Work and energy


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Work and energy

  1. 1. Topic 2 Work and Energy
  2. 2. Contents <ul><li>Work and Energy </li></ul><ul><li>Kinetic Energy </li></ul><ul><li>Potential Energy </li></ul><ul><li>Elastic Potential Energy </li></ul><ul><li>Conservation of Energy </li></ul><ul><li>Power </li></ul><ul><li>Centripetal Acceleration </li></ul>
  3. 3. Work and Energy <ul><li>Kinetic energy is defined as: </li></ul><ul><li>E k = ½ mv 2 </li></ul><ul><li>If a particle is moving freely with no unbalanced force acting on it : </li></ul><ul><ul><li>NI tells us that it will move with constant velocity. </li></ul></ul><ul><li>This means that kinetic energy will also be constant. </li></ul>
  4. 4. Work and Energy <ul><li>What happens however if an unbalanced force acts? </li></ul><ul><li>A constant unbalanced force produces : </li></ul><ul><ul><li>a constant acceleration. </li></ul></ul><ul><li>One of the kinematic equations that can be used in this circumstance is: </li></ul>
  5. 5. Work and Energy <ul><li>v 2 – u 2 = 2 as </li></ul><ul><li>T o find the K.E. multiply both sides by ½m. </li></ul><ul><li>½ mv 2 - ½ mu 2 = mas </li></ul><ul><li>From NII, F = ma </li></ul><ul><li>½ mv 2 - ½ mu 2 = Fs </li></ul>
  6. 6. Work and Energy <ul><li> K.E. = Fs </li></ul><ul><li>The term on the RHS of the equation is called WORK . </li></ul>
  7. 7. Work and Energy <ul><li>The work done by a constant unbalanced force acting on a particle : </li></ul><ul><ul><li>which is moving in one dimension is given by , </li></ul></ul><ul><ul><li>the product of the unbalanced force and , </li></ul></ul><ul><ul><li>the displacement produced. </li></ul></ul><ul><li>W = Fs </li></ul>
  8. 8. Work and Energy <ul><li>This equation shows us that if an unbalanced force acts : </li></ul><ul><ul><li>there will always be a change in kinetic energy and , </li></ul></ul><ul><ul><li>an amount of work done. </li></ul></ul><ul><li>A glider moving at constant velocity on an air track has : </li></ul><ul><ul><li>no unbalanced force acting on it. </li></ul></ul>
  9. 9. Work and Energy <ul><li>However, if it is on a slope ; </li></ul><ul><ul><li>t here is an unbalanced force , </li></ul></ul><ul><ul><li>of gravity (weight), </li></ul></ul><ul><ul><li>acting on it and it will accelerate. </li></ul></ul><ul><li>This weight can be resolved into two components, </li></ul><ul><ul><li>parallel and perpendicular to the motion. </li></ul></ul>
  10. 10. Work and Energy
  11. 11. Work and Energy <ul><li>The perpendicular component of the weight : </li></ul><ul><ul><li>is balanced by the reaction force , </li></ul></ul><ul><ul><li>of the air track on the glider , </li></ul></ul><ul><ul><ul><li>air on the glider. </li></ul></ul></ul>
  12. 12. Work and Energy <ul><li>The unbalanced force is therefore the parallel component of the weight. </li></ul><ul><li>This force : </li></ul><ul><ul><li>multiplied by the displacement along the track gives , </li></ul></ul><ul><ul><li>the work done on the glider. </li></ul></ul>
  13. 13. Work and Energy <ul><li>What part does the angle of inclination play in calculating the work done? </li></ul>
  14. 14. Example <ul><li>A Woolworths supermarket trolley (that does move in the direction you push it), is pushed with a force of 200 N acting at an angle of 40 o to the ground. Find the effective horizontal force pushing the trolley along. </li></ul>
  15. 15. Solution <ul><li> = 40 o </li></ul><ul><li>I F I = 200 N </li></ul><ul><li>Draw vector diagram </li></ul>
  16. 16. Solution
  17. 17. Solution <ul><li>F H is the effective force pushing the trolley </li></ul><ul><li>F H = F cos  </li></ul><ul><li>F H = 200 x cos 40 o </li></ul><ul><li>F H = 200 x 0.7660444 </li></ul><ul><li>F H = 153 N Horizontally </li></ul>
  18. 18. Work and Energy <ul><li>Work can be determined by studying a force-displacement graph. </li></ul>Force (N) Displacement (m) 10 5
  19. 19. Work and Energy <ul><li>Area under graph = height x length </li></ul><ul><li>Area under graph = Force x displacement </li></ul><ul><li>Force x displacement = Work </li></ul><ul><li>Area under graph = Work done </li></ul><ul><li>Area under graph = 5 x 10 </li></ul><ul><li>Area under graph = 50 J </li></ul>
  20. 20. Work and Energy <ul><li>Work is easy to calculate when the force is constant. </li></ul><ul><li>What happens if the force is not constant? </li></ul><ul><li>Use a F vs. disp. graph. </li></ul>
  21. 21. Work and Energy Force (N) Displacement (m) 10 5
  22. 22. Work and Energy <ul><li>Work = Area under a F vs. Disp. Graph </li></ul><ul><li>Work = ½ (b x h) </li></ul><ul><li>Work = ½ (5 x 10) </li></ul><ul><li>Work = 25 J </li></ul>
  23. 23. Energy and Power <ul><li>Kinetic Energy </li></ul><ul><li>Push an object and it can move. </li></ul><ul><li>If an object moves: </li></ul><ul><ul><li>it is capable of doing work. </li></ul></ul><ul><li>The object has energy associated with its motion called: </li></ul><ul><ul><li>Kinetic Energy </li></ul></ul>
  24. 24. Energy and Power <ul><li>W = Fs </li></ul><ul><li>F = ma </li></ul><ul><li>W = mas </li></ul><ul><li>v 2 – u 2 = 2as </li></ul>
  25. 25. Energy and Power <ul><li>As W = mas </li></ul><ul><li>W = ½mv 2 – ½mu 2 </li></ul><ul><li>W =  ½mv 2 </li></ul><ul><li>The quantity  ½mv 2 is called: </li></ul><ul><ul><li>Kinetic Energy </li></ul></ul>
  26. 26. Energy and Power <ul><li>Kinetic Energy, E k , can be defined as: </li></ul><ul><ul><li>The product of half the object’s mass m , </li></ul></ul><ul><ul><li>and the square of its speed v . </li></ul></ul>
  27. 27. Energy and Power <ul><li>Potential Energy </li></ul><ul><li>Kinetic energy is the ‘energy of motion’. </li></ul><ul><li>We can develop an expression for the energy that is dependent on position; </li></ul><ul><ul><li>potential energy. </li></ul></ul>
  28. 28. Energy and Power <ul><li>Consider an object that is dropped from a height above the floor, h t : </li></ul><ul><ul><li>where the floor is at height h o . </li></ul></ul><ul><li>Displacement is given by s = h t - h o . </li></ul><ul><li>The unbalanced force is given by : </li></ul><ul><ul><li>the weight of the object m g . </li></ul></ul>
  29. 29. Energy and Power <ul><li>As W = Fs </li></ul><ul><li>W = m g ( h t - h o ) </li></ul><ul><li>or W = m g  h </li></ul><ul><li>This gives the work done in terms of the objects position. </li></ul><ul><li>This quantity mgh , is defined as the gravitational potential energy. </li></ul>
  30. 30. Energy and Power <ul><li>P.E. = m gh </li></ul><ul><li>Work can also be defined as : </li></ul><ul><ul><li>the change in gravitational potential energy. </li></ul></ul><ul><li>When an object falls : </li></ul><ul><ul><li>it loses gravitational potential energy, </li></ul></ul><ul><ul><li>and gains kinetic energy. </li></ul></ul>
  31. 31. Energy and Power <ul><li>Work can be calculated by the change in either of these two terms. </li></ul><ul><li>Generally, work is defined as the change in energy. </li></ul>
  32. 32. Energy and Power <ul><li>The relationship between E k , E p and work can be shown using a downhill skier. </li></ul>
  33. 33. Energy and Power <ul><li>Energy transformation can be shown using a roller coaster. </li></ul>
  34. 34. Energy and Power <ul><li>Elastic Potential Energy </li></ul><ul><li>Consider a spring that has been compressed. </li></ul><ul><li>When released for time t , </li></ul><ul><ul><li>the spring will return to , </li></ul></ul><ul><ul><li>the uncompressed position. </li></ul></ul>
  35. 35. Energy and Power <ul><li>This means there must be an unbalanced force acting. </li></ul><ul><li>This force is given by Hooke’s Law. </li></ul><ul><li>The restoring force in a spring is : </li></ul><ul><ul><li>proportional to its extension or compression. </li></ul></ul><ul><li>Graphically, it can be described as: </li></ul>
  36. 36. Energy and Power
  37. 37. Energy and Power <ul><li>Mathematically, it can be described as: </li></ul><ul><li>F = - kx </li></ul><ul><li>Where k is the slope of the graph. </li></ul>
  38. 38. Energy and Power <ul><li>The elastic potential energy can also be calculated. </li></ul><ul><li>E.P.E. = ½ kx 2 . </li></ul><ul><li>This suggests that as a spring is compressed or extended : </li></ul><ul><ul><li>the energy increases. </li></ul></ul>
  39. 39. Energy and Power <ul><li>Conservation of Energy </li></ul><ul><li>Consider a ball thrown vertically into the air. </li></ul><ul><li>It begins its motion with kinetic energy. </li></ul><ul><li>As it reaches it’s highest point : </li></ul><ul><ul><li>The E k is zero. </li></ul></ul>
  40. 40. Energy and Power <ul><li>At the same time, the G.P.E. has : </li></ul><ul><ul><li>increased. </li></ul></ul><ul><li>The loss of one type of energy : </li></ul><ul><ul><li>is balanced by the gain in another. </li></ul></ul><ul><li>Total Energy = mgh + ½mv 2 . </li></ul><ul><li>If a glass of whisky is pushed along a bar to a waiting gunslinger : </li></ul><ul><ul><li>is energy conserved ? </li></ul></ul>
  41. 41. Energy and Power <ul><li>In this case, </li></ul><ul><ul><li>the G.P.E. has not increased : </li></ul></ul><ul><ul><li>when the K.E. has decreased. </li></ul></ul><ul><li>This however is not an isolated system. </li></ul><ul><li>Energy has been lost to friction. </li></ul><ul><li>The total energy in any isolated system : </li></ul><ul><ul><li>is constant . </li></ul></ul>
  42. 42. Energy and Power <ul><li>A dart is fired out of a gun using a spring. </li></ul>
  43. 43. Energy and Power <ul><li>A 3 kg cart moves down the hill. </li></ul><ul><li>Calculate the E p lost and E k gained. </li></ul>
  44. 44. Energy and Power <ul><li>E p = mgh </li></ul><ul><li>E p = 3 x 9.8 x (0.40 – 0.05) </li></ul><ul><li>E p = 10.3 J </li></ul><ul><li>E k = ½ mv 2 </li></ul><ul><li>E k = ½ x 3 x 2.62 2 </li></ul><ul><li>E k = 10.3 J </li></ul><ul><li>Energy is conserved. </li></ul>
  45. 45. Energy and Power <ul><li>Energy can be expended to perform a useful function. </li></ul><ul><li>A device that turns energy into some useful form of work is called a: </li></ul><ul><ul><li>Machine </li></ul></ul>
  46. 46. Energy and Power <ul><li>Machines cannot turn all the energy used to run the machine into useful work. </li></ul><ul><li>In any machine, some energy goes to: </li></ul><ul><ul><li>atomic or molecular kinetic energy. </li></ul></ul><ul><li>This makes the machine warmer. </li></ul><ul><ul><li>Energy is dissipated as heat. </li></ul></ul>
  47. 47. Energy and Power <ul><li>The amount of energy converted into: </li></ul><ul><ul><li>useful work by the machine is called, </li></ul></ul><ul><ul><li>The efficiency . </li></ul></ul><ul><li>An example of a simple machine is: </li></ul><ul><ul><li>A pulley system. </li></ul></ul><ul><li>We can do 100 J of work. </li></ul>
  48. 48. Energy and Power <ul><li>Friction turn the pulleys which in turn rub on the axles. </li></ul><ul><li>This may dissipate 40 J of energy as heat. </li></ul><ul><li>The system is 60% efficient. </li></ul>
  49. 49. Energy and Power <ul><li>Efficiency can be expressed mathematically: </li></ul>
  50. 50. Energy and Power <ul><li>Power </li></ul><ul><li>Power is defined as : </li></ul><ul><ul><li>the rate at which work is done . </li></ul></ul><ul><li>Units: </li></ul><ul><li>Js -1 or Watts. </li></ul>
  51. 51. Energy and Power <ul><li>The work in this equation could be : </li></ul><ul><ul><li>the change in kinetic energy or , </li></ul></ul><ul><ul><li>the work done on a mass that has been lifted. </li></ul></ul><ul><li>It does not matter what form the energy takes : </li></ul><ul><ul><li>it is just the rate at which work is done. </li></ul></ul>
  52. 52. Energy and Power <ul><li>A 100W light globe produces 100 J of energy every second. </li></ul><ul><li>To give an idea of the size of 1 W, </li></ul><ul><ul><li>a jumping flea produces 10 -4 W, </li></ul></ul><ul><ul><li>a person walking 300 W and , </li></ul></ul><ul><ul><li>a small car 40 000 W. </li></ul></ul>
  53. 53. Uniform Circular Motion <ul><li>Centripetal Acceleration </li></ul><ul><li>A particle undergoing uniform circular motion is continually changing velocity : </li></ul><ul><ul><li> acceleration is changing. </li></ul></ul>
  54. 54. Uniform Circular Motion
  55. 55. Uniform Circular Motion <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>
  56. 56. Uniform Circular Motion <ul><li>The acceleration, </li></ul><ul><ul><li>which produces these velocity changes in a direction which is , </li></ul></ul><ul><ul><li>always towards the centre of the circular motion, is called : </li></ul></ul><ul><ul><li>centripetal (centre seeking) acceleration. </li></ul></ul>
  57. 57. Uniform Circular Motion <ul><li>Newton’s 2 nd law tells us that : </li></ul><ul><ul><li>a centripetal acceleration can only happen if , </li></ul></ul><ul><ul><li>there is an unbalanced force. </li></ul></ul>
  58. 58. Uniform Circular Motion <ul><li>Any particle undergoing uniform circular motion is acted upon by : </li></ul><ul><ul><li>an unbalanced force which is , </li></ul></ul><ul><ul><li>constant in magnitude and , </li></ul></ul><ul><ul><li>directed towards the centre of the circle. </li></ul></ul><ul><li>This is called Centripetal Force . </li></ul>
  59. 59. Uniform Circular Motion
  60. 60. Uniform Circular Motion <ul><li>With a centripetal force, the object moves in a circular path. </li></ul>
  61. 61. Uniform Circular Motion <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>
  62. 62. Uniform Circular Motion <ul><li>Examples include: </li></ul><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>
  63. 63. Uniform Circular Motion <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>
  64. 64. Uniform Circular Motion <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>
  65. 65. Uniform Circular Motion <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>
  66. 66. Uniform Circular Motion <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>