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force-on-vehicle-on-banked-road-problem

🚀 In circular motion, when a vehicle travels along a banked road, the forces acting on it must be analyzed to ensure it can navigate the curve without slipping. The banking angle helps to provide the necessary centripetal force required for circular motion. The forces involved include gravitational force, normal force, and frictional force. The banking angle is designed such that the vehicle can maintain its path with minimal reliance on friction, which is particularly important in wet or slippery conditions. The relationship between the banking angle, speed of the vehicle, and radius of the curve can be derived using Newton's laws of motion and the concept of centripetal force.

Theory Explanation

Understanding Forces on a Banked Curve

When a vehicle moves on a banked road, the forces acting on it include the gravitational force acting downwards, the normal force acting perpendicular to the surface, and the frictional force that can act either up or down the slope depending on the speed of the vehicle. The net force must provide the necessary centripetal force to keep the vehicle moving in a circular path.

\[ F_{net} = F_{centripetal} = \frac{mv^2}{r} \]
Deriving the Banking Angle

To derive the banking angle, we can set up the equations of motion. The vertical component of the normal force balances the weight of the vehicle, while the horizontal component provides the centripetal force. By resolving the forces, we can derive the formula for the banking angle \( \theta \) as follows: \( \tan(\theta) = \frac{v^2}{rg} \), where \( v \) is the speed of the vehicle, \( r \) is the radius of the curve, and \( g \) is the acceleration due to gravity.

\[ \theta = \tan^{-1}\left(\frac{v^2}{rg}\right) \]
Applying the Concept to Real Problems

When solving problems involving vehicles on banked roads, it is essential to identify the given values such as speed, radius, and mass of the vehicle. Use the derived formulas to find the banking angle or the required speed for a given angle. Always check if friction is needed based on the speed of the vehicle and the angle of the bank.

Key Points

  • 🎯 The banking angle reduces reliance on friction for maintaining circular motion.
  • 🎯 The centripetal force required for circular motion is provided by the horizontal component of the normal force.
  • 🎯 The formula for the banking angle is derived from the balance of forces acting on the vehicle.

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Examples:💡

A car of mass 1000 kg is traveling at a speed of 20 m/s on a banked road with a radius of 50 m. Calculate the banking angle required for the car to navigate the curve without relying on friction.

Solution:

Step 1: Identify the given values: mass (m) = 1000 kg, speed (v) = 20 m/s, radius (r) = 50 m.

Step 2: Use the formula for the banking angle: \( \theta = \tan^{-1}\left(\frac{v^2}{rg}\right) \). Here, \( g = 9.81 m/s^2 \).

\[ \theta = \tan^{-1}\left(\frac{20^2}{50 \times 9.81}\right) \]

Step 3: Calculate the value: \( \theta = \tan^{-1}\left(\frac{400}{490.5}\right) \approx 38.66^\circ \).

A motorcycle is negotiating a banked curve with a radius of 30 m at a speed of 15 m/s. Determine if the motorcycle can make the turn without slipping if the banking angle is 30 degrees.

Solution:

Step 1: Identify the given values: radius (r) = 30 m, speed (v) = 15 m/s, banking angle (\( \theta \)) = 30 degrees.

Step 2: Calculate the required centripetal force: \( F_{centripetal} = \frac{mv^2}{r} \). Assume mass (m) = 200 kg.

\[ F_{centripetal} = \frac{200 \times 15^2}{30} = 1500 N \]

Step 3: Calculate the gravitational force: \( F_{gravity} = mg = 200 \times 9.81 = 1962 N \).

Step 4: Resolve the forces: The normal force can be calculated using the angle and the gravitational force. Check if the normal force can provide the required centripetal force.

Common Mistakes

  • Mistake: Confusing the direction of the forces acting on the vehicle. Students often misinterpret the direction of the normal force and friction.

    Correction: Always draw a free-body diagram to visualize the forces acting on the vehicle. Remember that the normal force acts perpendicular to the surface.

  • Mistake: Neglecting the effect of friction when it is necessary for the problem.

    Correction: Evaluate the speed of the vehicle and the banking angle. If the speed is too high for the angle, friction will be needed to prevent slipping.

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