How Should the Radius Change for Higher Speeds on Banked Circular Ramps?

[watsup91749]

hello, I am new and I am really just starting out in physics, but i have a question that my textbook doesn't address yet gives some problems on. This is a sample problem, it deals with banking during circular motion. "Circular freeway entrace and exit ramps are commonly banked to handle a car moving at 13 m/s. To design a similar ramp for 26 m/s, one should:
A) increase radius by a factor of 2
B) decrease radius by a factor of 2
C) increase radius by a factor of 4
D) decrease radius by a factor of 4
E) increase radius by a factor of squareroot of 2

the only circular motion equations i have are a= v^2/R and F=m V2/R
Does anyone know any equations that i can use to solve this? i have other problems like this but give angles of banking, and so does anyone know other equations i can use to figure these problems out?

 

[arunma]

Well, you know that the net force along the incline is:

$$F = \frac{mv^2}{r} \cos (\theta) - mg \sin (\theta)$$

This is of course in the frictionless case. Can you rewrite the expression when considering friction? After you do that, it should be obvious how to modify a bank so as to allow the new speed.

 

[belliott4488]

You don't really need to do the explicit calculations to answer this. This problem is asking you to look at how key quantities vary with respect to one another so that you can say how one changes if the other does.

You're looking at a curve and a car with doubled speed. Well, you've given the expression for the centripetal acceleration required for the car to follow the curved ramp, so how will this force change if the speed is doubled? Okay, now you know how this force has changed - has the gravitational force changed (hint: it's due only to the mass of the car)? Recall that the banking is done in order to balance the pull of gravity down the slope with the inertial (centrifugal) force up the slope. To keep them balanced if the speed is doubled, what do you need to do to the radius? (Look at the equations you stated - they're all you need.)

 

[metalInferno]

i think this is how it should be :
the eq for banking is : $$tan (\theta)= \frac{v^2}{rg}$$

where r is the radius and g is acc due to gravity . tan theta remains constant leaving u with this
$$\frac{13^2}{r} = \frac{26^2}{r'}$$

now $$r' =4r$$
this should be the answer.