Designing a Banked, Circular Highway Exit | Help with Homework

[Yellowkies_3275]

Homework Statement

The task is to design a highway. I did a similar project last year but I have completely forgotten and have no clue what to do.

You have to design a banked, circular, clockwise highway exit. you knwo you have 5000 sq. meters. of land to build it on..you also know that the average stretch before entering a highway exit is typically 300 m. you know the highways speed limits are 100 km/h but can reach 120 km/h (27.777 m/s and 33.333 m/s). you also know that vehicles can range in weight from 100 kg to 100,000 kg.

We have to determine the speed limit for the curve as well as determine the angle of the bank on the curve. and aslo take into account different road conditions such as rain and snow.

I've been looking at the problem for hours and tried some stuff but I only got ridiculous numbers because i used the highways speed limits because I don't know what the speed limit of the curve should be. I don't know what to do

Homework Equations

I don't even know

I think v= √g*r*tanθ

The Attempt at a Solution

[/B]
I have none for you right nowplease give me some guidance (I shouldn't have taken physics I'm so helpless)

 

[haruspex]

Draw a free body diagram, i.e. a vertical section through the vehicle and banked road, viewed end-on.
What forces act in this plane?
You will need some data regarding coefficients of static friction for rubber on road under different conditions.

 

[Yellowkies_3275]

where

Draw a free body diagram, i.e. a vertical section through the vehicle and banked road, viewed end-on.
What forces act in this plane?
You will need some data regarding coefficients of static friction for rubber on road under different conditions.

do i get this information from. it does not come with the lab? is it appropriate to search up
i know what forces act but i still can't see what to do both without a defined velocity and without an agle

 

[Yellowkies_3275]

Draw a free body diagram, i.e. a vertical section through the vehicle and banked road, viewed end-on.
What forces act in this plane?
You will need some data regarding coefficients of static friction for rubber on road under different conditions.

I also am curious, in the equations i know (V= √g*r*tanθ) (V=√[tanθ+μ/1-μtanθ]*r*g) weight isn't a factor, so why am i given the weight of vehicles?

 

[Yellowkies_3275]

Draw a free body diagram, i.e. a vertical section through the vehicle and banked road, viewed end-on.
What forces act in this plane?
You will need some data regarding coefficients of static friction for rubber on road under different conditions.

okay i think i have something that works could you just check if these numbers sound okay?

so knowing the radius would be 35 sq meters I tested out various angles because I really wasn't sure how to find a maximum logical one without just guessing and checking...because if you wanted you could make a 70 degree bank so that the car could continue to go 120 km/h even on the curve...but that's a bit extreme... so like many things were possible I think so I just tried to use common sense to figure out what was most reasonable...I ended up deciding to use a 45 degree angle for the curve, a bit steep but when you check the amount people would have to slow down to from the 100~120 km/h highway to drive the 45 degree curve the velocity still seems reasonable...fast maybe but efficient and still safe

r= 35 sq meters
θ=45°
V= on a frictionless surface (ice on road or something) 18.52 m/s or 66.667 km/h. Then on more regular conditions speeds of upto 20% increase can be expected so my max speed limit is 80 km/h (22.222 m/s) which would allow you to travel the curve of 45 degrees with a μ of 0.18

i still have no clue what to do with the info about the weight of vehivles traveling (range from 100 kg to 100,000 kg) logically in the real world i can understand why that info would be important you don't want any trucks tipping over, but nowhere in these banked curve equations is mass mentioned, mass gets mentioned seperatly in x or y direction but it cancels out so where is it relevent?

 

[haruspex]

if you wanted you could make a 70 degree bank

No, that's why I mentioned coefficients of friction.
You cannot impose a minimum speed. With a steep angle and ice a car could slide down at low speeds.
So the minimum guaranteed friction coefficient sets a maximum angle. Then you can find the maximum centripetal acceleration, giving you a relationship between speed and radius.

Not sure how to use the speed data in the question. I think they are suggesting the 300m is to be used to slow from the typical highway speed given to a speed to embark on the exit ramp. But then you need to know a reasonable deceleration rate.
Once on the ramp, they probably expect you to use a constant radius and speed, but in principle you could have the speed continuing to decline, allowing a tightening curve.

And you are right, the vehicle mass is irrelevant.

 

[Yellowkies_3275]

No, that's why I mentioned coefficients of friction.
You cannot impose a minimum speed. With a steep angle and ice a car could slide down at low speeds.
So the minimum guaranteed friction coefficient sets a maximum angle. Then you can find the maximum centripetal acceleration, giving you a relationship between speed and radius.

Not sure how to use the speed data in the question. I think they are suggesting the 300m is to be used to slow from the typical highway speed given to a speed to embark on the exit ramp. But then you need to know a reasonable deceleration rate.
Once on the ramp, they probably expect you to use a constant radius and speed, but in principle you could have the speed continuing to decline, allowing a tightening curve.

And you are right, the vehicle mass is irrelevant.

How do i find the minimum guaranteed friction coefficient?

 

[Yellowkies_3275]

No, that's why I mentioned coefficients of friction.
You cannot impose a minimum speed. With a steep angle and ice a car could slide down at low speeds.
So the minimum guaranteed friction coefficient sets a maximum angle. Then you can find the maximum centripetal acceleration, giving you a relationship between speed and radius.

Not sure how to use the speed data in the question. I think they are suggesting the 300m is to be used to slow from the typical highway speed given to a speed to embark on the exit ramp. But then you need to know a reasonable deceleration rate.
Once on the ramp, they probably expect you to use a constant radius and speed, but in principle you could have the speed continuing to decline, allowing a tightening curve.

And you are right, the vehicle mass is irrelevant.

isnt the minimum coefficient of friction let's say for a slippery ice covered road zero? so with 0 friction i would have to rely on v= √g*r*tanθ for which i don't know velocity or theta

on a reguar road without velocity or theta, where i have to judge for myself what is best how would i come up with friction, if my previous friction wasn't correct?

 

[CWatters]

Perhaps I missed it but how do you know the radius is 35m?

Can you check the OP contains the problem statement word for word or that you have been told to make your own assumptions?

For example what shape is the 5000sqm? Was a diagram provided?

 

[jbriggs444]

isnt the minimum coefficient of friction let's say for a slippery ice covered road zero? so with 0 friction i would have to rely on v= √g*r*tanθ for which i don't know velocity or theta

The concern is that a vehicle going around the curve dead-slow over glare ice could slide down off the roadway to the inside. The relevant parameter is the coefficient of friction on glare ice. You should not assume that this coefficient is zero -- that would lead to your designing the curve with no bank angle at all.

You are the designer. You choose v, you choose r, you choose theta. You want choose them so that certain constraints are met. For instance, you might want to optimize for max permitted v by modifying road width, curve radius and bank under the constraints that land use not exceed 5000 m^2, and that cars neither slide off the inside of the roadway under glare ice conditions nor off the outside of the roadway in the wet at the posted speed limit.

You need at least some bank angle so that rain drains from the roadway. https://en.wikipedia.org/wiki/Drainage_gradient. You can also find guidance in https://en.wikipedia.org/wiki/Interstate_Highway_standards.

I found this link with various coefficients of friction. http://www.iaeng.org/publication/WCE2011/WCE2011_pp2381-2384.pdf

One approach that could be used would be to make the surface of the ramp concave upward with an inner shoulder at a lower bank angle than the traveled portion of the roadway -- if cars slide off inward, they only slide onto the shoulder.