Banked Turn

Overview

Engineering reference data for Banked Turn in dynamics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Momentum

$p = mv$

Mass × velocity.

Work

$W = Fd\cos\theta$

Force × displacement × cos(angle).

Variables

References

• Original Source

Practical Example: Train on a Railway Curve

To demonstrate the application of the banked turn formulas, consider a train traveling on a curved track.

Given Parameters:

• Velocity (v): 70 km/h
• Curve Radius (r): 1000 m
• Acceleration of Gravity (g): 9.81 m/s²

Step 1: Convert Velocity to SI Units (m/s)

First, convert the train's speed from kilometers per hour to meters per second.

$$v = 70 \, \frac{\text{km}}{\text{h}} \times \frac{1000 \, \text{m}}{1 \, \text{km}} \times \frac{1 \, \text{h}}{3600 \, \text{s}} \approx 19.44 \, \frac{\text{m}}{\text{s}}$$

Step 2: Calculate the Banked Angle

Using the banked angle formula for radians:

$$\Theta_{\text{rad}} = \tan^{-1}\left(\frac{v^2}{r \cdot g}\right)$$

Substituting the values:

$$\Theta_{\text{rad}} = \tan^{-1}\left(\frac{(19.44 \, \text{m/s})^2}{1000 \, \text{m} \cdot 9.81 \, \text{m/s}^2}\right) = \tan^{-1}\left(\frac{377.9}{9810}\right) \approx \tan^{-1}(0.0385) \approx 0.0385 \, \text{rad}$$

Step 3: Convert Angle to Degrees

For a more intuitive understanding, convert the angle from radians to degrees:

$$\Theta_{\text{degrees}} = \Theta_{\text{rad}} \times \left(\frac{360}{2\pi}\right) \approx 0.0385 \, \text{rad} \times 57.2958 \frac{^\circ}{\text{rad}} \approx 2.2^\circ$$

Conclusion: To safely navigate this curve at 70 km/h, the outer rail of the track should be elevated to create a banking angle of approximately 2.2 degrees. This inclination generates the necessary centripetal force component, reducing the lateral thrust on the wheels and enhancing passenger comfort and safety.

Physical Principles

In a banked turn, two forces act on the vehicle: the centripetal force directed toward the center of the circular path, and the centrifugal force (a pseudo-force in the rotating frame of the vehicle) that appears to push the vehicle outward. Banking the track or road uses a component of the vehicle's weight to provide the necessary centripetal force, reducing reliance on friction between the tires and the road surface. This increases stability and allows for higher safe speeds.

Additional Formulas

The required banked angle for a vehicle to navigate a curve without lateral friction is defined by the following relationships:

Banked Angle (Radians): $\Theta_{rad} = \tan^{-1}\left( \frac{v^2}{r \cdot g} \right)$

Banked Angle (Degrees): $\Theta_{deg} = \frac{180}{\pi} \cdot \tan^{-1}\left( \frac{v^2}{r \cdot g} \right)$

General Force Relationship: The ideal banked angle $\Theta$ relates velocity $v$, radius $r$, and gravity $g$ such that the net horizontal force provides the centripetal acceleration $a_c = \frac{v^2}{r}$.

Example: Car on a Highway Curve

Consider a car traveling at 100 km/h on a highway curve with a radius of 500 m. The required banked angle to eliminate reliance on friction is:

$v = 100 \, \text{km/h} \times \frac{1000 \, \text{m/km}}{3600 \, \text{s/h}} \approx 27.78 \, \text{m/s}$ $\Theta = \tan^{-1}\left( \frac{(27.78)^2}{500 \times 9.81} \right) = \tan^{-1}(0.157) \approx 0.156 \, \text{rad} \approx 8.95^\circ$

A bank of about 9° would allow the car to navigate the curve comfortably at that speed.

Diagram Note

The original source includes a diagram illustrating the relationship between velocity, curve radius, and the required banked angle. This visual aid helps engineers quickly estimate design parameters for road and track engineering.

Banked Turn Calculator

This calculator determines the centripetal acceleration and required banked angle for a vehicle negotiating a curve.

Centripetal Acceleration

The centripetal acceleration $a_c$ experienced by the vehicle is given by:

$$a_c = \frac{v^2}{r}$$

Banked Angle in Degrees

While the primary formula uses radians, the banked angle $\Theta$ in degrees can be calculated as:

$$\Theta_{\text{degrees}} = \tan^{-1}\left(\frac{v^2}{r \cdot a_g}\right) \cdot \left(\frac{360}{2\pi}\right)$$

where:

• $v$ is the vehicle velocity (m/s),
• $r$ is the curve radius (m),
• $a_g$ is the acceleration due to gravity (≈9.81 m/s²).

Required Banked Angle Reference Table

This table illustrates the relationship between velocity, curve radius, and the required banked angle to counteract centrifugal forces on a horizontal curve.