## Physics Problem on Banked Curve for a Freeway### Problem StatementCalculate the angle at which a freeway must be banked at a curve to eliminate sideways forces parallel to the road. Given:- The curve has a radius of 100 ft.- Cars are expected to move at 40 ft/sec.- Acceleration due to gravity, \( g = 32 \, \text{ft/sec}^2 \). ### Diagram ExplanationThe provided diagram illustrates the forces acting on a vehicle moving along a banked curve. Key elements include:- **Track Surface:** The inclined plane representing the banked angle \( \theta \).- **Vehicle:** Moving on the banked track at an angle \( \theta \).- **Forces Acting on the Vehicle:** - **Weight (mg):** The downward force due to gravity. - **Centrifugal Force (\(\frac{mv^2}{r}\)):** The outward force as the vehicle moves in a circular path. - **Components of Forces:** - The weight force is decomposed into two components: - \( mg \, \sin \theta \): The component of weight parallel to the track surface. - \( mg \, \cos \theta \): The component of weight perpendicular to the track surface. - The centrifugal force is decomposed into two components: - \( \frac{mv^2}{r} \, \cos \theta \): The component along the direction perpendicular to the track surface. - \( \frac{mv^2}{r} \, \sin \theta \): The component parallel to the track surface. **Diagram Labels:**- **\( \theta \):** Angle of the banked curve.- **\( \frac{mv^2}{r} \):** The centrifugal force where \( m \) is mass, \( v \) is velocity, and \( r \) is the radius of curvature.- **\( mg \):** Gravitational force where \( g \) is the acceleration due to gravity.- **\( \sin \theta \):** Mathematical sine function representing a component of the forces due to the angle \( \theta \).- **\( \cos \theta \):** Mathematical cosine function representing another component of the forces.### GoalTo find the banking angle \( \theta \) that negates the sideways forces, thus ensuring

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At what angle must a freeway be banked at a curve to eliminate sideway forces parallel to the road, given that the curve has a radius of 100 ft. and cars are expected to move at 40 fA / sec?(g = 32 f sec²). cos 0 mu? (Centrifugal force, r is curve radius) mg sin 0 Track surface mg (Weight)

At what angle must a freeway be banked at a curve to eliminate sideway forces parallel to the road, given that the curve has a radius of 100 ft. and cars are expected to move at 40 fA / sec?(g = 32 f sec²). cos 0 mu? (Centrifugal force, r is curve radius) mg sin 0 Track surface mg (Weight)

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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