What Angle Should a Curve Be Banked to Allow Safe Travel Without Friction?

[mikefitz]

A car can negotiate an unbanked curve safely at a certain maximum speed when the coefficient of static friction between the tires and the ground is 0.84. at what angle should the same curve be banked for the car to negotiate the curve safely at the same maximum speed without relying on friction ?

Honestly, I have no idea where to begin. I can't calculate speed, I don't know the mass of the car, the radius of the curve, anything.

How do I compare the turning angle and speed with a banked curve as opposed to an unbanked curve?

 

[mikefitz]

Ok, still trying to figure this one out...

I know I need to use Newton's second law F=ma; but how do I analyze the unbanked curve in x and y directions with only the coefficient of static friction?

Also - does centripetal acceleration change for a banked curve?

thanks!

 

[Andrew Mason]

A car can negotiate an unbanked curve safely at a certain maximum speed when the coefficient of static friction between the tires and the ground is 0.84. at what angle should the same curve be banked for the car to negotiate the curve safely at the same maximum speed without relying on friction ?

Honestly, I have no idea where to begin. I can't calculate speed, I don't know the mass of the car, the radius of the curve, anything.

How do I compare the turning angle and speed with a banked curve as opposed to an unbanked curve?

Can you work out the (maximum) speed from the first part? All you have to know is the expression for the horizontal force (think circular motion). Since the horizontal force is supplied entirely by friction, you will be able to work out the speed.

Establish the equations for the vertical and horizontal forces on the banked curve (no friction) and that will give you the expression for angle in terms of the car's speed.

[Hint: Do a free body diagram of the car. What is the vertical acceleration (does it accelerate vertically?). What are the forces (as vectors) acting on the car vertically? What is the horizontal acceleration as the car is rounding the curve? What are the forces acting on the car horizontally? ]

AM

 

[mikefitz]

looking through my notes, I don't have any equations relating to circular motion that include coefficient of static friction in them; this is why I'm having trouble coming up with speed for example. The only equations I have to find the bank angles require speed and radius of the curve...

 

[mikefitz]

If any of you have additional tips you can pass my way I will check this topic in the morning; again, thanks for your time.

 

[Andrew Mason]

If any of you have additional tips you can pass my way I will check this topic in the morning; again, thanks for your time.

The horizontal force is supplied by friction. The maximum speed without sliding is determined by the maximum centripetal force that can be supplied by friction:

$$F_c = \mu_s N = \mu_s mg$$

That gives you the speed, v (when you used the correct expression for centripetal force).

For the second part, I'll get you started:

$$\vec N\sin\theta + m\vec g = ?$$ (vertical acceleration)

$$\vec N\cos\theta = ?$$ (horizontal acceleration)

Solve that for $\theta$ in terms of v and plug in the value of v from the first part.

AM

 

[mikefitz]

So, the vertical acceleration will have to be zero (since the car isn't actually accelerating upward, just horizontally).

What about the first equation http://img147.imageshack.us/img147/5912/10944360ce2.png - how do you get v from this equation when the variable v isn't part of the equation? also - how am I supposed to substitute a value for m if m isn't given?

sorry for the elementary questions !

 

[Andrew Mason]

So, the vertical acceleration will have to be zero (since the car isn't actually accelerating upward, just horizontally).

What about the first equation http://img147.imageshack.us/img147/5912/10944360ce2.png - how do you get v from this equation when the variable v isn't part of the equation? also - how am I supposed to substitute a value for m if m isn't given.

I said you can work out v from the correct expression for centripetal force. You are supposed to know what that is. What is the centripetal acceleration when an object moves in a curved path or radius R with tangential speed v?

AM