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.

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.