Optimizing Velocity for Motion on a Banked Track

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To determine the ideal velocity for an object on a banked track, it is essential to consider the forces acting on the object, including gravity and the normal force, which must be analyzed at the angle of the incline. A free-body diagram is crucial for visualizing these forces and applying Newton's laws in both the x and y directions. The incline affects the distribution of forces, making it necessary to adjust calculations compared to a flat track scenario. Understanding the relationship between the banking angle and the required velocity is key to preventing the object from sliding. Properly accounting for these factors allows for accurate determination of the ideal velocity on a banked track.
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Homework Statement


An object travels around a circular track with a given radius that is banked/inclined (inward) at a given angle. Find the ideal velocity for the object to travel around the track (so that it does not slide up or down but stays in the middle) with no friction or with a given friction coefficient.


Homework Equations



I have no idea how to begin to solve this or where to begin.

The Attempt at a Solution

 
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Start by drawing a free-body diagram, identifying all forces, and writing Newton's law for both the x and the y directions.
 
I have done this, but I want to know how i would take the incline of the track into account. The problem would be simple enough if it was on a flat track.
 
You need to account for the incline to draw gravity and the normal force at the correct angle to each other. Just draw the incline, draw the car, and analyze away.
 
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