Angle for a banked curve (friction-less surface)

In summary, the conversation discusses finding the angle at which a roadway should be banked to allow cars to negotiate a curve at a given speed and radius, assuming a frictionless surface. The formula used is β=arctan(v^2/r*g), and the correct answer is 16.4°. The answer key is incorrect and the conversation also touches on the concept of normal force being greater than gravity, which may be unintuitive to some.
  • #1
physgrl
138
0

Homework Statement



At what angle should the roadway on a curve with a 50 m radius be banked to allow cars to negotiate the curve at 12 m/s even if the roadway is frictionless?

a. 0°
b. 12.2°
c. 17.1°
d. 35.0°
e. 73.2°

Homework Equations



β=arctan(v^2/r*g)

The Attempt at a Solution



im using β=arctan(v^2/r*g) but the key says the answer is b?
can anyone help me figure out why its not 16.4??
 
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  • #2
Why not try to derive the formula?
 
  • #3
i did...i though of the triangle as if the gravity was pointing downward the centripetal force outward and then the angle should be the same as the angle for the curve right?
 
  • #4
If your acceleration is towards the centre the force will be too. You need to find the angle of the road that gives a normal force with the horizontal component equal to the force needed to keep the circular motion described by v = 12 m/s and r = 50m

I'm aware that my response is a mouthful :smile:, feel free to ask me for clarification.
 
  • #5
do you mean like a triangle with the vertical 9.81*m and the horizontal (12^2/50)*m ??
 
  • #6
physgrl said:
do you mean like a triangle with the vertical 9.81*m and the horizontal (12^2/50)*m ??

Yes. But the horizontal component will be a fraction of the normal force, which depends on the angle.
 
  • #7
Welcome to PF, physgrl! :smile:

I derived the same formula as you have.
When I fill in the values I also get 16.4 degrees.
So I believe your answer is correct (assuming you did not make a typo).
 
  • #8
I like Serena said:
Welcome to PF, physgrl! :smile:

I derived the same formula as you have.
When I fill in the values I also get 16.4 degrees.
So I believe your answer is correct (assuming you did not make a typo).

I believe your answer is incorrect, as I got 17.1 as my answer. You should investigate why you are using arctan.
 
  • #9
dacruick said:
I believe your answer is incorrect, as I got 17.1 as my answer. You should investigate why you are using arctan.

Here's my interpretation of the forces.

attachment.php?attachmentid=40391&stc=1&d=1319734802.gif


We have:
[tex]F_{resultant} = {m v^2 \over r}[/tex]
[tex]\tan \beta = {F_{resultant} \over m g }[/tex]
The OP's formula follows from this.
 

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  • #10
16.4° is correct, the answer key is wrong. It happens quite often.

ehild
 
  • #11
Oooh I see where I went wrong. My apologies to all. I just always have trouble with the normal force being more than gravity. Its very unintuitive to me.
 
  • #12
Thanks to everyone!
 
  • #13
You're welcome! :smile:
 

1. What is the angle for a banked curve on a friction-less surface?

The angle for a banked curve on a friction-less surface is determined by the relationship between the centripetal force and the gravitational force. It is given by the equation θ = tan^-1 (v^2 / rg), where v is the speed of the object, r is the radius of the curve, and g is the acceleration due to gravity.

2. How does the angle for a banked curve affect the speed of an object?

The angle for a banked curve affects the speed of an object by providing the necessary centripetal force to keep the object moving along the curve. As the angle increases, the required speed for the object to stay on the curve decreases.

3. Is there a limit to the angle for a banked curve?

Yes, there is a limit to the angle for a banked curve. If the angle is too steep, the object will experience a higher centripetal force than the gravitational force, causing it to slide off the curve. This is known as "over-banking" and can be dangerous for vehicles or objects traveling on the curve.

4. How does friction affect the angle for a banked curve?

Friction plays a crucial role in determining the angle for a banked curve. On a friction-less surface, the angle is directly related to the speed of the object. However, on a surface with friction, the angle must be adjusted to compensate for the frictional force and maintain the correct balance of forces on the object.

5. Can the angle for a banked curve be changed?

Yes, the angle for a banked curve can be changed by altering the radius of the curve, the speed of the object, or the coefficient of friction between the object and the surface. These changes can affect the required centripetal force and therefore, the angle needed for the object to stay on the curve.

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