If a car takes a banked curve at less than the ideal speed,...

If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). (a) Calculate the ideal speed to take a 100 m radius curve banked at $15.0^{\circ} .$ (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at $20.0 \mathrm{km} / \mathrm{h}$ ?

Key Concepts

Banked Curve

A banked curve refers to a road or track that is tilted at an angle relative to the horizontal to help vehicles negotiate turns at higher speeds. The banking of the surface provides a component of the normal force that contributes to the centripetal force required for circular motion, reducing the reliance on friction alone to prevent sliding.

Centripetal Force

Centripetal force is the net inward force required to keep an object moving in a circular path at a constant speed. For a car negotiating a curve, this force is provided by a combination of components of the normal force and friction, which act towards the center of the circular path.

Ideal Banked Curve Speed

The ideal speed on a banked curve refers to the specific speed at which a vehicle can navigate the curve without relying on friction, meaning that the horizontal component of the normal force alone exactly provides the needed centripetal force. This speed is determined by the banking angle, gravitational acceleration, and the radius of the curve.

Role of Friction in Banked Curves

Friction is the resistive force that acts parallel to the contact surface between the vehicle’s tires and the road. In a banked curve, if a vehicle is traveling slower or faster than the ideal speed, friction will either oppose or assist the lateral component of the forces to help maintain the circular motion, preventing the vehicle from sliding inward or outward on the curve.

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