Problem 18 - Basic Engineering Mathematics

Question

A locomotive moves around a curve of radius, \(r=500 \mathrm{~m}\). The angle of banking, \(\theta\), is given by: \(\theta=\tan ^{-1}\left(\frac{v^{2}}{r g}\right)\) where \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\) and \(v\) is the speed in \(\mathrm{m} / \mathrm{s}\). Calculate the angle of banking when the speed of the locomotive is \(30 \mathrm{~km} / \mathrm{h}\).

Step-by-Step Solution

Answer

The angle of banking is approximately \(0.81^\circ\).

Step 1: Convert Speed Units

First, convert the speed from kilometers per hour (km/h) to meters per second (m/s). Recall: \[ \text{1 km/h} = \frac{1}{3.6} \text{ m/s} \] So, \[ v = 30 \text{ km/h} \times \frac{1}{3.6} = 8.33 \text{ m/s} \] (rounded to two decimal places). This is the speed to use in calculations.

Step 2: Recall the Banking Angle Formula

Recall the formula for the banking angle: \[ \theta = \tan^{-1}\left(\frac{v^2}{rg}\right) \] where \( v = 8.33 \text{ m/s} \), \( r = 500 \text{ m} \), and \( g = 9.81 \text{ m/s}^2 \).

Step 3: Plug Values Into the Formula

Substitute the known values into the formula: \[ \frac{v^2}{rg} = \frac{(8.33 \text{ m/s})^2}{500 \text{ m} \times 9.81 \text{ m/s}^2} \] Calculate the numerator and denominator separately.

Step 4: Calculate the Result

Calculate \( v^2 \) first: \[ 8.33^2 = 69.39 \text{ m}^2/\text{s}^2 \]Then calculate \( rg \): \[ 500 \text{ m} \times 9.81 \text{ m/s}^2 = 4905 \text{ m}^2/\text{s}^2 \] Now divide to find: \[ \frac{69.39}{4905} = 0.01415 \] (rounded to five decimal places).

Step 5: Determine the Angle

Use the arctangent function to find \( \theta \): \[ \theta = \tan^{-1}(0.01415) \] Using a calculator, \( \theta \approx 0.81^\circ \). This is the angle of banking required.

Key Concepts

Curve Radius

In engineering mathematics, the concept of curve radius is crucial when discussing motion along curved paths, such as the path of a locomotive on a track. The radius of a curve, denoted by \( r \), is the distance from the center of the circular path to any point on the curve itself. A larger radius indicates a gentler curve, while a smaller radius signifies a sharper curve.
Understanding the curve radius is essential when calculating forces and designing safe travel paths for vehicles, ensuring they negotiate curves without skidding or overturning.

Angle of Banking

The angle of banking is an engineering concept used to facilitate the safe travel of vehicles on a curve. It refers to the inclination or tilt of a surface with respect to the horizontal. By designing roads or tracks with a suitable banking angle, vehicles can navigate curves more safely at higher speeds.
This is because banking helps counteract the lateral forces acting on a vehicle due to the curve, allowing it to maintain traction and minimize the risk of sliding. Additionally, an angle of banking is tailored to specific speeds to ensure the force balance is optimal for safe maneuvering.

Unit Conversion

Unit conversion plays a vital role in engineering calculations, especially when dealing with speed and measurements in different units. In our context, converting speed from kilometers per hour (km/h) to meters per second (m/s) is essential for applying formulas correctly.

• To convert km/h to m/s, you divide the speed value by 3.6.
• This conversion is necessary because many engineering formulas are derived using standard metric units, particularly meters and seconds.

Performing accurate unit conversions ensures that calculations maintain their integrity and results are reliable.

Banking Angle Formula

The banking angle formula is an important equation used to determine the angle required for safe negotiation of a curve. This formula involves the variables of speed \( v \), curve radius \( r \), and gravitational acceleration \( g \).
From the formula \( \theta = \tan^{-1}(\frac{v^2}{rg}) \), we see how the angle of banking is determined by these factors. Here's a breakdown:

• \( v^2 \): The square of the speed of the vehicle, reflecting how higher speeds increase the required banking angle.
• \( rg \): The product of the radius and gravitational acceleration, which provides a measure of the centripetal force requirements.

Understanding this formula allows engineers to design curves that accommodate specific vehicle speeds and track conditions safely.