A speedway turn, with radius of curvature \(R\), is banked at an angle \(\theta\) above the horizontal. a) What is the optimal speed at which to take the turn if the track's surface is iced over (that is, if there is very little friction between the tires and the track)? b) If the track surface is ice-free and there is a coefficient of friction \(\mu_{s}\) between the tires and the track, what are the maximum and minimum speeds at which this turn can be taken? c) Evaluate the results of parts (a) and (b) for \(R=400 . \mathrm{m}\), \(\theta=45.0^{\circ},\) and \(\mu_{\mathrm{s}}=0.700 .\)

Understanding the role of centripetal force is critical when analyzing the physics of a banked curve. Centripetal force is not an independent force but refers to the 'center-seeking' component of force that keeps an object moving in a circular path. It's always directed towards the center of the circle. The formula for centripetal force is given by \( F_c = \frac{mv^2}{R} \),where \(F_c\) is the centripetal force, \(m\) is the mass of the object, \(v\) is the speed of the object, and \(R\) is the radius of curvature of the path.

In the context of a car on a banked track, the centripetal force is provided by the horizontal component of the normal force when there is no friction. If friction is present, it also contributes to centripetal force. It's essential for vehicles taking a turn on a banked curve because it allows the vehicle to take the turn at higher speeds without skidding outward. Without sufficient centripetal force, the car would not follow the curved path and would tend to move in a straight line due to inertia, potentially leading to an accident.

Frictional force plays a pivotal role in real-world driving conditions. It arises due to the interaction between the surface of the track and the car's tires. The formula for the maximum frictional force that can occur without slippage is given by \(f_{max} = \mu_sN\), where \(f_{max}\) is the maximum static frictional force, \(\mu_s\) is the coefficient of static friction, and \(N\) is the normal force exerted by the track on the car.

In a banked curve, friction can help the car stay on the track by providing additional centripetal force. When the surface is icy, friction is nearly negligible, and the optimal speed depends only on gravity and the banking angle. However, on a dry track, friction becomes a crucial factor that allows a vehicle to take the curve at speeds higher or lower than the optimal speed without skidding. The maximum speed is limited by the static friction before it starts sliding, while the minimum speed is where the frictional force is needed to keep the car moving inwards and prevent it from slipping down the slope.

The normal force is the force exerted by a surface to support the weight of an object resting on it. It acts perpendicular to the surface which, in the case of a banked curve, would not be vertically upwards but rather at an angle from the horizontal. The magnitude of the normal force can be altered by an object's weight (mass times the gravitational acceleration) and the angle of the bank.

When a car is on a banked curve, the normal force gets divided into two components — the vertical component balances the weight of the car, while the horizontal component provides the necessary centripetal force (in the absence of friction). A correct understanding of the normal force is essential for calculating both the optimal speed on a frictionless banked curve and the maximum and minimum speeds on a curve where friction is present. It ensures the necessary forces are met for a car to remain on the path without veering off the track or slipping.