UCM from AP B Exam (1977 B2) : Confused with solution to #3

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SUMMARY

The discussion focuses on the physics problem from the 1977 AP B Exam regarding a box on a truck traveling on a circular roadway. The key condition for the coefficient of static friction, μ, is established as μ ≥ v²/(Rg), where v is the truck's speed, R is the radius of curvature, and g is the acceleration due to gravity. The participants clarify that when the roadway is banked, the normal force (N) acting on the box must be greater than the gravitational force (mg) to prevent slippage. The confusion arises from the relationship between the normal force and the forces acting on the box, particularly in relation to centripetal force.

PREREQUISITES
  • Understanding of Newton's laws of motion
  • Knowledge of centripetal force concepts
  • Familiarity with free body diagrams
  • Basic principles of static friction
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  • Study the derivation of centripetal force in circular motion
  • Learn about the effects of banking on circular motion
  • Explore static friction and its role in preventing slippage
  • Review free body diagram techniques for inclined planes
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Students preparing for AP Physics exams, educators teaching mechanics, and anyone interested in understanding the dynamics of objects in circular motion.

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Homework Statement



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1977 B2. A box of mass M, held in place by friction, rides on the flatbed of a truck which is traveling with constant speed v. The truck is on an unbanked circular roadway having radius of curvature R.
a.On the diagram provided above, indicate and clearly label all the force vectors acting on the box.
b.Find what condition must be satisfied by the coefficient of static friction [Symbol] between the box and the truck bed. Express your answer in terms of v, R, and g.
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If the roadway is properly banked, the box will still remain in place on the truck for the same speed v even when the truck bed is frictionless.

c.On the diagram above indicate and clearly label the two forces acting on the box under these conditions
d.Which, if either, of the two forces acting on the box is greater in magnitude?

Homework Equations



1977B2
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b.Friction, f ≤ [Symbol]N where N = Mg. Friction provides the necessary centripetal force so we have f = Mv2/RMv2/R ≤ [Symbol]Mg, or [Symbol] ≥ v2/Rg
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c.
d.from the diagram below, a component of the normal force N[Symbol] balances gravity so N[Symbol] must be greater than mg
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The Attempt at a Solution


I am befuddled by the solution. I can see how Fn is greater than mg by virtue of Fn*cos(theta)=mg, But if I incline the coordinate system, I get Mg*cos(theta)=Fn, making Fn smaller than mg. I would expect Fn to be smaller because the incline is getting closer to vertical, thereby decreasing Fn and leading to slippage. I know this has something to do with Centripetal Force=Fn*sin(theta) but that force is not included on the free body diagram and I can't quite see how that would change the actual normal force. Looking for clarification. Thank you.
 

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I've attached the question and college board solution
 

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