A banked circular highway curve is designed for traffic...

A banked circular highway curve is designed for traffic moving at $95 \mathrm{~km} / \mathrm{h}$. The radius of the curve is $210 \mathrm{~m}$. Traffic is moving along the highway at $52 \mathrm{~km} / \mathrm{h}$ on a stormy day. ( $a$ ) What is the minimum coefficient of friction between tires and road that will allow cars to negotiate the turn without sliding? $(b)$ With this value of the coefficient of friction, what is the greatest speed at which the curve can be negotiated without sliding?

Key Concepts

Static Equilibrium

Static equilibrium refers to the state where an object is not accelerating in any direction, meaning that the sum of forces in both the horizontal and vertical directions is zero. In the analysis of banked curves, establishing static equilibrium ensures that the vehicle does not move vertically (i.e., it neither lifts off nor sinks into the road) while it is negotiating the curve, even as it experiences centripetal acceleration.

Force Decomposition

Force decomposition involves breaking down the forces acting on an object into their respective components, typically parallel and perpendicular to the surface. This is crucial in analyzing banked curve problems, as it allows the separation of the gravitational, frictional, and normal forces into components that contribute to both vertical equilibrium and the required centripetal force.

Coefficient of Friction

The coefficient of friction is a dimensionless quantity that represents the ratio of the frictional force resisting motion to the normal force between two surfaces in contact. In the context of a banked curve, it determines how much additional grip is available for vehicles, especially when traveling at speeds that differ from the design speed, or under adverse conditions such as wet or stormy weather.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the curve. For vehicles on a banked curve, the combination of frictional force and the horizontal component of the normal force must provide this centripetal force to prevent the vehicle from sliding outward.

Banked Curve

A banked curve is a road or track that is tilted with respect to the horizontal plane. This design allows a component of the normal force to provide the necessary centripetal force, reducing reliance on friction and allowing vehicles to negotiate the curve at a design speed without sliding, even in the absence of frictional forces.

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