Problem 38 - Master Resource Book in JEE Main Physics

Question

The maximum speed with which a car is driven round a curve of radius \(18 \mathrm{~m}\) without skidding (where, \(g=10 \mathrm{~ms}^{-2}\) and the coefficient of friction between rubber tyres and the roadway is \(0.2\) ) is (a) \(36.0 \mathrm{kmh}^{-1}\) (b) \(18.0 \mathrm{kmh}^{-1}\) (c) \(21.6 \mathrm{kmh}^{-1}\) (d) \(14.4 \mathrm{kmh}^{-1}\)

Step-by-Step Solution

Answer

The correct answer is option (c) 21.6 \(\mathrm{km/h}\).

Step 1: Understand the problem context

We need to find the maximum speed a car can travel around a curve without skidding. This occurs when the frictional force provides the necessary centripetal force.

Step 2: Write down known values and formulas

The radius of the curve, \( r \), is \( 18\, \mathrm{m} \). The acceleration due to gravity, \( g \), is \( 10\, \mathrm{ms}^{-2} \). The coefficient of friction, \( \mu \), is \( 0.2 \). The formula relating these is the frictional force, \( f = \mu mg \), and the centripetal force, \( f = \frac{mv^2}{r} \). Here, the frictional force must equal the centripetal force to prevent skidding.

Step 3: Set up the equation for maximum velocity

By equating the frictional force to the centripetal force, \( \mu mg = \frac{mv^2}{r} \). Here, \( m \) (the mass of the car) cancels out, simplifying the equation to \( \mu g = \frac{v^2}{r} \).

Step 4: Solve for velocity \( v \)

Rearrange the equation to get \( v^2 = \mu gr \). Substitute the known values: \( \mu = 0.2 \), \( g = 10\, \mathrm{ms}^{-2} \), and \( r = 18\, \mathrm{m} \) to find \( v^2 = 0.2 \times 10 \times 18 \).

Step 5: Calculate \( v \)

Calculate \( v^2 = 36 \) so \( v = \sqrt{36} = 6 \mathrm{ms}^{-1} \).

Step 6: Convert \( v \) from m/s to km/h

Since 1 \( \mathrm{m/s} \) equals 3.6 \( \mathrm{km/h} \), convert 6 \( \mathrm{ms}^{-1} \) to \( 6 \times 3.6 = 21.6 \mathrm{km/h} \).

Step 7: Choose the correct answer from the options given

The calculated speed 21.6 \( \mathrm{km/h} \) matches option (c). Hence, the correct answer is (c) 21.6 \( \mathrm{km/h} \).

Key Concepts

Frictional Force

Frictional force is an essential concept when understanding how a car can move around a curve without skidding. This force arises from the interaction between the tires of a vehicle and the road surface. It acts to prevent slipping or sliding motions. In the context of a car on a curved road, frictional force provides the needed grip, which keeps the vehicle on its desired path.
The more friction between the car tires and the road, the less likely it is for the car to skid. - Frictional force is calculated using the formula: \ f = \mu mg \ where \( \mu \) is the coefficient of friction, \( m \) the mass of the object, and \( g \) the acceleration due to gravity.
Remember that for a car traveling on a curve, the frictional force must be greater than or equal to the required centripetal force to keep the car moving in a circle.

Coefficient of Friction

The coefficient of friction (\( \mu \) is a measure that indicates how much frictional resistance exists between two surfaces in contact. It varies depending on the materials involved.
For a car on a roadway, the coefficient of friction between the rubber tires and the road is a crucial factor in determining the maximum speed the car can travel without losing traction.

• A higher coefficient of friction means more grip and thus, higher possible traveling speeds without skidding.
• The coefficient of friction is a unitless number, typically falling between 0 (no friction) and 1 (very high friction), although it can be greater than 1 in some cases.

In problems involving circular motion, like a car on a curve, knowing the coefficient of friction helps calculate the critical speed that prevents skidding.

Circular Motion

Circular motion is a fundamental topic that describes the movement of an object along a circular path. For an object like a car moving around a curve, maintaining this motion requires a force that keeps it directed toward the center of the curve. This is known as the centripetal force.
Here’s how it works:- The centripetal force is not an independent force, rather it is the net force that arises from other forces like tension, gravity, or friction.- For a car on a curve, the necessary centripetal force is provided primarily by frictional force.- The formula for calculating centripetal force is given by: \ f = \frac{mv^2}{r} \ where \( m \) is the mass of the car, \( v \) is its velocity, and \( r \) is the radius of the curve.Understanding circular motion and the forces at play ensures that designs for roadways and vehicles promote safe travel at appropriate speeds.