Banked Curve - Minimum Turn Radius

[Devtycoon]

Homework Statement

A car traveling 10 m/s is moving along a track banked at 5 degrees. The tire-road friction coefficient is .3 What is the minimum radius it can travel without slipping?

v₀= 10 m/s
Bank Angle Θ = 5°
μ=0.3

Note I am working through prep material for the exam. The solution in the book skips steps and doesn't make sense.

Homework Equations

Normal and Tangential Kinetics for Planar Problems - FE Exam Reference Guide Page 70 - Dynamics
∑F_t=ma_t
∑F_n=mv_t/ρ , where ρ is the radius of curvature

The Attempt at a Solution

∑F_x=F_c-F_μ
∑F_x=mv²/ρ-μmgcosΘ=0, where Θ=5°
ρ=v²/μgcosΘ=100/.3⋅*9.81⋅cos(5°)=34.1 mNote:

Why is the review manual saying to use the following solution? I do not understand the F_Θ term or how it was derived.

F_c=F_μ+F_θ
mv²/ρ=μmg+mg⋅tanΘ
ρ=26m

Note: Thanks for the help. I don't understand the solution and my attempt is the wrong answer.

 

[olivermsun]

$F_c = F_\mu + F_\theta$ appears to decompose the centripetal force into a part caused by friction with coefficient $\mu$ and a part caused by the inward radial component of the normal force exerted by banked track.

So you know that the banked track is pushing normal to its force with some force $F$ (acting at angle $\theta$relative to vertical). The vertical component of $F$ balances downward gravity, so
$$F \cos\theta = mg.$$
Meanwhile, the inward radial component of $F$ is what's left, namely
$$F_\theta = F\sin\theta.$$
Now combine them to get an expression for $F_\theta$ as a function of $mg$ and $\theta$.

Also, I wonder if you are supposed to use the small angle approxation to get $F_\mu = \mu mg$.

 

[Devtycoon]

I am confused. Is my FBD messed up? I see a gravitational force in the direction of the centripetal force.

 

[ehild]

I am confused. Is my FBD messed up? I see a gravitational force in the direction of the centripetal force.

View attachment 87216

The centripetal force must be horizontal, as the car moves along a horizontal circle.

 

[Devtycoon]

Car is on a 5 degree bank. Did I still mess up the FBD?

 

[ehild]

There is a normal force F_N which is perpendicular to the road and static friction F_s that is parallel with it. The force of friction acts inward in case of smallest possible radius, to prevent the car sliding out.

 

[Devtycoon]

I am confused because the solutions states:

F_c = F_f + F_θThe overhead view looks like this. Right?

 

[ehild]

I do not understand your notations.
The friction is not horizontal.
Decompose all forces into horizontal and vertical components.

 

[Devtycoon]

First of all. Thanks for your help. I clearly am having a physics fundamental issue here. I have a fundamental question before I rehash the scenario here.Centripetal force is sucking the car in towards the instantaneous center of rotation right?

 

[ehild]

The centripetal force points towards the centre of curvature. But it does not "suck the car". It is not an operating force: it is the resultant of all forces acting on the car.
What do you mean on IoCp?

 

[Devtycoon]

The centripetal force points towards the centre of curvature. But it does not "sucks the car". It is not an operating force: it is the resultant of all forces acting on the car.
What do you mean on IoCp?

I got sloppy with the paint tool. Sorry. It was supposed to be I.C. , the acronym for instantaneous center.

Okay next fundamental question, there are 3 additional forces acting on the particle. The friction force, force due to bank angle and the normal force due to gravity.

1) Is the force of friction preventing the car from slipping towards or away from the center of curvature?

2) What does the book mean when it states "The bank angle is the centrifugal force contributing to the normal force."

 

[ehild]

Okay next fundamental question, there are 3 additional forces acting on the particle. The friction force, force due to bank angle and the normal force due to gravity.

There is no force due to bank angle, and the normal force is not due to gravity.
The forces are gravity, normal force (due to the road) and friction.

1) Is the force of friction preventing the car from slipping towards or away from the center of curvature?

You have to figure it out. The force of static friction is less or equal to μN.

2) What does the book mean when it states "The bank angle is the centrifugal force contributing to the normal force."

That statement has no sense. The bank angle is not a force.

 

[Devtycoon]

How can I thank you for your help?

 

[ehild]

How can I thank you for your help?

Show your (correct) solution :)