Lecture Notes: Car on a Banked Road

Topic Overview

• Discussion about the movement of cars on a banked road, particularly focusing on the calculation of the banking angle to maintain maximum speed without skidding.
• Assumption: No frictional force acting on the car.

Problem Statement

• Determine the banking angle for a car moving on a banked curve with:
  • Radius of the curve: 150 meters
  • Car speed: 75 km/h
• Objective: Calculate the maximum speed at which the car can move without skidding.

Key Concepts

• Banking Angle (Theta): The angle at which the road is inclined to help cars move through the curve safely.
• Frictionless Environment: Assumption that there is no friction to consider.
• Free Body Diagram Analysis: Used to understand the forces acting on the car.

Forces Involved

• Weight (mg): Acts downward.
• Normal Force (N): Perpendicular to the road surface, resolved into two components:
  • N sin(Theta): Provides the necessary centripetal force.
  • N cos(Theta): Balances the weight of the car.
• Centripetal Force: Required for circular motion, calculated as (\frac{mv^2}{R}).

Mathematical Derivation

1. Analyze the equilibrium:
   • (N \cos(\Theta) = mg)
   • (N \sin(\Theta) = \frac{mv^2}{R})
2. Eliminate N by dividing the equations:
   • (\tan(\Theta) = \frac{v^2}{Rg})
3. Solve for (\Theta):
   • (\Theta = \tan^{-1}\left(\frac{v^2}{Rg}\right))

Calculation

• Convert speed from km/h to m/s: 75 km/h = 20.8 m/s
• Use the given values:
  • (v = 20.8 \text{ m/s}), (R = 150 \text{ m}), (g = 9.8 \text{ m/s}^2)
• Calculate the banking angle:
  • (\Theta \approx 16.4^\circ)

Conclusion

• At an angle of approximately 16.4 degrees, the car can safely navigate the curved path at a speed of 75 km/h.

Closing Remarks

• Encouragement to ask questions and engage in further discussion via comments.
• Reminder to like, share, and subscribe to the channel.