A park has a radius of 10 m. If a vehicle goes round it at an average speed of 18 km/hr, what should be the proper angle of banking?

To determine the proper angle of banking for a vehicle traveling around a circular path, we can use some principles from physics, particularly those related to circular motion. The banking angle helps to counteract the forces acting on the vehicle, allowing it to navigate the curve safely without relying solely on friction. Let's break this down step by step.

Understanding the Problem

We have a circular park with a radius of 10 meters, and a vehicle is moving around it at an average speed of 18 kilometers per hour. To find the angle of banking, we need to consider the forces acting on the vehicle as it moves in a circular path.

Key Concepts

• Centripetal Force: This is the force required to keep an object moving in a circular path, directed towards the center of the circle.
• Banking Angle: This is the angle at which the road is inclined, which helps in providing the necessary centripetal force through the component of gravitational force acting on the vehicle.
• Speed Conversion: Since our speed is given in kilometers per hour, we need to convert it to meters per second for our calculations.

Calculating the Speed in Meters per Second

First, let's convert the speed from kilometers per hour to meters per second. The conversion factor is:

1 km/hr = 1/3.6 m/s

So, for 18 km/hr:

Speed = 18 km/hr × (1/3.6 m/s) = 5 m/s

Applying the Banking Angle Formula

The formula for the banking angle (θ) when there is no friction involved is given by:

tan(θ) = v² / (r * g)

Where:

• v = speed of the vehicle (in m/s)
• r = radius of the circular path (in meters)
• g = acceleration due to gravity (approximately 9.81 m/s²)

Plugging in the Values

Now we can substitute the values into the formula:

• v = 5 m/s
• r = 10 m
• g = 9.81 m/s²

So, we have:

tan(θ) = (5 m/s)² / (10 m * 9.81 m/s²)

This simplifies to:

tan(θ) = 25 / 98.1

Calculating this gives:

tan(θ) ≈ 0.2548

Finding the Angle

To find the angle θ, we take the arctangent (inverse tangent) of 0.2548:

θ = arctan(0.2548)

Using a calculator, we find:

θ ≈ 14.5°

Final Thoughts

The proper angle of banking for a vehicle traveling at 18 km/hr around a circular path with a radius of 10 meters is approximately 14.5 degrees. This angle helps ensure that the vehicle can navigate the curve safely, minimizing the risk of skidding off the path. Understanding these principles of physics not only helps in solving such problems but also provides insight into real-world applications, like road design and vehicle safety.