Answer :
The required angle of banking for a train going at 12 m/s on a 400 m radius curve is approximately 2.11°. This angle allows the train to negotiate the curve without slipping, only relying on the normal force for the required centripetal force.
To calculate the necessary angle of banking for a train moving at 12 m/s on tracks 1.5 m apart on a curve of radius 400 m, we can use the principles of circular motion. The force providing the centripetal acceleration in this scenario is the horizontal component of the normal force exerted by the track, which depends on the angle of banking (θ) and the speed (v) of the train, as well as the radius (r) of the curve and the acceleration due to gravity (g).
The formula for the ideal bank angle is given by tan(θ) = v^2 / (r × g), where v is the velocity of the train, r is the radius of the curve, and g is the acceleration due to gravity (9.8 m/s2). We can substitute the values to find the angle θ. For v = 12 m/s, r = 400 m, and g = 9.8 m/s2, we get:
tan(θ) = 122 / (400 × 9.8)
tan(θ) = 144 / 3920
tan(θ) = 0.03673
θ = arctan(0.03673)
θ ≈ 2.11°
Therefore, the banking angle should be approximately 2.11° to prevent slipping of the train on the curve without the need for additional frictional forces.
