Problem 37 - Master Resource Book in JEE Main Physics

Question

If a car is to travel with a speed \(v\) along the frictionless banked circular track of radius \(r\), the required angle of banking so that the car does skid is [J\&K 2010] (a) \(\theta=\tan ^{-1}\left(\frac{v^{2}}{r g}\right)\) (b) \(\theta=\tan ^{-1}\left(\frac{v}{r g}\right)\) (c) \(\theta=\tan ^{-1}\left(\frac{r^{2}}{v g}\right)\) (d) \(\theta<\tan ^{-1}\left(\frac{\partial^{2}}{r g}\right)\)

Step-by-Step Solution

Answer

(a) \( \theta = \tan^{-1}\left(\frac{v^2}{rg}\right) \)

Step 1: Understanding the Problem

The problem requires us to find the angle of banking \( \theta \) such that a car moving at speed \( v \) along a frictionless banked circular track of radius \( r \) does not skid. The options provided are various expressions for \( \theta \).

Step 2: Identifying Forces Acting

In this case, the car is subjected to gravitational force and the normal force of the track. Since the track is banked and frictionless, the component of the normal force parallel to the centripetal direction must provide the necessary centripetal force to maintain the circular motion.

Step 3: Using Newton's Second Law

Applying Newton's Second Law in the radial (horizontal) direction, the centripetal force required is \( \frac{mv^2}{r} \). This force is provided entirely by the horizontal component of the normal force \( N \cos(\theta) \). Meanwhile, the vertical component \( N \sin(\theta) \) balances the gravitational force \( mg \).

Step 4: Equating Forces

For no skidding to occur, the sum of the forces should satisfy the equations: \( N \sin(\theta) = mg \) and \( N \cos(\theta) = \frac{mv^2}{r} \).

Step 5: Solving for Angle of Banking

Divide the two forces equations: \( \frac{N \sin(\theta)}{N \cos(\theta)} = \frac{mg}{\frac{mv^2}{r}} \), simplifying to \( \tan(\theta) = \frac{v^2}{rg} \). Therefore, \( \theta = \tan^{-1}\left(\frac{v^2}{rg}\right) \).

Step 6: Selecting the Correct Option

Based on the calculation \( \theta = \tan^{-1}\left(\frac{v^2}{rg}\right) \), the correct answer is option (a).

Key Concepts

Centripetal Force

Centripetal force is a key concept when discussing the motion of a car on a banked circular track. It is the force required to keep an object moving in a circular path, countering the inertia that would otherwise cause it to move in a straight line. For a car traveling in a frictionless, banked curve, the centripetal force needed to maintain its circular path is provided entirely by the normal force of the road. Specifically, its horizontal component acts as the centripetal force. Consider the following points about centripetal force:

• The centripetal force is calculated using the formula \( F_c = \frac{mv^2}{r} \), where \( m \) is mass, \( v \) is velocity, and \( r \) is the radius of the circle.
• This force acts toward the center of the circle, altering the direction of the car's velocity and thus keeping it in a circular orbit.
• In scenarios with no friction, like on a perfectly smooth banked track, the angle and the speed of the car must be just right to generate this force naturally via bank angle design.

Understanding this force explains how cars negotiate turns without skidding off the track. It is the foundational force at play in safe and effective banked turns.

Angle of Banking

The angle of banking is essential for designing roads or tracks meant to facilitate turns without reliance on friction. It is the angle at which the road surface is inclined relative to the horizontal. This incline helps direct parts of the normal force towards providing the necessary centripetal force.Here's what happens when a road is banked:

• The road is tilted at an angle \( \theta \), which means a component of the normal force will act as the centripetal force.
• This angle ensures that even in the absence of friction, vehicles can maintain their motion along the desired circular path.
• Mathematically, it is determined by the formula \( \tan(\theta) = \frac{v^2}{rg} \), where \( \theta \) is the angle of banking, \( v \) is the velocity, \( r \) is the radius, and \( g \) is the acceleration due to gravity.

Choosing the correct angle of banking ensures that the gravitational component aligns perfectly with the required motion direction, reducing the risk of slipping or skidding. It illustrates an elegant combination of physics and engineering principles in road design.

Newton's Second Law

Newton's Second Law of Motion forms the basis for understanding the dynamics of a vehicle on a banked track. It states that the force acting on an object is equal to the mass of the object multiplied by its acceleration, summarized as \( F = ma \).In the context of banked circular motion, we apply this law to two main components:

• Vertical Component: The gravitational force pulling the car down is balanced by the vertical component of the normal force \( N \sin(\theta) \), keeping the car from falling straight downward.
• Horizontal Component: The centripetal force required for circular motion \( N \cos(\theta) \) is derived from the same normal force but acts horizontally towards the track center, enabling the car to turn.

By solving these force equations derived from Newton's Second Law, we find that the balance leads to the determination of the correct angle of banking. Each component has a precise role, demonstrating the practical application of this fundamental physical law in real-world scenarios. This understanding helps in designing banked curves that are both safe and allow for efficient motion of vehicles at specific speeds.