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March 28, 2017

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The roller coaster starts with a velocity of 16 m/s. One of the riders is a small girl of mass 26 kg. Find her apparent weight when the roller coaster is at locations B and C. At these two locations, the track is circular, with the radii of curvature given.

rB = 10 m,
rC = 20 m,

The heights at points A, B, and C are
hA = 22 m,
hB = 32 m,
and
hC = 0.

Assume friction is negligible and ignore the kinetic energy of the wheels. (The figure is not necessarily drawn to scale.)

What is the weight at B?

  • Physics - ,

    Point A: h₁= 22 m, v₁=16 m/s;
    Point B: h₂= 32 m, r₂= 10 m, v₂=?;
    Δh₁₂ = h₂-h₁=32-22 = 10 m;
    Point C: h₃=0, r₃= 20 m, v₃=?;
    Δh₂₃ = h₂-h₃ = 32 – 0 = 32 m.

    KE₁+PE₁=PE₂ + KE₂,
    mv₁²/2 +mgh₁=mgh₂ +mv₂²/2,
    mv₁²/2= mv₂²/2+ mgh₂ - mgh₁=
    =mv₂²/2+ mg Δh₁₂.
    v₂ = sqrt{ v₁²- 2gΔh₁₂} =
    =sqrt{16² - 2•9.8•10} =7.75 m/s.

    For point B:
    ma₂=mg-N₂,
    N₂=mg-ma₂=m[g- (v₂²/r₂)]=
    = 26[ 9.8 – (7.75²/10)]= 98.8 N,
    N₂ (normal force) = W₂ (weight)
    W₂= 98.8 N.

    PE₂ +KE₂ = PE₃+KE₃,
    mgh₂ +mv₂²/2 = mgh₃ +mv₃²/2,
    mv₃²/2 = mgh₂ +mv₂²/2 - mgh₃=
    =mv₂²/2 + mg Δh₂₃ ,
    v₃ = sqrt{v₂² +2g Δh₂₃}=
    =sqrt{ 7.75² + 2•9.8•32) = 26.2 m/s.

    For point C:
    ma₃= N₃ – mg,
    N₃= mg+ma₃=m[g+ (v₃²/r₃)]=
    = 26[ 9.8 + (26.2²/20)]= 1148.2 N.
    N₃ (normal force) = W₃ (weight)
    W₃= 1148.2 N.

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