On Sun, 21 Aug 2005 14:40:42 -0400, Ned Simmons
wrote:
In article ,
says...
Ned Simmons wrote:
In any case, I just can't accept that the car travels anything other
than 2*pi*r per rev (as before, r is the distance between the axle and
the road), regardless of what the tread does.
As I explained in my previous post (and as others have subsequently
explained), the equation c=2*pi*r applies only if (a) you're dealing
with a circle and (b) r is measured from the center of the circle.
Neither of those are true in this case. Why would you expect to be
able to use that equation when neither of the basic premises are met?
You *are* dealing with a circle; r is constant regardless
of the shape the tire assumes.
You must be using the distance from axel to pavement as r. That
distance is constant, but the distance from axel to periphery of the
tire is not constant.
Would you argue that you
can't use the distance from the driven axle of a tracked
vehicle to the road surface to calculate the speed of the
vehicle as a function of axle RPM simply because the track
isn't circular in shape?
The problem with this analogy is that the wheel on a tracked vehicle
is a rigid circle with constant radius. A tire is an elastic wheel
with varying radius. Not all points on the periphery of the tire are
moving at the same angular velocity wrt the axel. The (scalar)
surface speed stays pretty constant if the tread doesn't stretch
circumferentially, but some stretching and compression obviously goes
on in the sidewalls. If surface speed remains constant while radius
varies, then angular velocity must vary over the course of a rev. It
all averages out over a rev because the tread makes no net progress
relative to the periphery of the rigid steel wheel.
Think about it from the standpoint of torque.
Sorry, but an analysis of the physics isn't going to do you much good
unless you first get the geometry right.
Then where is the propelling force F acting, if it's not at
the interface between the tire and road, i.e.,
perpendicular to r and distance r from the axle?
Force acts on the road at the point(s) of contact, but torque is
applied all round the tire as tangential force transmitted from the
wheel to the tread thru the sidewalls. Net torque is the sum of the
moments, i.e. the definite integral of the differential moments.
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