Problem 7 - Master Resource Book in JEE Main Physics

Question

When the road is dry and coefficient of friction is \(\mu\), the maximum speed of a car in a circular path is \(10 \mathrm{~ms}^{-1}\). If the road becomes wet and \(\mu^{\prime}=\mu / 2\), what is the maximum speed permitted? (a) \(5 \mathrm{~ms}^{-1}\) (b) \(10 \mathrm{~ms}^{-1}\) (c) \(10 \sqrt{2} \mathrm{~ms}^{-1}\) (d) \(5 \sqrt{2} \mathrm{~ms}^{-1}\)

Step-by-Step Solution

Answer

The maximum permitted speed is \(5 \sqrt{2} \mathrm{~ms}^{-1}\), option (d).

Step 1: Understand the Context and Initial Conditions

When a car is moving in a circular path, the maximum speed is determined by the friction between the tires and the road, described by the coefficient of friction \( \mu \). The given maximum speed on a dry road with a friction coefficient \( \mu \) is \( 10 \mathrm{~ms}^{-1} \).

Step 2: Use Physics Formula for Maximum Speed

The formula to determine the maximum speed \( v \) on a circular path is given by:\[v = \sqrt{\mu g r}\]Where \( g \) is the acceleration due to gravity, and \( r \) is the radius of the circular path. On a dry road, this gives us:\[10 = \sqrt{\mu g r}\]This implies:\[100 = \mu g r\]

Step 3: Adjust Conditions for a Wet Road

If the road is wet, the coefficient of friction changes to \( \mu' = \frac{\mu}{2} \). Substituting this into the maximum speed formula, we get:\[v' = \sqrt{\mu' g r} = \sqrt{\frac{\mu}{2} g r}\]

Step 4: Substitute Known Values to Find New Maximum Speed

Since \( \mu g r = 100 \), substituting into the adjusted formula gives:\[v' = \sqrt{\frac{1}{2} \times 100} = \sqrt{50} = 5 \sqrt{2} \mathrm{~ms}^{-1}\]

Step 5: Select the Correct Answer from Options

From the given options, the value \( 5 \sqrt{2} \mathrm{~ms}^{-1} \) corresponds to option (d).

Key Concepts

Coefficient of Friction

In physics, the coefficient of friction, denoted by \( \mu \), is a measure of how much resistance one surface or material exerts on another. It might sound complex, but it's actually quite easy to understand. Imagine trying to slide a heavy box across a wooden floor. The amount of effort you need to exert depends on the coefficient of friction between the box and the floor. A higher coefficient means more resistance and thus more effort is needed.

In the context of circular motion, such as a car navigating a curve, the coefficient of friction is crucial. It determines the maximum speed at which the car can safely move without skidding.

• A higher coefficient of friction means the car can travel faster without losing grip.
• A lower coefficient indicates that the car must slow down to maintain traction.

In the given scenario, for a dry road, the coefficient of friction allows a speed of \(10 \mathrm{~ms}^{-1}\). When the road condition changes, so does the coefficient, impacting the maximum allowable speed.

Maximum Speed

When a car is rounding a curve, the maximum speed it can achieve without slipping is determined by a simple yet powerful physical formula. This is:

• \[ v = \sqrt{\mu g r} \]
• Where \( v \) is the maximum speed, \( \mu \) is the coefficient of friction, \( g \) is the gravitational acceleration (\( \approx 9.8 \mathrm{~m/s}^{2} \)), and \( r \) represents the curve's radius.

This equation demonstrates the relationship between these variables. If any changes occur, such as a decrease in \( \mu \), it results in a lower permissible speed to avoid skidding off the road.

In this exercise, the dry road allows a speed of \( 10 \mathrm{~ms}^{-1} \). If road conditions worsen, our equation shows that the maximum speed must be recalculated based on the new \( \mu \). Here, with the reduced coefficient due to wet conditions, we find that the safe speed decreases to \( 5 \sqrt{2} \mathrm{~ms}^{-1} \).

Wet Road Conditions

Wet road conditions significantly change the dynamics of driving. Water acts as a potential lubricant between the car tires and the road, which reduces the coefficient of friction, \( \mu \). As the friction decreases, so does the traction, making it essential to reduce speed to maintain safe handling.

For example, when the road is wet, driving at the same speed as you would on a dry road can result in losing control. The concept of hydroplaning is an extreme form of this, where the tire loses contact with the road entirely.

• This is why the maximum speed is lower on a wet road—it ensures the tires maintain proper contact with the road.
• The exercise illustrates this clearly: the reduction of \( \mu \) to half its original value means the car needs to travel slower, specifically at \( 5 \sqrt{2} \mathrm{~ms}^{-1} \), to stay safe.

Considering how real-world driving is affected, wet conditions are not just an academic exercise but a crucial consideration for road safety.