Circular Motion, Banked Turn Without Friction

[mobwars]

1.Traffic travels north on Newton blvd and turns east onto the road einstein way. The speed of the traffic traveling north and east is 55 mph (24.6m/s). The banked turn must fit within an area (shown in the picture attached). The illustration is not drawn to scale.

You are to design an exit ramp which is banked such that it requires no friction in the radial direction. Also, you are to design the ramp so the vehicles maintain their speed and are not required to slow down.

2. Fnet=ma, r=v^2/(g*tan(theta)), r=(v*T)/(2*pi), v^2=g*r*tan(theta)


3. What I have attempted so far is this:

First I set r equal to v^2/(g*tan(theta)), and r equal to (v*T)/(2*pi), and then set the two equations equal to each other. My thought process is that since there are two unknowns, the radius and the angle of the banked turn, I should set up a system of equations in order to solve for one variable so that I can find the other. The point in time where I run into trouble is when I don't know the period of time to make one revolution (T).

Am I making a dumb mistake, or am I using the wrong equations? Also, is it safe to assume that, since the vehicles will stay at a constant velocity of 24.6 m/s, that the velocity given is also the critical velocity?

http://s296.photobucket.com/albums/mm179/mobwars911/?action=view&current=PHYSICS.jpg"

 

[SammyS]

Get the maximum r from analyzing the figure. Maybe give Bubba a little extra room.

You don't need to find T.

Assuming your equations are for the situation shown here, find θ once you get r.

 

[mobwars]

Get the maximum r from analyzing the figure. Maybe give Bubba a little extra room.

You don't need to find T.

Assuming your equations are for the situation shown here, find θ once you get r.

Okay. I analyzed the figure and I'm still a little confused. The maximum r would be found by seeing that the dimensions make an imaginary square 18m x 18m. I'm just so lost.

I know this isn't for just getting the answer, but I've been staring at this picture for past hour. I know I'm over thinking it.

 

[mobwars]

Here's what I have thought of as far as maximum radius is concerned.

If the space above bubba's is 18 m and the space beside bubba's is 18m, then the two dimensions make a triangle (in the form of a^2+b^2=c^2) (18+x)^2+(18+x)^2=c^2. Am I on the right track?

 

[SammyS]

Finding r is basically a geometry or a trigonometry problem

Inscribe a circle in a square. The left side & top side of the square represent the two streets. The upper left 1/4 of the circle represents the ramp.

Now inscribe a smaller square in the circle. Bubba's BBQ sits in the upper left corner of this smaller square. The two squares are separated by 18 meters.

Find the radius of the circle.

 

[SammyS]

If you don't want the maximum radius try this:

Draw a 36×36 square, the left side & the top along the two streets. The diagonal from the upper right to the lower left, just touches Bubba's. Draw the upper left quarter of the circle with radius 36 meters and centered at the lower right corner of this square. It easily misses Bubba's.

 

[SammyS]

If you don't want the maximum radius try this:

Draw a 36×36 square, the left side & the top along the two streets. The diagonal from the upper right to the lower left, just touches Bubba's. Draw the upper left quarter of the circle with radius 36 meters and centered at the lower right corner of this square. It easily misses Bubba's.

 

[mobwars]

If you don't want the maximum radius try this:

Draw a 36×36 square, the left side & the top along the two streets. The diagonal from the upper right to the lower left, just touches Bubba's. Draw the upper left quarter of the circle with radius 36 meters and centered at the lower right corner of this square. It easily misses Bubba's.

Yes, okay thank you. That makes so much more sense. This seems like a problem that you have to be able to visualize more than anything else, and the way you worded that make it extremely clear. Thank you thank you thank you. Have a great night!

 

[SammyS]

You're welcome & good luck.

 

[mobwars]

You're welcome & good luck.

Everything else is going smoothly, as I outlined every other process that I would need to go through for the later calculations. Pretty insane angle for the banked turn though...59.8 degrees. Certainly would be interesting to try and drive on that.

 

[SammyS]

Even at the very maximum radius, I get about 45°.

The Daytona Speedway's turns are only banked at 31°, and that is very extreme.

 

[mobwars]

Even at the very maximum radius, I get about 45°.

The Daytona Speedway's turns are only banked at 31°, and that is very extreme.

Hmm...what equation did you use?

I used (theta)=tan^-1(v^2/rg)

 

[SammyS]

Your answer is fine.

It's possible to use a radius of about 61 meters. That gives about 45°.

I don't know what figures Bill France used at Daytona. LOL

 

[mobwars]

Your answer is fine.

It's possible to use a radius of about 61 meters. That gives about 45°.

I don't know what figures Bill France used at Daytona. LOL

radius MUST be less than 62 right?

EDIT: and indeed. lol.

 

[SammyS]

Yes, a little less than 62. Actually the max radius is:

$$r_{max}=\left(2+\sqrt{2}\,\right)\cdot18$$

 

[mobwars]

Yes, a little less than 62. Actually the max radius is:

$$r_{max}=\left(2+\sqrt{2}\,\right)\cdot18$$

okay, cool. thank you again for all of this, you've been a great help!