Finding Safe Speeds for a Curved Road

[zaddyzad]

Homework Statement

A curve of radius 78 m is banked for a design speed
of If the coefficient of static friction is 0.30 (wet
pavement), at what range of speeds can a car safely make
the curve?

Homework Equations

The Attempt at a Solution

I calculated theta being 36.1... (85/3.6)^2/78=gTan(theta)

Now, I'm wondering how to calculate the range of acceptable velocitys.. could someone explain it to me.

I think for max velocity the centripetal force = Sliding force + Frictional force = 104kph correct?

Can explain how I need to think for minimum velocity?

 

[imiuru]

Homework Statement

A curve of radius 78 m is banked for a design speed
of If the coefficient of static friction is 0.30 (wet
pavement), at what range of speeds can a car safely make
the curve?

Homework Equations

The Attempt at a Solution

I calculated theta being 36.1... (85/3.6)^2/78=gTan(theta)

Now, I'm wondering how to calculate the range of acceptable velocitys.. could someone explain it to me.

I think for max velocity the centripetal force = Sliding force + Frictional force = 104kph correct?

Can explain how I need to think for minimum velocity?

Your equation is not correct. I suppose this could be due to your understanding of circular motion.

You mentioned sliding force. What is the sliding force? Why is it there?

 

[zaddyzad]

Sorry not sliding force, but instead the force that points to the centre of the circle.

 

[imiuru]

Good. Let's see if we can workout the correct equation. But first you need to have a good idea of all the forces involved.

In your scenerio, is there such a force that points to the centre of the circle?

What is the direction of the frictional force then?

 

[zaddyzad]

opposing the force pointing to the centre

 

[imiuru]

Yes, the direction of the frictional force is opposite the other force, but the latter does not points to the centre of the circle. In fact it is a fictitious force, which arises due to the non-inertial frame of the object in circular motion. (Have you wonder why all motorcyclists bend inwards when they turn?)

Check out centrifugal and centripedal force in wiki or other sources.

However, you can still solve the problem without that part of the knowledge.

In your case, there is only one physical force involved, and that is frictional force. So what do you think supply the centrifugal force needed to maintain circular motion?

 

[ehild]

It is a banked curve, so not only the force of friction keeps the car on track.
Force involved are gravity, normal force and static friction.

When the speed is greater than the allowed one, the car slides outward and up. When it is less, it slides down and inward.

The car moves along a horizontal circle of radius R so the resultant of all forces is equal to the centripetal force, mv²/R, a horizontal force pointing inward, towards the centre of the circle.
Draw free body diagram and figure out the relation between forces and the banking angle.

ehild

 

[imiuru]

Thanks ehild for pointing out the keyword "banked" here.

Yes, since it is a banked curve, the force involved are those 3 mentioned.

In my previous post, I mistakenly mentioned "centrifugal force needed to maintain circular motion", it should be centripetal force.