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8. Centripetal Forces & Gravitation
Banked Curves

8. Centripetal Forces & Gravitation

Banked Curves

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Textbook Question
An airplane feels a lift force L perpendicular to its wings. In level flight, the lift force points straight up and is equal in magnitude to the gravitational force on the plane. When an airplane turns, it banks by tilting its wings, as seen from behind, by an angle from horizontal. This causes the lift to have a radial component, similar to a car on a banked curve. If the lift had constant magnitude, the vertical component of L would now be smaller than the gravitational force, and the plane would lose altitude while turning. However, you can assume that the pilot uses small adjustments to the plane's control surfaces so that the vertical component of L continues to balance the gravitational force throughout the turn. a. Find an expression for the banking angle θ needed to turn in a circle of radius r while flying at constant speed v.
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Textbook Question

A banked curve of radius R in a new highway is designed so that a car traveling at speed v₀ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, it will slip away from the center of the circle. If the coefficient of static friction increases, it becomes possible for a car to stay on the road while traveling at a speed within a range from vₘᵢₙ to vₘₐₓ . Derive formulas for vₘᵢₙ and vₘₐₓ as functions of μₛ, v₀ and R.

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