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8 July, 03:06

You're investigating a subway accident in which a train derailed while rounding an unbanked curve of radius 150 m, and you're asked to estimate whether the train exceeded the 35-km/h speed limit for this curve. you interview a passenger who had been standing and holding onto a strap; she noticed that an unused strap was hanging at about a 15 â angle to the vertical just before the accident.

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  1. 8 July, 06:46
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    Refer to the diagram shown below.

    v = 35 km/h, the speed limit of the train

    r = 150 m, the radius of the curve

    ω = angular velocity

    m = the mass of the strap

    θ° = 15, the angle the strap makes with the vertical

    T = tension in the strap

    Note that

    v = 35 km/h = 35*0.2778 m/s = 9.7223 m/s

    The tangential velocity is v = rω, therefore the angular vcelocity is

    ω = (9.7223 m/s) / (150 m) = 0.0648 rad/s

    The centripetal force tending to make the train derail causes the strap to make an angle of 15 with the vertical.

    Let θ = the maximum allowable angle at 35 km/h.

    For horizontal equilibrium,

    Tsin (θ) = mrω²

    For vertical equilibrium,

    Tcos (θ) = mg

    Therefore

    tan (θ) = (rω²) / g

    = [ (150 m) * (0.0648 rad/s) ] / (9.8 m/s²)

    = 0.0643

    θ = tan⁻¹ 0.0643 = 3.7°

    Because 15 > 3.7, we conclude that the train exceeded the 35 km/h speed limit when rounding the curve.

    Answer: The train exceeded the 35 km/h speed limit.
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