Solve Banked Curve Problem: tan θ = ν^2/rg

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For a banked curve without friction, the relationship tan θ = ν^2/rg needs to be proven, focusing on the forces involved. The normal force acts perpendicular to the road surface, while gravity acts vertically, and the centripetal force is directed horizontally towards the center of the curve. The component of the normal force provides the necessary centripetal force, which can be visualized as part of a force triangle involving gravitational force. Understanding the geometry of these forces is crucial, especially when the road is banked at an angle, such as 45 degrees. Clarifying the direction of the centripetal force relative to the normal force and gravity is essential for solving the problem accurately.
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A problem states " For a banked curve, ignoring friction, prove that tan θ = ν^2/rg". I tried to prove but I thought that as the normal force is at right angle to track then how could be the component of normal force provide the centripetal force as my book is saying. Please someone help. I would be greatful.
 
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I have solved it but still can't understand how is component of normal force providing centripetal force.
 
Think about the resultant of the normal force and centripetal force being equal to the gravitational force.
 
I want to confirm this that is centripetal force of curved bank directed towards its center which is perpendicular to normal force or is directed towards the axis of curved road. Please reply
 
Normal is perpendicular to road surface. Gravity is vertical. Centripetal is in horizontal plane. The road is banked at 45 degrees according to your post.
You caught me, didn't you. Normal would be hypotenuse of force triangle, centripetal would be one leg, and half of gravitational would be other. Best double-check me on that.
 

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