Re: Centripetal F from friction

[kowalskil <kowalskil@MAIL.MONTCLAIR.EDU>]

Referring to this:

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 there is no tilting.


"John S. Denker" wrote:
> You had it 98% solved in your original note.High-school geometry
> suffices.  Extend each axle to be a line not just a line segment.
> If necessary (to account for different-radius wheels) consider
> the orthogonal projection of each line onto the plane of motion.
> The point of intersection of these axle-line-shadows is the
> center of the motion.  What's the problem?
>
> Tilting of otherwise-ideal wheels does not even cause any trouble.

Now I think that a simple tricycle will never be able to
follow a circular trajectory without skipping or sliding.
Suppose the back-wheels axis is oriented along the radius
of a circle traced by the midpoint between these two wheels.
Assuming (for a moment) the front wheel slides along a
circle without friction I can focus on back wheels. The
inner wheel must follow a smaller circle than the outer
wheel. Thus their RPM must differ. This already implies
sliding. It is impossible to follow a circle without
sliding.

Suppose each of the back wheels has its own short axis
and that a mechanism is provided to allow different RPM.
I am assuming that short axes are collinear at any time
and point toward the center of my circle. The midpoint
of the front wheel (no longer sliding without friction)
can not possibly be on the same circle as the midpoint
between the back wheels. Just make a drawing to see
that this is true. (The center of three wheels make an
isosceles triangle.) It means that additional sliding
is imposed; the tricycle must constantly rotate about
the vertical axis passing through the CM.

The second version of this problem (see below) is directly
related with what I am going to teach today. Unfortunately,
I will again "wave hands" and try to avoid the problem.
If asked I would say "I do not know." Here is the second
version of the problem again:
-------------------------------------------------------
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 is 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.
Ludwik Kowalski