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Normal force on a banked curve????
In every textbook I've looked at, I can't find an answer to this question. When determining the centripetal force on an object on a banked curve, it is stated that the banking angle for a given speed and radius is found by tan θ = v^2/rg
It is found as follows (there is an attachment as well):
The normal force on the object is resolved into components. The xcomponent (the one providing the centripetal force) is:
Fn * sin θ = mv^2/r (1)
Then, the y component is set equal to mg and it is found that the normal force:
Fn = mg / cos θ (2)
Substitute Fn from (2) into (1) and get:
tan θ = v^2/rg
I'm fine with that.

Here is where I'm confused...When resolving the forces of an object resting on an inclined plane:
The component down and parallel to the plane due to gravity, Fp, is as follows:
Fp = mg * sin θ
The component representing the force of gravity into (perpendicular) to the plane is:
mg * cos θ
The normal force is equal to this component into the plane by Newton's 3rd Law, so Fn = mg * cos θ

Why in the first scenario (banked curve) is Fn = mg / cos θ, HOWEVER, in the second (block on plane) Fn = mg * cos θ ????
How can this be? There are two different values for Fn
I'm stumped. I have no solution.
Thank you for any help.
In every textbook I've looked at, I can't find an answer to this question. When determining the centripetal force on an object on a banked curve, it is stated that the banking angle for a given speed and radius is found by tan θ = v^2/rg
It is found as follows (there is an attachment as well):
The normal force on the object is resolved into components. The xcomponent (the one providing the centripetal force) is:
Fn * sin θ = mv^2/r (1)
Then, the y component is set equal to mg and it is found that the normal force:
Fn = mg / cos θ (2)
Substitute Fn from (2) into (1) and get:
tan θ = v^2/rg
I'm fine with that.

Here is where I'm confused...When resolving the forces of an object resting on an inclined plane:
The component down and parallel to the plane due to gravity, Fp, is as follows:
Fp = mg * sin θ
The component representing the force of gravity into (perpendicular) to the plane is:
mg * cos θ
The normal force is equal to this component into the plane by Newton's 3rd Law, so Fn = mg * cos θ

Why in the first scenario (banked curve) is Fn = mg / cos θ, HOWEVER, in the second (block on plane) Fn = mg * cos θ ????
How can this be? There are two different values for Fn
I'm stumped. I have no solution.
Thank you for any help.
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