Forces on a Banked Surface

Circular motion on banked surfaces is a fascinating aspect of physics that combines concepts of circular motion, forces, and inclined planes. To understand circular motion on banked surfaces, we must first analyse the forces at play. Let’s consider an object moving in a circular path on a banked surface without friction.

The key forces acting on the object are:

Unlike circular motion on a flat surface, the normal force is not parallel to the weight. This angle between the normal force and the vertical is crucial for maintaining circular motion.

Resolving Forces

We need to resolve the normal force into its horizontal and vertical components:

$$N_x=Nsin\theta$$ $$N_y=Ncos\theta$$

Where,

What is the Ideal Velocity for Uniform Circular Motion?

To determine the ideal velocity for uniform circular motion on a banked surface, we analyse the forces in both horizontal and vertical directions.

Horizontal Forces

The horizontal component of the normal force provides the centripetal force required for circular motion:

$$N\sin\theta = \frac{mv^2}{r} … (1)$$

Where:

Vertical Forces

The vertical component of the normal force balances the weight of the object:

$$Ncos θ = mg … (2)$$

Where,

Deriving the Ideal Velocity

Dividing equation $(1)$ by equation $(2)$:

$$\frac{N\sin\theta}{N\cos\theta} = \frac{mv^2/r}{mg}$$

Simplifying:

$$\tan\theta = \frac{v^2}{rg}$$

Rearranging for $v$:

$$v = \sqrt{rg\tan\theta}$$

This equation gives us the ideal velocity for an object to maintain uniform circular motion on a banked surface without the need for friction.

What are the Factors Affecting Circular Motion on Banked Surfaces?

The ideal velocity depends on three key factors:

Comparison with Flat Surfaces

It’s important to note that the velocity required for circular motion on a banked surface is lower than that needed on a flat surface of the same radius. This is because part of the normal force contributes to the centripetal force, reducing the reliance on friction.