When we calculate friction force in an inclined plane, the normal reaction N is taken as mg cos theta but when we calculate friction in banking of roads than mg is taken as Ncos theta where theta is angle of inclination. If Ncos theta is mg than R is more than mg, how it happens

To understand the differences in how we calculate friction forces on an inclined plane versus a banked road, we need to delve into the concepts of normal force and the forces acting on an object in both scenarios. Let's break this down step by step.

Inclined Plane Scenario

When dealing with an inclined plane, the forces acting on an object resting on the slope can be analyzed using basic physics principles. The weight of the object, denoted as mg (where m is mass and g is the acceleration due to gravity), acts vertically downward. The normal force (N) acts perpendicular to the surface of the incline.

Calculating the Normal Force

On an inclined plane, the normal force can be calculated using the angle of inclination (θ). The component of the weight acting perpendicular to the incline is given by:

• N = mg cos(θ)

This means that the normal force is less than the weight of the object because it only counteracts the component of the weight acting perpendicular to the incline. The frictional force (f) can then be calculated using:

• f = μN = μ(mg cos(θ))

Here, μ represents the coefficient of friction between the surfaces in contact.

Banked Road Scenario

Now, let’s consider a banked road. In this case, the road is inclined at an angle θ to help vehicles negotiate curves without relying solely on friction. The forces acting on a vehicle on a banked curve are a bit different.

Understanding Forces on a Banked Curve

For a vehicle moving in a circular path on a banked road, the normal force (N) and gravitational force (mg) work together to provide the necessary centripetal force to keep the vehicle moving in a circle. The normal force is not just balancing the weight; it also contributes to the centripetal acceleration required for circular motion.

Components of Forces

In this scenario, the normal force can be expressed as:

• N = mg / cos(θ)

This indicates that the normal force is greater than the weight of the vehicle when the road is banked. The vertical component of the normal force balances the weight:

• N cos(θ) = mg

However, the horizontal component of the normal force provides the centripetal force needed for circular motion:

• N sin(θ) = mv²/r

Here, v is the velocity of the vehicle and r is the radius of the curve. This means that the normal force is indeed greater than the weight of the vehicle, allowing for the necessary centripetal force to be generated.

Bringing It All Together

In summary, the key difference lies in the role of the normal force in each scenario. On an inclined plane, the normal force directly opposes the weight component perpendicular to the incline, while on a banked road, the normal force not only balances the weight but also provides the centripetal force needed for circular motion. This is why we see different expressions for the normal force in these two cases, and it highlights the importance of understanding the context of the forces at play.