SOLUTION: On a banked race track, the smallest circular path on which cars can move has a radius of 114 m, while the largest has a radius of 166 m, as the drawing illustrates. The height of

Question 1099405: On a banked race track, the smallest circular path on which cars can move has a radius of 114 m, while the largest has a radius of 166 m, as the drawing illustrates. The height of the outer wall is 14.6 m. Find (a) the smallest and (b) the largest speed at which cars can move on this track without relying on friction.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The centripetal acceleration needed to stay in circular motion is related to
circle radius, , linear velocicty and angular velocity by

.
If we cannot count on friction,
the centripetal acceleration needed will have to be supplied by the banking of the track.

The sketch below shows a vertical cross-section of the banked track,
an object upon that track, and a representation of some of the accelerations acting on that object.
The only force acting on the plane of the cross section is weight,
so the only acceleration is the acceleration of gravity.
We can consider it as made of components parallel and perpendicular to the track surface.

The acceleration of gravity and its component parallel to the track surface is shown in the sketch.
The acceleration of gravity has a (not shown) component perpendicular to the track surface,
which would make the car sink into the track,
but a reaction force from the track (called the normal force) keeps that from happening.

For a car to stay on a circular path,
the acceleration component parallel to the track surface must be

There are two similar right triangles in the sketch,
the wedge representing the cross-section of the track,
and the triangle formed by and ,
so,

.
Using ,

,
and

For ,

--->

.
For ,

--->

.

NOTES:
If I did not make a mistake, I believe those are the answers a teacher would want. (I am not a physics teacher. I am a chemist, but I did some substitute physics teaching 45 years ago, when I was in college).

If the track was a circle, and covered with ice, It would be a good idea, to keep your speed between 63 and 75 kph while entering the track and driving on it.
Lower speeds would cause the car to drift towards the inner lane, and eventually off the track.
Higher speeds would cause the car to drift towards the outside lane, and eventually fly off the track.
However, race tracks are not all curves, and drivers can go faster or slower than calculated by letting the car drift through the curve, and correcting that drift as they exit the curve. There is also friction (with track surface and hair) that can be harnessed to help.
RELATED QUESTIONS
On a banked race track, the smallest circular path on which cars can move has a radius of (answered by ikleyn)

a circular ground of radius 7 m has a path of width 7 m around it on its outside. the... (answered by Cromlix)

A Gardener wishes to but a circular water feature (pool) in a right - angled triangular... (answered by addingup)

A race-car starts at the 3 o'clock position and drives counterclockwise on a circular... (answered by Alan3354)

Hello, I have a question that im extremely stuck on. The question is as follows:- A... (answered by stanbon)

Hello, I have a question that im extremely stuck on.The question is as follows:- A... (answered by ptaylor,ankor@dixie-net.com)