Re: Centripetal F from friction

[Ludwik Kowalski <kowalskiL@MAIL.MONTCLAIR.EDU>]

OK, let approach the same problem from the other end. The
radius R is given and I want to know by how much should
the front wheel (or wheels) be turned to follow the circle at
a given speed.

The angle iz zero when R is infinity; R decreases when the
angle (between the planes of front and back wheels) becomes
larger. Driving experience tells me that there is only one
answer for a given car on a uniform horizontal surface.

I know that the resultant of frictional forces acting on all
wheels must be equal to centripetal force. Suppose the
speed and the mass of the vehicle are given. Then I can
calculate the centripetal force needed as Fc=m*v^2/R. At
what angle should the front wheels (or wheel) be turned to
generate this force? The distances between the wheels are
presumably known. Specify other details, if necessary.
I have no idea how to solve this problem.

Hugh Haskell wrote:

> At 22:28 -0500 10/31/01, Ludwik Kowalski wrote:
> > Is this OK?
> > A tricycle is moving along a straight line on a horizontal floor.
> > All three wheels are parallel. Then the front wheel is turned
> > by 10 or 20 degrees and the path becomes circular. The wheels
> > roll without sliding or skidding. What is the radius of the path?
> >
> > My guess is that R does not depend on the speed. To estimate
> > the radius I would replace (on my drawing only) the back
> > wheels by a single wheel in the middle. Then I would say
> > that the plane of the back wheel is tangent to the circle and
> > the plane of the front wheel is tangent to the circle. Knowing
> > the distance between the centers of two wheels I can trace
> > the circle and determine R (or derive a formula for R). Would
> > this produce a good prediction? Probably not. Why not?
> >
> > I think that a tricycle would be easier to analyze than a bicycle
> > because the is no tilting.
>
> Assuming no tilting, and you have stated that the tires do not slip
> or skid. Then you cannot predict the radius of the circle, because
> the friction that is providing the centripetal force is static
> friction, whose law is an inequality. So as you go around the circle,
> you can go at any radius or any speed as long as the centripetal
> force necessary is less than the maximum possible static friction
> force between the wheels and the floor.
>
> What you can do is predict the minimum radius for a given speed, or a
> maximum speed for a given radius. Just plug in the maximum friction
> force for the surfaces and solve for the missing item.
>
> Its the same if you allow tilting, only figuring out the friction
> force is a bit more tricky.
>
> An almost as interesting problem that is solvable is to find the
> turning radius for an airplane in balanced flight at a given speed
> and given angle of bank. Turns out to be independent of the mass of
> the airplane. Complicating factor: when you bank the plane the amount
> of lift available to maintain the plane's altitude decreases, since
> it is always perpendicular to the wings (assume no dihedral for
> simplicity). Hence in order to maintain altitude, and airspeed, you
> have to add power, because you need to raise the nose a bit to
> increase the lift of the wings, which slows down the aircraft. If you
> are going to slow for the angle of bank, the airplane can stall. Fun
> to do at altitude, but really scary if it happens as you turn onto
> your final approach during landing when you are only a couple of
> hundred feet up.
>
> I'm sure John D. can tell you of at least one instance where a
> student did it to him. Probably only one, though, since that is a
> lesson that you learn very well the first time-if you survive. I'm
> sure that was the kind of stall that little girl who was trying to
> solo across the country a couple of years ago, when she spun in soon
> after takeoff from Cheyenne, Wyo. Turning when you are low, slow and
> heavy can get you into really big trouble very fast, especially if
> you are inexperienced.