Question

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.

Answer 1

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.