Chapter 4: Problem 34

When two trains, going in opposite directions, are passing on tracks that are laid out close together, the train cars can often be seen to be leaning in toward one another where they are in proximity. How might the air passing through the narrow gap separating the two trains contribute to the observed attraction between their cars?

Short Answer

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Answer: The train cars appear to lean towards each other due to a pressure difference created by the high velocity of the air passing through the narrow gap between the train cars. This pressure difference results in a net inward force that causes the train cars to lean towards each other.

Step by step solution

(Understanding Bernoulli's Principle)

(Bernoulli's principle states that for an incompressible, steadily flowing fluid, the sum of its pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline. Mathematically, it is given by: \(P + \frac{1}{2}\rho v^2 + \rho gh = constant\), where \(P\) is the pressure, \(\rho\) is the density of the fluid, \(v\) is the fluid velocity, \(g\) is the acceleration due to gravity, and \(h\) is the height above a reference point.)

(Applying Bernoulli's Principle to the Problem)

(In this problem, we will apply Bernoulli's principle to the air flowing between the two train cars. As the trains pass each other, the air in the gap between them is forced to flow at a higher velocity. According to the Bernoulli's principle, if the velocity of the fluid (air) increases, the pressure will decrease, causing a pressure difference between the gap and the surrounding air.)

(Understanding the Pressure Difference and Attraction between Train Cars)

(The pressure inside the gap is lower than the pressure outside the gap due to the high velocity of the air passing through the gap. This pressure difference causes a net inward force on the train cars, making them appear to lean towards each other. The higher the velocity of the air through the gap, the greater the pressure difference and attraction between the train cars.)

(Considering the Effect of Train Car Shapes)

(It is important to note that the actual effect of this pressure difference on the train cars also depends on their shape. Aerodynamic designs can minimize the pressure difference and thus reduce the effect of the air passing through the gap. Modern train designs often incorporate such aerodynamic features to minimize resistance and improve energy efficiency.) In conclusion, the air passing through the narrow gap between two trains moving in opposite directions leads to a decrease in pressure within the gap due to Bernoulli's principle, causing the train cars to lean towards each other as a result of the net inward force created by the pressure difference.

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Short Answer

Step by step solution

(Understanding Bernoulli's Principle)

(Applying Bernoulli's Principle to the Problem)

(Understanding the Pressure Difference and Attraction between Train Cars)

(Considering the Effect of Train Car Shapes)

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