# Maximum Speed on a Banked Curve with Friction

• Jacky Lee
In summary, a car can travel at a maximum speed of 34.5 m/s on a highway curve banked at a 15° angle with a static friction coefficient of 1.
Jacky Lee

## Homework Statement

A concrete highway curve of radius 70 m is banked at a 15° angle. What is the maximum speed with which a 1500 kg rubber tired car can take this curve without sliding? The static friction coefficient is 1.

## Homework Equations

Centripetal Force = ## (m v^2) / r ##
Maximum Friction Force = ## μN ##

## The Attempt at a Solution

## (m v^2) / r ## = centripetal force

Friction_max = 1 * normal force = ## 9.8 * 1500 * cos(15) = 14199.11 N ##

Plug back into equation:

## (1500 v^2) / 70 = 14199.11 ##

Which yields v = 25.7 m/s

However, the answer is 34.5 m/s. What am I doing wrong? I don't understand where I am going wrong in my steps.

Are you sure that the normal force is the only force you have to worry about? No other forces? None at all?

I mean the frictional force

AlephNumbers said:
Are you sure that the normal force is the only force you have to worry about? No other forces? None at all?

I figured that when the centripetal force exceeds the maximum frictional force, the car would skid. Solving for the maximum frictional force only requires the normal force, so I didn't include any other forces. Where would I have to use the friction force?

Look, I phrased that statement poorly. Reevaluate the forces acting on the car tire.

Everything else is correct, your steps, your concept of friction, all correct.

Forces on the tires: friction, normal, centripetal

Am I missing anything?

Yes. Yes, you are missing something very important. A force that is in every inclined plane problem you have ever seen...

Ignore the centripetal forces. Just think about a tire on an incline. What forces act on it?

AlephNumbers said:
Ignore the centripetal forces. Just think about a tire on an incline. What forces act on it?

OH, I'm missing gravity, aren't I?

I do believe you are.

AlephNumbers said:
I do believe you are.

Hmm, I still don't get the right answer. I tried adding the force of gravity in the direction of the incline to the frictional force (which would just double it), so I got:

## (m v^2) / r = 28398.22 ##
## (1500 v^2 / 70) = 28398.22 ##
## v = 36.4 ##

Don't just give me numbers. Keep it symbolic. What are the two factors that make up the 28398.22N?

The gravitational force directed down the incline would not be cosθmg

AlephNumbers said:
Don't just give me numbers. Keep it symbolic. What are the two factors that make up the 28398.22N?

I made an error, it should be 18003.75N

Maximum Frictional Force: ## 9.8 * 1500 * cos(15) = 14199.11 ##
Force due to gravity: ## 9.8 * 1500 * sin(15) = 3804.64 ##
Added together: ## 14199.11 + 3804.64 = 18003.75 N ##

## (m v^2) / r = 18003.75 ##
## (1500 v^2 / 70) = 18003.75##
## v = 29.0 m/s ##

I would wait until you have the solution to start plugging numbers in. You made a sign error.

AlephNumbers said:
I would wait until you have the solution to start plugging numbers in. You made a sign error.

Should the friction force be negative?

Even if I changed the sign on any of the numbers, my answer would still be incorrect. If I change the sign on either forces, I would get a smaller number and therefore a smaller velocity as an answer.

You are right. Your solution above looks good. I think that you either made a calculation mistake, or that the answer you claim is correct, is not actually correct.

AlephNumbers said:
You are right. Your solution above looks good. I think that you either made a calculation mistake, or that the answer you claim is correct, is not actually correct.

Hmm, okay. Thank you for all the help! :) I really appreciate it.

Any time.

## What is "Max Speed on an Incline"?

"Max Speed on an Incline" refers to the maximum speed that an object can reach when moving up or down an incline.

## How is "Max Speed on an Incline" calculated?

"Max Speed on an Incline" is calculated by taking into account the angle of the incline, the mass of the object, and the force of gravity acting on the object.

## Why is "Max Speed on an Incline" important to know?

Knowing the "Max Speed on an Incline" can help determine the safety and stability of an object on an incline, as well as its potential for acceleration or deceleration.

## What factors can affect "Max Speed on an Incline"?

The factors that can affect "Max Speed on an Incline" include the angle and surface of the incline, the mass and shape of the object, and the presence of any external forces such as friction or air resistance.

## How can "Max Speed on an Incline" be increased?

"Max Speed on an Incline" can be increased by decreasing the angle of the incline, reducing the mass of the object, and minimizing external forces such as friction.

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