find the expression for maximum speed and hence optimum speed of a vehicle which can be achieved while taking a turn on a bend road

To determine the maximum speed a vehicle can achieve while navigating a turn on a curved road, we need to consider several factors, including the radius of the turn, the friction between the tires and the road, and the gravitational force acting on the vehicle. This involves applying concepts from physics, particularly those related to circular motion and friction.

Understanding Circular Motion

When a vehicle takes a turn, it moves in a circular path. The force that keeps the vehicle moving in this circular path is called centripetal force. This force is provided by the friction between the tires and the road surface. The maximum speed at which a vehicle can safely navigate a turn without skidding can be derived from the balance of forces acting on the vehicle.

The Formula for Maximum Speed

The centripetal force required to keep the vehicle moving in a circle is given by the formula:

• F_c = (mv^2) / r

Where:

• F_c = centripetal force
• m = mass of the vehicle
• v = speed of the vehicle
• r = radius of the turn

The frictional force, which provides the necessary centripetal force, can be expressed as:

• F_f = μN

Where:

• F_f = frictional force
• μ = coefficient of friction between the tires and the road
• N = normal force (which, on a flat surface, equals the weight of the vehicle, mg)

Setting Forces Equal

For the vehicle to navigate the turn without skidding, the centripetal force must equal the maximum frictional force:

• (mv^2) / r = μmg

We can simplify this equation by canceling out the mass (m) from both sides, assuming it is not zero:

• v^2 / r = μg

Deriving Maximum Speed

Now, we can solve for the maximum speed (v) as follows:

• v^2 = μgr
• v = √(μgr)

Optimum Speed for Safe Turning

The expression we derived, v = √(μgr), gives us the maximum speed at which a vehicle can safely navigate a turn without skidding. This speed is often referred to as the optimum speed for taking a turn. It ensures that the vehicle maintains traction and stability while cornering.

Factors Influencing Optimum Speed

Several factors can influence this optimum speed:

• Radius of the Turn (r): A larger radius allows for a higher speed since the required centripetal force decreases.
• Coefficient of Friction (μ): A higher coefficient indicates better grip between the tires and the road, allowing for higher speeds.
• Road Conditions: Wet or icy roads reduce the coefficient of friction, lowering the optimum speed.

Practical Application

When driving, it's essential to consider these factors to maintain safety. For instance, if you're approaching a bend with a radius of 50 meters and the coefficient of friction is 0.7, you can calculate the maximum speed:

• g (acceleration due to gravity) = 9.81 m/s²
• v = √(0.7 * 9.81 * 50) ≈ 12.5 m/s

This means that to safely navigate the turn, you should aim to keep your speed below approximately 12.5 m/s.

In summary, understanding the relationship between speed, friction, and the radius of a turn is crucial for safe driving. By applying the derived formula, drivers can make informed decisions about their speed when approaching curves on the road.