Representation of Motion <ul><li>Graphical Representation of Velocity </li></ul><ul><li>Consider graph a) from above. </li></ul><ul><li>From the information from the graph, a graph of the person’s velocity can be drawn. </li></ul>
  50. 50. Graphical Representation of Motion
  51. 51. Graphical Representation of Motion <ul><li>The graph is slightly idealised as : </li></ul><ul><ul><li>the person could not travel at the constant velocity , </li></ul></ul><ul><ul><li>at every instant of the journey. </li></ul></ul>
  52. 52. Graphical Representation of Motion <ul><li>In a real situation ; </li></ul><ul><ul><li>velocity does not change in zero time and a more likely description is , </li></ul></ul><ul><ul><li>the one shown above. </li></ul></ul>
  53. 53. Graphical Representation of Motion <ul><li>What about v/t graphs for the two cars previously? </li></ul>
  54. 54. Graphical Representation of Motion <ul><li>V vs t graph animation </li></ul>
  55. 55. Graphical Representation of Motion <ul><li>Multiflash photography can be used to obtain : </li></ul><ul><ul><li>direct values of average velocity. </li></ul></ul><ul><li>This is because time intervals are : </li></ul><ul><ul><li>very short and , </li></ul></ul><ul><ul><li>constant. </li></ul></ul>
  56. 56. Graphical Representation of Motion <ul><li>The distance from one image to the next is : </li></ul><ul><ul><li>the change in position of the object in , </li></ul></ul><ul><ul><li>a specific interval of time. </li></ul></ul><ul><li>Changes in position in equal intervals of time are : </li></ul><ul><ul><li>direct measures of the average velocity , </li></ul></ul><ul><ul><li>over those intervals. </li></ul></ul>
  57. 57. Graphical Representation of Motion <ul><li>The area under the Velocity vs. Time graph can : </li></ul><ul><ul><li>be used to obtain further information. </li></ul></ul><ul><li>This information relates to : </li></ul><ul><ul><li>the change in position of the object. </li></ul></ul>
  58. 58. Graphical Representation of Motion <ul><li>Rearranging this equation : </li></ul><ul><ul><li>change in position is given by the product; </li></ul></ul><ul><li>average velocity x time taken </li></ul><ul><li>The graph on the next slide deals with constant velocity . </li></ul>
  59. 59. Graphical Representation of Motion
  60. 60. Graphical Representation of Motion <ul><li>As the velocity is constant ; </li></ul><ul><ul><li>the average velocity must be v . </li></ul></ul><ul><li>Hence, the change in position : </li></ul><ul><ul><li>will be vt . </li></ul></ul><ul><li>This is also equal to the : </li></ul><ul><ul><li>area under graph in time interval t . </li></ul></ul><ul><li>This relationship holds for all v/t graphs ; </li></ul><ul><ul><li>as in the example on the next slide : </li></ul></ul>
  61. 61. Graphical Representation of Motion
  62. 62. Graphical Representation of Motion <ul><li>The motion in the graph above describes : </li></ul><ul><ul><li>motion under non-zero constant acceleration </li></ul></ul><ul><ul><ul><li>eg motion under the force of gravity. </li></ul></ul></ul>
  63. 63. Graphical Representation of Motion <ul><li>Since velocity changes in a regular way ; </li></ul><ul><ul><li>the average velocity is , </li></ul></ul><ul><ul><li>the average of the initial velocity and , </li></ul></ul><ul><ul><li>the final velocity. </li></ul></ul><ul><li>The horizontal line through P represents this. </li></ul>
  64. 64. Graphical Representation of Motion <ul><li>The areas a and b are equal : </li></ul><ul><ul><li>one being under the graph and , </li></ul></ul><ul><ul><li>the other, above the line. </li></ul></ul>
  65. 65. Graphical Representation of Motion <ul><li>The product of v av t is : </li></ul><ul><ul><li>the area under the graph and , </li></ul></ul><ul><ul><li>gives the change in position. </li></ul></ul>
  66. 66. Graphical Representation of Motion <ul><li>Another way to calculate the change in position is : </li></ul><ul><ul><li>to divide the area under the graph into , </li></ul></ul><ul><ul><li>a rectangle and triangle and , </li></ul></ul><ul><ul><li>add the two solutions. </li></ul></ul>
  67. 67. Graphical Representation of Motion <ul><li>Any area below the axis corresponds to : </li></ul><ul><ul><li>motion in the opposite direction where , </li></ul></ul><ul><ul><li>the change in position is opposite in sign. </li></ul></ul>
  68. 68. Graphical Representation of Motion <ul><li>Graphical Representation of Acceleration </li></ul>
  69. 69. Graphical Representation of Motion <ul><li>In the example, the slope, </li></ul><ul><ul><li>and therefore the rate at which velocity is changing, </li></ul></ul><ul><ul><li>is constant. </li></ul></ul><ul><li>Constant Acceleration Graph </li></ul>
  70. 70. Graphical Representation of Motion <ul><li>Acceleration is defined as : </li></ul><ul><ul><li>the rate of change of velocity. </li></ul></ul><ul><li>Mathematically, this can be written as: </li></ul>
  71. 71. Graphical Representation of Motion <ul><li>The slope of the graph is given by: rise/run . </li></ul><ul><li>In the example , the rise =  v and the run as  t. </li></ul>
  72. 72. Graphical Representation of Motion <ul><li>This gives the formula: </li></ul><ul><li>This corresponds to our formula for acceleration. </li></ul>
  73. 73. Graphical Representation of Motion <ul><li>This corresponds to our formula for acceleration. </li></ul><ul><li>The slope of a v/t graph gives : </li></ul><ul><ul><li>acceleration. </li></ul></ul><ul><li>Acceleration is measured in m s -2 and ; </li></ul><ul><ul><li>is a vector quantity so , </li></ul></ul><ul><ul><li>direction must always be included. </li></ul></ul>
  74. 74. Graphical Representation of Motion <ul><li>A car is stationary at the lights when the lights change to green. </li></ul><ul><li>Another car is moving when the lights turn green. </li></ul><ul><li>What is the displacement of each car after 3 seconds? </li></ul>
  75. 75. Graphical Representation of Motion
  76. 76. Graphical Representation of Motion <ul><li>Red car </li></ul><ul><li>Area of triangle = ½ b x h </li></ul><ul><li>Displacement = ½ x 3 x 12 </li></ul><ul><li>Displacement = 18 m </li></ul>
  77. 77. Graphical Representation of Motion <ul><li>Blue car </li></ul><ul><li>Area of rectangle = b x h </li></ul><ul><li>Displacement = 3 x 10 </li></ul><ul><li>Displacement = 30 m </li></ul>
  78. 78. Graphical Representation of Motion <ul><li>Watch the animation again. </li></ul><ul><li>What is the acceleration of the red car after 3 seconds? </li></ul>
  79. 79. Graphical Representation of Motion <ul><li>Slope = ryse/run </li></ul><ul><li>Acceleration = 12/3 </li></ul><ul><li>Acceleration = 4 m s -1 </li></ul><ul><li>Watch the animation again and determine the displacement of both cars after 9 seconds. </li></ul>
  80. 80. Graphical Representation of Motion
  81. 81. Graphical Representation of Motion <ul><li>Red Car </li></ul><ul><li>Displacement = area of triangle + area of rectangle. </li></ul><ul><li>Displacement = (½ x 3 x 12) + (9 x 12) </li></ul><ul><li>Displacement =18 + 72 </li></ul><ul><li>Displacement = 90 m </li></ul>
  82. 82. Graphical Representation of Motion <ul><li>Blue Car </li></ul><ul><li>Displacement = area of rectangle </li></ul><ul><li>Displacement = 9 x 10 </li></ul><ul><li>Displacement = 90 m </li></ul><ul><li>Watch the animation again. </li></ul><ul><li>When do the two cars pass each other? </li></ul><ul><li>Does it agree with your calculations? </li></ul>
  83. 83. Graphical Representation of Motion
  84. 84. Graphical Representation of Motion <ul><li>Graphs of position, velocity and acceleration can be drawn for the same object. </li></ul>
  85. 85. Graphical Representation of Motion <ul><li>What happens if it is traveling backwards? </li></ul>
  86. 86. Graphical Representation of Motion <ul><li>What happens when it is being pushed forward? </li></ul>
  87. 87. Graphical Representation of Motion <ul><li>Traveling in the opposite direction? </li></ul>
  88. 88. Graphical Representation of Motion <ul><li>What happens when it is pulled backward? </li></ul>
  89. 89. Graphical Representation of Motion <ul><li>Traveling in the opposite direction? </li></ul>
  90. 90. Graphical Representation of Motion <ul><li>Very complicated motion can be studied using graphs. </li></ul><ul><li>Watch the two stage rocket as it is launched, run out of fuel and returns to Earth. </li></ul>
  91. 91. Graphical Representation of Motion
  92. 92. Graphical Representation of Motion <ul><li>The Moving Man </li></ul><ul><li>Motion Graphs </li></ul><ul><li>If you would like more practice at drawing motion graphs diagrams, try this web site: </li></ul><ul><li> </li></ul>
  93. 93. Equations for Uniformly Accelerated Motion <ul><li>The previous equations used can : </li></ul><ul><ul><li>be applied to all types of motion. </li></ul></ul><ul><li>However, when acceleration is constant : </li></ul><ul><ul><li>our mathematical description can be taken further. </li></ul></ul>
  94. 94. Equations for Uniformly Accelerated Motion
  95. 95. Equations for Uniformly Accelerated Motion <ul><li>In the graph on the previous slide , as the slope is constant ; </li></ul><ul><ul><li>the acceleration is also constant. </li></ul></ul><ul><li>If we let u and v be the velocity at the start and end of the time interval t ; </li></ul><ul><ul><li>average velocity can be described as; </li></ul></ul>
  96. 96. Equations for Uniformly Accelerated Motion <ul><li>and from previously, </li></ul><ul><li>Comparing these equations, we can say: </li></ul>
  97. 97. Equations for Uniformly Accelerated Motion <ul><li>The average acceleration can also be given as: </li></ul><ul><li>Rearranging the equation gives: </li></ul>
  98. 98. Equations for Uniformly Accelerated Motion <ul><li>These two equations describe motion : </li></ul><ul><ul><li>at constant acceleration , </li></ul></ul><ul><ul><li>in terms of five variables; </li></ul></ul><ul><ul><li>u , v , t , s , a . </li></ul></ul>
  99. 99. Equations for Uniformly Accelerated Motion <ul><li>Equation  does not use a and ; </li></ul><ul><ul><li> does not use s . </li></ul></ul><ul><li>Using algebra, we can derive 3 more equations, </li></ul><ul><ul><li>each one not using one variable listed above. </li></ul></ul>
  100. 100. Equations for Uniformly Accelerated Motion <ul><li>rearranging gives; </li></ul>
  101. 101. Equations for Uniformly Accelerated Motion <ul><li>Equation 2: v = u + at rearranged becomes; </li></ul><ul><li>substituting for t we have: </li></ul>
  102. 102. Equations for Uniformly Accelerated Motion <ul><li>v 2 - u 2 = 2as  </li></ul>
  103. 103. Equations for Uniformly Accelerated Motion <ul><li>Equation  v = u + at </li></ul><ul><li>Substituting for v from  we get; </li></ul>
  104. 104. Equations for Uniformly Accelerated Motion s=ut + ½at 2 
  105. 105. Equations for Uniformly Accelerated Motion <ul><li>Exercise: </li></ul><ul><li>Try and derive  , which is independent of u . </li></ul>
  106. 106. Equations for Uniformly Accelerated Motion VARIABLES EQUATION u v t s a u v t s a v = u + at u v t s a v 2 - u 2 = 2as u v t s a u v t s a
  107. 107. Vertical Motion of Objects <ul><li>The Earth, near its surface, has a uniform gravitational field. </li></ul><ul><li>Objects that move perpendicular to the Earth’s surface, </li></ul><ul><ul><li>move parallel to the gravitational field. </li></ul></ul>
  108. 108. Vertical Motion of Objects <ul><li>The acceleration experienced by the object will be constant. </li></ul><ul><li>a = 9.81 m s -2 down. </li></ul><ul><li>If the object is moving towards the Earth, </li></ul><ul><ul><li>the object’s speed will increase at the rate of, </li></ul></ul><ul><ul><li>9.81 m s -1 every second. </li></ul></ul><ul><li>This is assuming that air resistance is negligible. </li></ul>
  109. 109. Vertical Motion of Objects
  110. 110. Vertical Motion of Objects <ul><li>True or False? </li></ul><ul><li>1. The elephant encounters a smaller force of air resistance than the feather and therefore falls faster. </li></ul><ul><li>2. The elephant has a greater acceleration of gravity than the feather and therefore falls faster.   </li></ul>
  111. 111. Vertical Motion of Objects <ul><li>3. Both elephant and feather have the same force of gravity, yet the acceleration of gravity is greatest for the elephant. </li></ul><ul><li>4. Both elephant and feather have the same force of gravity, yet the feather experiences a greater air resistance. </li></ul>
  112. 112. Vertical Motion of Objects <ul><li>5. Each object experiences the same amount of air resistance, yet the elephant experiences the greatest force of gravity. </li></ul><ul><li>6. Each object experiences the same amount of air resistance, yet the feather experiences the greatest force of gravity. </li></ul>
  113. 113. Vertical Motion of Objects <ul><li>7. The feather weighs more than the elephant, and therefore will not accelerate as rapidly as the elephant. </li></ul><ul><li>8. Both elephant and feather weigh the same amount, yet the greater mass of the feather leads to a smaller acceleration. </li></ul>
  114. 114. Vertical Motion of Objects <ul><li>9. The elephant experiences less air resistance and than the feather and thus reaches a larger terminal velocity. </li></ul><ul><li>10. The feather experiences more air resistance than the elephant and thus reaches a smaller terminal velocity. </li></ul>
  115. 115. Vertical Motion of Objects <ul><li>11. The elephant and the feather encounter the same amount of air resistance, yet the elephant has a greater terminal velocity. </li></ul><ul><li>If you answered True to any of these questions, you need to review your understanding. </li></ul>
  116. 116. Vertical Motion of Objects
  117. 117. Vertical Motion of Objects <ul><li>Objects which move vertically upwards: </li></ul><ul><ul><li>will slow down at the rate of 9.81 m s -1 , </li></ul></ul><ul><ul><li>every second until it is stationary. </li></ul></ul><ul><li>It will then start to accelerate towards the earth at: </li></ul><ul><ul><li>the rate of 9.81 m s -1 every second. </li></ul></ul><ul><li>Vertical Motion </li></ul>
  118. 118. Air Resistance <ul><li>Objects falling in the Earth’s uniform gravitational field have two opposing forces acting on it. </li></ul><ul><ul><li>Gravity acts towards the Earth, pulling the object downward. </li></ul></ul><ul><li>Any resistance force opposes motion. </li></ul>
  119. 119. Air Resistance <ul><li>This means, in this case ; </li></ul><ul><ul><li>air resistance acts upwards. </li></ul></ul><ul><li>The faster the object falls ; </li></ul><ul><ul><li>the greater the air resistance. </li></ul></ul><ul><li>As the object accelerates under the force of gravity ; </li></ul><ul><ul><li>the greater the air resistance. </li></ul></ul>
  120. 120. Air Resistance <ul><li>This slows the rate at which the object ; </li></ul><ul><ul><li>accelerates towards the earth. </li></ul></ul><ul><li>Eventually, the two forces cancel each other out. </li></ul><ul><li>This means there is no acceleration and ; </li></ul><ul><ul><li>the velocity becomes constant. </li></ul></ul><ul><li>This is known as terminal velocity. </li></ul>
  121. 121. Air Resistance <ul><li>The terminal velocity is different : </li></ul><ul><ul><li>for different objects. </li></ul></ul><ul><li>Sky diver’s have a terminal velocity of : </li></ul><ul><ul><li>about 150 to 200 km h -1 depending on ; </li></ul></ul><ul><ul><ul><li>the mass of the sky diver and , </li></ul></ul></ul><ul><ul><ul><li>their orientation. </li></ul></ul></ul>
  122. 122. Air Resistance <ul><li>Sky divers tend to try to: </li></ul><ul><ul><li>increase the air resistance thereby , </li></ul></ul><ul><ul><ul><li>reducing the terminal velocity. </li></ul></ul></ul><ul><li>This gives them a longer free fall. </li></ul><ul><li>The parachute : </li></ul><ul><ul><li>greatly increases air resistance and , </li></ul></ul><ul><ul><li>cuts the terminal velocity to , </li></ul></ul><ul><ul><ul><li>between 15 and 20 km h -1 . </li></ul></ul></ul>
  123. 123. Air Resistance
  124. 124. Forces and Dynamics <ul><li>To maintain the velocity of an object such as a wheelbarrow : </li></ul><ul><ul><li>a push or pull is required. </li></ul></ul><ul><li>To maintain acceleration : </li></ul><ul><ul><li>either by changing the speed of an object or changing its direction, </li></ul></ul><ul><ul><li>a push or pull is required. </li></ul></ul>
  125. 125. Forces and Dynamics <ul><li>The name we use for this push or pull is : </li></ul><ul><ul><li>force . </li></ul></ul><ul><li>A force is required to : </li></ul><ul><ul><li>keep things moving or , </li></ul></ul><ul><ul><li>change their motion. </li></ul></ul><ul><li>A force can also be used to : </li></ul><ul><ul><li>deform an object. </li></ul></ul><ul><li>This happens when a tennis racquet hits a tennis ball. </li></ul>
  126. 126. Forces and Dynamics <ul><li>Using an air track, we can eliminate (or very nearly) ; </li></ul><ul><ul><li>the force of friction on a glider. </li></ul></ul><ul><li>This leaves the force of gravity and : </li></ul><ul><ul><li>the force of the hand that pushes a glider. </li></ul></ul>
  127. 127. Forces and Dynamics <ul><li>How can we determine the effect of a single force when ; </li></ul><ul><ul><li>there are two forces acting? </li></ul></ul><ul><li>We must make an assumption. </li></ul><ul><li>Gravitational forces being vertical : </li></ul><ul><ul><li>have no effect on motion , </li></ul></ul><ul><ul><li>in the horizontal plane. </li></ul></ul>
  128. 128. Forces and Dynamics <ul><li>Recall the air track practical : </li></ul><ul><ul><li>once the hand released the glider, </li></ul></ul><ul><ul><li>it moved with a constant velocity. </li></ul></ul><ul><li>This indicates that the assumption is true. </li></ul>
  129. 129. Forces and Dynamics <ul><li>There are cases when more than one force acts on a object. </li></ul><ul><li>These forces may oppose each other: </li></ul><ul><ul><li>Pushing an object forward and, </li></ul></ul><ul><ul><li>Friction backward </li></ul></ul><ul><ul><li>Or a tug of war </li></ul></ul>
  130. 130. Forces and Dynamics <ul><li>Or they may act at some other angle: </li></ul><ul><ul><li>Running for the football and, </li></ul></ul><ul><ul><li>being hip and shouldered. </li></ul></ul><ul><li>A diagram can be drawn to show these forces. </li></ul><ul><li>It is called a Free-body Diagram. </li></ul>
  131. 131. Forces and Dynamics <ul><li>Consider a statue resting on a table. </li></ul><ul><li>What forces are acting on it? </li></ul><ul><li>As it is at rest: </li></ul><ul><ul><li>there are no unbalanced forces. </li></ul></ul><ul><li>Gravity ( F g ) is acting downwards. </li></ul><ul><li>What force balances gravity? </li></ul>
  132. 132. Forces and Dynamics <ul><li>The table exerts an upward force. </li></ul><ul><li>The table is compressed by the statue </li></ul><ul><ul><li>Due to its elasticity it pushes upwards. </li></ul></ul><ul><li>This is called a contact force. </li></ul><ul><ul><li>As it occurs when two objects are in contact. </li></ul></ul>
  133. 133. Forces and Dynamics <ul><li>When the contact force acts: </li></ul><ul><ul><li>perpendicular to the common surface of contact, it is called, </li></ul></ul><ul><ul><li>the Normal Force ( F N ). </li></ul></ul><ul><li>A diagram of the objects and their forces can be drawn. </li></ul>
  134. 134. Forces and Dynamics
  135. 135. Forces and Dynamics <ul><li>We are interested in the forces and so: </li></ul><ul><ul><li>we will only include vectors for the forces. </li></ul></ul>F g F N
  136. 136. Forces and Dynamics <ul><li>Notice: </li></ul><ul><li>The vectors are of the same length: </li></ul><ul><ul><li>Indicating same magnitude of force. </li></ul></ul><ul><li>The vectors are in opposite directions: </li></ul><ul><ul><li>Indicating the forces oppose each other. </li></ul></ul><ul><li>The vectors have been labeled: </li></ul><ul><ul><li>with appropriate symbols. </li></ul></ul>
  137. 137. Forces and Dynamics <ul><li>What happens when you add a third force? </li></ul><ul><li>Consider a box at rest on a table. </li></ul><ul><li>What forces are acting on it? </li></ul><ul><ul><li>F g and: </li></ul></ul><ul><ul><li>F N </li></ul></ul>
  138. 138. Forces and Dynamics F g F N
  139. 139. Forces and Dynamics <ul><li>What if you were to push down on the box with a force of 40 N? </li></ul>F N F g 40 N
  140. 140. Forces and Dynamics <ul><li>Notice that the combined length of: </li></ul><ul><ul><li>the F g and 40 N vectors, </li></ul></ul><ul><ul><li>equal the F N vector. </li></ul></ul><ul><li>This indicates the box is stationary. </li></ul><ul><li>What happens if you pull up with a force of 40 N? </li></ul>
  141. 141. Forces and Dynamics F g F N 40 N
  142. 142. Forces and Dynamics <ul><li>Notice that the combined length of: </li></ul><ul><ul><li>the F N and 40 N vectors, </li></ul></ul><ul><ul><li>equal the F g vector. </li></ul></ul><ul><li>This indicates the box is stationary. </li></ul>
  143. 143. Forces and Dynamics <ul><li>What forces act on a shopping trolley? </li></ul>
  144. 144. Forces and Dynamics
  145. 145. Forces and Dynamics <ul><li>As F N = F g </li></ul><ul><li>The force making the trolley move is equal to: </li></ul><ul><li>F p = Force supplied by the person. </li></ul><ul><li>This is an unbalanced force and: </li></ul><ul><ul><li>it will cause an acceleration. </li></ul></ul>
  146. 146. Forces and Dynamics <ul><li>What happens when the ( 10 kg) box in the earlier example is: </li></ul><ul><ul><li>connected to another box (12 kg), </li></ul></ul><ul><ul><li>by a string and both, </li></ul></ul><ul><ul><li>are pulled along a table? </li></ul></ul><ul><li>What do the Free Body Diagrams look like? </li></ul>
  147. 147. Forces and Dynamics
  148. 148. Forces and Dynamics Box 1 F N F g T F P
  149. 149. Forces and Dynamics <ul><li>Notice: </li></ul><ul><li>The vectors F g and F N are of the same length: </li></ul><ul><ul><li>Indicating same magnitude of force. </li></ul></ul><ul><ul><li>No vertical motion </li></ul></ul><ul><li>The vector F p is longer than T indicating: </li></ul><ul><ul><li>acceleration to the right </li></ul></ul>
  150. 150. Forces and Dynamics Box 2 F N F g T
  151. 151. Forces and Dynamics <ul><li>Free Body Diagram </li></ul><ul><li>Another one </li></ul><ul><li>If you would like more practice at drawing free body diagrams, try this web site </li></ul><ul><li> </li></ul>
  152. 152. Forces and Dynamics <ul><li>Suppose you wanted to see what was in the box in the earlier example. </li></ul><ul><li>You pull it towards you: </li></ul><ul><ul><li>using a string attached with, </li></ul></ul><ul><ul><li>a force of 40 N at, </li></ul></ul><ul><ul><li>an angle of 30 o above the horizontal.. </li></ul></ul>
  153. 153. Forces and Dynamics
  154. 154. Forces and Dynamics <ul><li>This becomes more difficult to analyse . </li></ul><ul><li>Notice F N  F g </li></ul><ul><li>This is because some of the force supplied by the person is: </li></ul><ul><ul><li>acting in an upward direction. </li></ul></ul><ul><li>Forces can be resolved into components. </li></ul>
  155. 155. Forces and Dynamics <ul><li>Forces can be resolved into two components : </li></ul><ul><ul><li>horizontal and vertical. </li></ul></ul>
  156. 156. Forces and Dynamics <ul><li>This is useful when forces are applied at an angle but : </li></ul><ul><ul><li>the effective force is in a particular direction , </li></ul></ul><ul><ul><li>such as pushing a roller </li></ul></ul>
  157. 157. Forces and Dynamics <ul><li>Forces can combine to give very different results. </li></ul><ul><li>A 10 N block hanging vertically from one horizontal string : </li></ul><ul><ul><li>when measured by a spring balance would read , </li></ul></ul><ul><ul><li>10N. </li></ul></ul>
  158. 158. Forces and Dynamics
  159. 159. Forces and Dynamics <ul><li>As the block is stationary : </li></ul><ul><ul><li>the force of the scale pulling up balances , </li></ul></ul><ul><ul><li>the force downwards supplied by , </li></ul></ul><ul><ul><li>the weight. </li></ul></ul><ul><li>If 2 spring balances support the block : </li></ul><ul><ul><li>the total weight will be the same. </li></ul></ul>
  160. 160. Forces and Dynamics <ul><li>However each spring balance will support equal amounts : </li></ul><ul><ul><li>ie 5N. </li></ul></ul>
  161. 161. Forces and Dynamics <ul><li>Again the weight down is balanced by : </li></ul><ul><ul><li>the two springs supplying a force up. </li></ul></ul><ul><li>In all situations the two forces vectors : </li></ul><ul><ul><li>one vertically downward and , </li></ul></ul><ul><ul><li>the other vertically upwards , </li></ul></ul><ul><ul><li>must balance. </li></ul></ul>
  162. 162. Forces and Dynamics <ul><li>This looks logical for a vertical orientation but how do they work for non-vertical situations? </li></ul>
  163. 163. Forces and Dynamics <ul><li>The vertically up vector must still equal 10 N and so by vector addition : </li></ul><ul><ul><li>the two scales will read more than 5 N each , </li></ul></ul><ul><ul><li>perhaps 10 N. </li></ul></ul><ul><li>If the angle is increased to 60 o from the vertical : </li></ul><ul><ul><li>120 o between the scales , </li></ul></ul><ul><ul><li>The reading is much higher. </li></ul></ul>
  164. 164. Forces and Dynamics
  165. 165. Forces and Dynamics <ul><li>If we continue to do this until the angle is 90 o , </li></ul><ul><ul><li>the horizontal rope must support a force that is , </li></ul></ul><ul><ul><li>much greater than the original weight. </li></ul></ul>
  166. 166. Forces and Dynamics <ul><li>Gymnasts who hold their arms out horizontally from the rings : </li></ul><ul><ul><li>must supply a force that is considerably greater than their own body weight. </li></ul></ul><ul><li>This is an extreme test of strength. </li></ul>
  167. 167. Forces and Dynamics <ul><li>To determine exact values for the components: </li></ul><ul><ul><li>Trigonometry must be used. </li></ul></ul>
  168. 168. Forces and Dynamics <ul><li>F vert = opposite side </li></ul><ul><li>F horiz = adjacent side </li></ul><ul><li>F = applied force </li></ul><ul><li>sin  = opposite/hypotenuse </li></ul><ul><li>Opposite = F vert = F sin  </li></ul><ul><li>cos  = adjacent/hypotenuse </li></ul><ul><li>Adjacent = F horiz = F cos  </li></ul>
  169. 169. Forces and Dynamics <ul><li>Springs </li></ul><ul><li>If you hang a weight from a spring: </li></ul><ul><ul><li>apply a force, </li></ul></ul><ul><ul><li>it stretches. </li></ul></ul><ul><li>Add more weight: </li></ul><ul><ul><li>It stretches more </li></ul></ul><ul><li>Remove the weights: </li></ul><ul><ul><li>It returns to its original length. </li></ul></ul>
  170. 170. Forces and Dynamics <ul><li>The spring is said to be elastic. </li></ul><ul><li>By stretching a spring: </li></ul><ul><ul><li>or compressing it, </li></ul></ul><ul><li>The amount it is stretched is: </li></ul><ul><ul><li>Directly proportional to the force applied. </li></ul></ul><ul><li>This was first noticed by the British Physicist: </li></ul><ul><ul><li>Robert Hooke </li></ul></ul><ul><ul><li>In the 17 th century </li></ul></ul>
  171. 171. Forces and Dynamics <ul><li>It is now remembered as Hooke’s Law. </li></ul><ul><li>The applied force F and: </li></ul><ul><ul><li>The extension (or compression) x </li></ul></ul><ul><li>Can be represented mathematically. </li></ul><ul><li>F  x </li></ul><ul><li>A graph of this can be shown. </li></ul>
  172. 172. Forces and Dynamics F x
  173. 173. Forces and Dynamics <ul><li>Compare the line to the equation of a straight line: </li></ul><ul><li>y = mx + c </li></ul><ul><li>y = F </li></ul><ul><li>x = x </li></ul><ul><li>c = 0 (as through the origin) </li></ul><ul><li>To turn the proportionality into an equation: </li></ul>
  174. 174. Forces and Dynamics <ul><li>F = mx </li></ul><ul><li>The slope is constant and so is given a special symbol: </li></ul><ul><ul><li>k </li></ul></ul><ul><li>All constants in Physics have special symbols. </li></ul><ul><li>The equation now becomes: </li></ul><ul><li>F = kx </li></ul>
  175. 175. Forces and Dynamics <ul><li>The full form of the equation is: </li></ul><ul><li>F = -kx </li></ul><ul><li>This is because the force is a restoring force. </li></ul><ul><ul><li>It always acts to try an return the spring to the original length. </li></ul></ul><ul><li>Hookes Law </li></ul>
  176. 176. Forces and Dynamics <ul><li>You will need to be able to: </li></ul><ul><li>draw a graph </li></ul><ul><ul><li>Given data </li></ul></ul><ul><li>Determine the spring constant k : </li></ul><ul><ul><li>For a particular spring. </li></ul></ul>
  177. 177. Forces and Dynamics <ul><li>Determine the force required to: </li></ul><ul><ul><li>extend a spring by a certain amount </li></ul></ul><ul><li>Determine the amount a spring is stretched by: </li></ul><ul><ul><li>for a given force. </li></ul></ul>
  178. 178. Forces and Dynamics <ul><li>An object that moves with constant velocity : </li></ul><ul><ul><li>requires no force. </li></ul></ul><ul><li>Although this is one experiment, it can be shown to be true : </li></ul><ul><ul><li>in all situations </li></ul></ul><ul><ul><li>whether it is a person ice-skating or , </li></ul></ul><ul><ul><ul><li>a person slipping on a banana peel. </li></ul></ul></ul>
  179. 179. Forces and Dynamics <ul><li>Newton investigated this phenomenon, </li></ul><ul><ul><li>which he embodied in his first law: </li></ul></ul><ul><li>Provided no external force acts, the velocity of any object will remain constant unless an unbalanced force acts upon the object. </li></ul>
  180. 180. Forces and Dynamics <ul><li>A body is resistant to change. </li></ul><ul><li>This resistance of a body to change is called inertia . </li></ul><ul><li>Does this hold for a stationary object? </li></ul><ul><li>A stationary object has zero velocity : </li></ul><ul><ul><li>and so it will remain at zero. </li></ul></ul>
  181. 181. Forces and Dynamics <ul><li>Plates on a tablecloth are at rest. </li></ul><ul><li>If you pull the tablecloth quickly enough from under the plates : </li></ul><ul><ul><li>t he small and brief force of friction , </li></ul></ul><ul><ul><li>between the plates and the tablecloth is , </li></ul></ul><ul><ul><li>not significant enough to , </li></ul></ul><ul><ul><li>appreciably move the dishes. </li></ul></ul>
  182. 182. Forces and Dynamics <ul><li>Does this hold for tug of war contest? </li></ul><ul><li>When the two sides are even : </li></ul><ul><ul><li>the forces are equal in magnitude and , </li></ul></ul><ul><ul><li>opposite direction. </li></ul></ul><ul><li>The resultant force will be zero. </li></ul><ul><li>This implies that the two sides will not move. </li></ul>
  183. 183. Forces and Dynamics <ul><li>Does this apply to a car moving on a straight road ; </li></ul><ul><ul><li>at a constant velocity? </li></ul></ul><ul><li>The car will slow to a stop unless : </li></ul><ul><ul><li>the force applied by the engine continues. </li></ul></ul>
  184. 184. Forces and Dynamics <ul><li>This appears to disobey the law : </li></ul><ul><ul><li>until friction is taken into account. </li></ul></ul><ul><li>We can ignore gravity because : </li></ul><ul><ul><li>it is acting in a vertical direction. </li></ul></ul><ul><li>Friction however is a retarding force acting : </li></ul><ul><ul><li>in the opposite direction to the motion. </li></ul></ul>
  185. 185. Forces and Dynamics <ul><li>This is an example of translational equilibrium. </li></ul><ul><li>Two forces are being applied in opposite directions. </li></ul><ul><li>As the forces cancel each other out, </li></ul><ul><ul><li>no acceleration occurs. </li></ul></ul><ul><li>The object continues to move at: </li></ul><ul><ul><li>a constant velocity. </li></ul></ul>