How Does a Car Maintain Stability on a Banked Curve?

[Swatch]

A car is turning on a banked curve. The coefficient static friction between the car and the pavement is 0.30 and the coefficient of kinetic friction is 0.25.
The angle of the banking is 25 degrees, and the radius of the curve is 50 m. What is the minimum speed the car can have before sliding down the banking. I have found the maximum speed the car can have without sliding up the banking.

For the sliding up the bank I did this:

b=angle
s= coefficient of static friction

Fx = n*sin (b) + sin (b)*n*s = ma

Fy = n*cos (b) - mg - cos(b)*n*s = 0

So this is when the car is in equlibrium.

From this I find the acceleration and then the speed

I haven't figured out a relation between the speed and movement down the banking. I tried to resolve the weight vector into components together with the friction but I didn't get anywhere with that. The weight has some part in this I know. I'm just really lost. Could someone please give me a hint to this problem. The coefficient of kinetic friction is given but I don't see I have to use it.

With thanks,
Swatch

 

[himanshu121]

max frictional force can be ::
$$f = \mu N$$

 

[OlderDan]

A car is turning on a banked curve. The coefficient static friction between the car and the pavement is 0.30 and the coefficient of kinetic friction is 0.25.
The angle of the banking is 25 degrees, and the radius of the curve is 50 m. What is the minimum speed the car can have before sliding down the banking. I have found the maximum speed the car can have without sliding up the banking.

For the sliding up the bank I did this:

b=angle
fs=static friction

Fx = n*sin (b) + sin (b)*n*fs = ma

Fy = n*cos (b) - mg - cos(b)*n*fs = 0

So this is when the car is in equlibrium.

From this I find the acceleration and then the speed

I haven't figured out a relation between the speed and movement down the banking. I tried to resolve the weight vector into components together with the friction but I didn't get anywhere with that. The weight has some part in this I know. I'm just really lost. Could someone please give me a hint to this problem. The coefficient of kinetic friction is given but I don't see I have to use it.

With thanks,
Swatch

Your notation needs to be clarified. fs usually means the force of friction. If that is how you mean it, then you should not have products of n*fs in your equations. If you mean fs is the coefficient of static friction, you should state that.

Your sines and cosines do not look correct. The minimum speed will be a condition where static friction is helping to keep the car from sliding down the incline, so the friction force on the car will be up the plane, proportional to the normal force acting on the car. Since the friction and the normal forces are perpendicular, you are not going to have just sines in the x equation or just cosines in the y equation.

Draw a diagram showing the forces acting on the car including weight, normal force, and frictional force. Assume the frictional force is maximum (because you are looking for the minimum speed) and write the friction force in terms of the normal force and the coefficient of friction. Resolve the three forces into x and y components, and try writing your equations again.

 

[Swatch]

I know my first post wasn't to clear. The work I displayed was for the question "What is the maximum speed before the car starts to slide up the banking" in that case the frictional force is pointed down the slope and I get only sine in Fx and cos in Fy. In the case of the question "What is the minimum speed" I did the work again as you asked me to do OlderDan and I succesfully got the right answer. Then I got sine and cos in Fx.

Thanks.

 

[OlderDan]

I know my first post wasn't to clear. The work I displayed was for the question "What is the maximum speed before the car starts to slide up the banking" in that case the frictional force is pointed down the slope and I get only sine in Fx and cos in Fy. In the case of the question "What is the minimum speed" I did the work again as you asked me to do OlderDan and I succesfully got the right answer. Then I got sine and cos in Fx.

Thanks.

Are you sure your first answer is correct? If your sines and cosines are all of the same angle, then you should have sines and cosines in both your x and y equations because the normal force and the frictional force are perpendicular in both problems.

 

[Swatch]

Thanks OlderDan. Of course you're right. I got the angle all mixed up. But the funny thig is I got an answer that was pretty close to the right one. So I made the assumption that I was right, makes you wonder how many times you could be wrong. Thanks for the help.