What Minimum Coefficient of Static Friction Prevents Skidding on a Banked Curve?

[PhysicsDerp]

Homework Statement

A curve with a 140m radius on a level road is banked at the correct angle for a speed of 20 m/s. If an automobile rounds this curve at 30 m/s, what is the minimum coefficient of static friction between tires and road needed to prevent skidding?

Homework Equations

F_fr = MkN
N = mgcosθ
∑F = ma
∑F_net-x = mv^2/r = Nsinθ

The Attempt at a Solution

I have drawn my xy coordinate system so that the x component is parallel to the curve, and , and found that the normal force is mgcosθ. However, I don't know how to combine my equations together to find an angle theta. Is finding an angle here even the proper approach if the problem does not ask for it? Should I be doing something else instead?

 

[collinsmark]

Hello PhysicsDerp,

Welcome to Physics Forums!

Homework Statement

A curve with a 140m radius on a level road is banked at the correct angle for a speed of 20 m/s. If an automobile rounds this curve at 30 m/s, what is the minimum coefficient of static friction between tires and road needed to prevent skidding?

Homework Equations

F_fr = MkN

Not sure what that equation is. Oh, wait. I get it: The force of friction, F_fr = μ_kN, where N is the normal force. Okay, that's good.

N = mgcosθ

Not so fast (see below).

∑F = ma

Yes, Newton's second law is important for this one.

∑F_net-x = mv^2/r = Nsinθ

Be careful there. It's not quite that simple.

The Attempt at a Solution

I have drawn my xy coordinate system so that the x component is parallel to the curve, and ,

That should work out. Personally, I would have simply stuck with the x-axis on the horizontal direction and the y-axis on the vertical.

I don't think changing the coordinate system such that the x-axis is parallel to the ramp will make the problem any easier. But whatever the case, it should still allow you to find the correct answer.

and found that the normal force is mgcosθ.

Something is not quite right there.

The normal force would be mgcosθ if the car were just sitting there at rest. But it's not; it's accelerating around the curve. The normal force is more complicated than that (regardless of the choice of coordinates).

However, I don't know how to combine my equations together to find an angle theta. Is finding an angle here even the proper approach if the problem does not ask for it? Should I be doing something else instead?

You might want to take another look at your free body diagram. Make sure that you have the following forces outlined on it:

• The force due to gravity
• The normal force
• The force of friction
• The resultant force (i.e., the net force)

The resultant (net) force is equal to ma. This is the force equal to the centripetal force.