Uniform circular motion worked examples

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Uniform circular motion worked examples

  1. 1. Uniform Circular Motionv is constantv is constantly changing r 1
  2. 2. Examples of UCMhttp://www.youtube.com/watch?v=a3N9BanDc6E&list=PL6413B12522DDEA5D 2
  3. 3. Tangential speedThe distance travelled in onerotation is the circumference s  2r rThe time to travel around thecircle is T. s 2r v  t T 2r v T 3
  4. 4. Example : Tangential SpeedA merry-go-round does 5 complete revolutions in aminute. It has a radius of 8 metres.Calculate the tangential speed. 4
  5. 5. 2 rv T 5
  6. 6. 2 rv T 6
  7. 7. 2 rv T 7
  8. 8. ∆ Centripetal Acceleration To find the instantaneous acceleration you must make t (the time interval) very small v a t   v  vf - vi  vf v  - vi t is small, approaching zero 8
  9. 9. ∆ Centripetal Acceleration At any instant the centripetal acceleration is towards the centre of the circle. SO- Using vector subtraction show centripetal acceleration is directed towards the centre of a circle over a small time interval. 9
  10. 10. Centripetal AccelerationMagnitude calculated by : 2 v ac  r 10
  11. 11. Centripetal AccelerationMagnitude calculated by : 2 v ac  r 11
  12. 12. Note 2 v if ac  then r ac  v 2If you double the tangential speed, youquadruple the centripetal acceleration 12
  13. 13. 1 also ac  rIf you double the radius, you halve thecentripetal acceleration 13
  14. 14. Example : The Effect of Radius on Centripetal Acceleration The bobsled track at the 1994 Olympics in Lillehammer, Norway, contained turns with radii of 33 m and 24 m. Find the centripetal acceleration at each turn for a speed of 34 m/s, a speed that was achieved in the two-man event. 14
  15. 15. 2 vac  r 15
  16. 16. v2ac  r 16
  17. 17. v2ac  r 17
  18. 18. Centripetal Force - FCBoth FC and aC are at right angles to thevelocity, directed towards the centre of thecircle. F  ma 2 va  c r 2 Quantity: force , F mv FC  Units: newtons, N r 18
  19. 19. Example: Centripetal ForceA model airplane has a mass of 0.90 kg andmoves at a constant speed on a circle that isparallel to the ground.Find the tensionin the 17.0m long stringif the plane circlesonce every 3 seconds. 19
  20. 20. mv 2FC  r 2 rv T 20
  21. 21. aC provided by the tension force aC provided by the normal force Describe situations in which the centripetal acceleration is caused by a tension force, a frictional force, a gravitational force, or a normal forceaC provided by the frictional force aC provided by the gravitational force 21
  22. 22. aC provided by the tension force aC provided by the normal forceA bob on a string being whirled A bobsled travelling around anaround . Olympic bobsled track.aC provided by the frictional force aC provided by the gravitational forceA car driving around a flat The Moon orbiting the Earth.roundabout. 22
  23. 23. Free body diagrams Box on a table FN    FgFNET  Fg  FN  0N• Draw a dot at the centre of mass• draw labelled force vectors from the centre of mass• If body is not accelerating the forces are balanced 23
  24. 24. Box on a slope, not moving   FN Ffriction   FNET  0 Fg• Normal force is perpendicular to the surface of contact 24
  25. 25. Hanging bob 25
  26. 26. Car stationary 26
  27. 27. Car constant v 27
  28. 28. Car accelerating 28
  29. 29. Draw free body diagrams for a fridge• Pushing against a fridge (not moving)• Pushing against a fridge (accelerating)• Pushing against a fridge (constant velocity) 29
  30. 30. Car going around a bend SO: Identify the vertical and horizontal forces on a vehicle moving with constant velocity on a flat horizontal road. 2 mv F friction  r 30
  31. 31. Factors affecting cornering 2 mv F friction  r r – radius of curvature m – mass of vehicle v - speed of vehicle Ffriction – depends on • tyres • road surface 31
  32. 32. Banked CurvesExplain that when a vehicle travels round a banked curve at thecorrect speed for the banking angle, the horizontal component ofthe normal force on the vehicle (not the frictional force on thetyres) causes the centripetal acceleration. 32
  33. 33. A car is going around a friction-free banked curve. The radius of the curve is r. FN sin  that points toward the center C 2 mv FC  FN sin   rFN cos  and, since the car does not accelerate in thevertical direction, this component must balance theweight mg of the car. FN cos  mg 33
  34. 34. FN sin  mv / r 2  FN cos mg 2 v tan   rgAt a speed that is too small for a given  , a carwould slide down a frictionless banked curve: ata speed that is too large, a car would slide off thetop. 34
  35. 35. Want to watch the derivation?http://www.youtube.com/watch?v=OlUlLglTEn4 Derive the equation relating the banking angle to the speed of the vehicle and the radius of curvature . 35
  36. 36. Banking Angle Example A curve has a radius of 50 m and a banking angle of 15o. What is the ideal speed (no friction required between cars tyres and the surface) for a car on this curve?Solve problems involving the use ofthe equation 36
  37. 37. Banking Angle ExampleA curve has a radius of 50 m and a banking angle of 15o.What is the ideal speed (no friction required between carstyres and the surface) for a car on this curve? 37
  38. 38. Banking Angle ExampleA curve has a radius of 50 m and a banking angle of 15o.What is the ideal speed (no friction required between carstyres and the surface) for a car on this curve? If the car negotiates the bend at 11.5ms-1 it can do so without 38 friction.
  39. 39. Conceptual QuestionIn a circus, a man hangs upside down from a trapeze, legs bentover the bar and arms downward, holding his partner. Is itharder for the man to hold his partner when the partner hangsstraight down and is stationary or when the partner is swingingthrough the straight-down position? 39
  40. 40. When they are moving in a circular arc they havecentripetal acceleration. The acrobat exerts anadditional pull compared to when they are stationary. 40
  41. 41. Proportionality examplesA car is traveling in uniform circular motion on a section of road whose radius is r. The road is slippery, and the car is just on the verge of sliding.(a) If the car’s speed was doubled, what would have to be the smallest radius in order that the car does not slide? Express your answer in terms of r.(b) What would be your answer to part (a) if the car were replaced by one that weighted twice as much? 41

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