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.
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.
