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Horizontal and Vertical Circles . Car on Curve Car on banked Curve Horizontal vs. Vertical Circle Vertical circle at top Examples . More complicated examples 1. Car on Track with Friction Example 5-6 μ mg = mv 2 /r. More complicated examples 2. Car on Banked Track Example 5-7

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horizontal and vertical circles
Horizontal and Vertical Circles
  • Car on Curve
  • Car on banked Curve
  • Horizontal vs. Vertical Circle
  • Vertical circle at top
  • Examples
more complicated examples 1
More complicated examples 1
  • Car on Track with Friction
    • Example 5-6
    • μ mg = mv2/r
more complicated examples 2
More complicated examples 2
  • Car on Banked Track
    • Example 5-7
    • Don’t use tilted axes
    • FNcosθ = mg,
    • FN sinθ = mv2/r
    • tanθ = v2/rg
horizontal vs vertical circle
Horizontal vs.Vertical Circle
  • Horizontal - circular perp. (┴) to gravity
  • Vertical – circular inline with gravity
  • Situations
    • Ball on a cord (tension only pulls in)
    • Car going over hump (normal pushes up)
vertical circle at the top
Vertical circle at the top
  • Inside Problem
    • Tension force (ball on string) - v starts high, slowing down

mg + FT = mv2/r

  • Outside Problem
    • - Normal force - (car on hump) v starts low, speeding up

mg - FN = mv2/r

FT

mg

FT adds to mg for high RHS. As RHS gets smaller, mg takes over and FT not needed

Required ma starts high, gets smaller as v decreases

FN

mg

Required ma starts low, gets larger as v increases

FN balances mg for low RHS. As RHS gets larger more mg required and FN not needed

vertical circle examples
Vertical Circle Examples
  • Situations
    • Ball on a cord (tension only pulls in)
    • Car going over hump (normal pushes up)
  • Example 5-4 (ball on vertical string)
    • (“inside problem”)
  • Problem 14 (sports car on hump)
    • (“outside problem”)
  • Problem 15 (ferris wheel)
    • (“outside problem”)
  • Problem 16 (bucket with water)
    • (“inside problem”)
examples inside vertical circle
Examples - “Inside” Vertical Circle
  • Example 5-4 (ball on vertical string)
    • 0.15 kg ball on 1.1 m cord
    • What speed does string go slack at top?
    • What is tension for v = 5 m/s?
  • Problem 16 (bucket with water)
    • 2 kg bucket on 1.1 m cord
    • Tension at bottom 25 N, what is speed at bottom?
    • What speed does string go slack at top?
examples outside vertical circle
Examples - “Outside” Vertical Circle
  • Problem 14 (sports car on hump)
    • 950 kg car, 95 m radius hump
    • Goes over top at 22 m/s
    • What is normal force of road?
    • What is normal force on 72 kg driver?
    • What speed does driver feel weightless?
  • Problem 15 (ferris wheel)
    • 7.5 m radius ferris wheel
    • weightless at top
example roto ride
Example – Roto-Ride
  • 4.6 m drum radius
  • 0.5 rev/sec frequency (14.45 m/s)
  • Radial
  • Vertical
  • Dividing vertical by Radial
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