The
trike moves __forward__! This seems counterintuitive to almost everyone
(certainly it is to us), but the result of an experiment with a real trike
is indeed that it moves forward.

You
can check it out - just find a trike. You do have to be careful not to let
that front wheel twist sideways. You need to do something to keep it from
twisting, but without pushing forward or backward on it. You can gently put
a finger against the side of the wheel. Or you could use duct tape (the all-purpose
American solution to all problems) to, say, strap the wheel's fender to the
main frame of the trike. Or you can use a bicycle instead of a tricycle, but
then you have to gently use your fingers to keep it from flopping over sideways.

__ not__ ride the trike, instead you stand
or kneel next to it and push on that pedal.)

__push__ gently __forward__
on the pedal that is initially pointing straight down:

Here's
an analysis of the trike problem, which (if you do it very carefully) is actually
a fairly stiff first-year physics problem. The trike starts at rest, we will
apply a force to the pedal, and we want to know which way it will __start__
to move. (After it's moved a bit, the pedal will no longer be directly down.
We won't worry about that - just concentrate on its velocity right after it
starts to move.) Forces cause __accelerations, __and since it starts at
rest, its initial acceleration will be in the same direction as the direction
of its velocity an instant after the beginning of our experiment. So what
we want to know is the __direction of its initial acceleration__.

The
important parts of the trike are the front wheel and the pedals. You can think
of the rest of the trike as superfluous stuff that is dragged along behind
and keeps the whole trike from tipping over. (If you're planning to use a
bike for your experiment, think about the pedals, the chain, and the rear
wheel, and ignore the rest of the bike.)

__hope__
they would say is either "Let me think about it" or "Let's
try it, where's my trike?") It's because we have all ridden on trikes
and (we think) know everything there is to know about trikes that most of
us just jump to the erroneous conclusion: Push forward on that lower pedal
and "of course" the trike will go backward. (Like the big guy in
the second panel of the bus placard: "I've known about trikes since I
was two.")

**F**__right__ in all these diagrams).
Forget the other symbols in that diagram for now. Since the pedal is rigidly
attached to the front wheel, it's really just part of the wheel, providing
a convenient place for you to push. Now the force **F** *tends*
to rotate that wheel __clockwise__ around the point of contact between
the wheel and the ground (the point directly below the center of the wheel).
And if the wheel does start to turn clockwise, then the wheel will __have__
to move to the right, forward. End of story.

**F**) on the pedal in the forward direction.
Now the ground will probably also exert a force in the horizontal direction
(call it **f**, it's really a "frictional" force),
but we don't know to begin with whether **f** is in the forward
or backward direction. Let's take the positive direction to be forward and
let F and f denote the components of those forces in the forward direction.
(If f turns out to be negative, then that simply means that** f, **a
vector, is in fact in the backward direction.) There are also forces (up and
down) in the vertical direction, exerted by the earth's gravity and also the
upward push of the ground, but we'll leave those out of our analysis because
we only care about the horizontal motion.

Newton's
second law gives

F
+ f = ma (1)

where
m is the mass and a is the component of the acceleration in the forward
direction. We want a, but f in Eq. (1) is unknown. We need more physics, which
we can get from the torque equation.

As
drawn, both F and f produce torques tending to turn the wheel counterclockwise.
(That is, counterclockwise around its __axle__, the __center of the wheel__.
That's what "clockwise" or "counterclockwise" refer to
in this discussion. By contrast, in the initial "semi-intuitive"
discussion given earlier, that force F tends to turn the wheel clockwise with
respect to the contact point with the ground, and f exerts no turning effect
at all with respect to that contact point.)

Care
with __signs__ is essential here. Let
a denote the angular acceleration in the __clockwise__
direction. Then a and a are
related by

a = a/R (2)

and
the torque equation (get the signs right!) is:

Fr
+ fR = - Ia (3)

where
I is the moment of inertia about the center of the wheel.

Now
comes a little algebra (which we'll omit), leading to the results:

a
= FR(R - r)/(I + mR² ) (4)

__square__ of
the radius. On some browsers, what should be a superscript "2" shows
up as a "?" (question mark). Similar peculiarities may show up in
several of the equations below. Ah, the idiosyncrasies of computers!)

All
that we care about is the __sign__ of a. Since R > r (the radius of the
wheel is greater than the length of the pedal arm), a is positive - the trike
moves __forward__.

That
surely is contrary to most people's "intuition", probably because
we have all ridden trikes and we know that the way to back up is to push forward
on the pedal that is pointing down. Remember, though, that in our problem,
we are not sitting on the trike, instead we're standing next to it. Those
are two __different__ situations, and there is no reason to expect their
solutions to be the same.

It's
still strange, though, to see the trike move as it does. As you push forward on
the downward pointing pedal, the trike really does move forward a bit and that
pedal rises up. Here are two sketches of trike-plus-pedal - first when it's at
rest, and then when it has moved a bit.

The
wheel as a whole moves forward. So does the pedal. Relative to the __wheel__,
it's true that the pedal moves backward. That is, though the wheel and pedal
both move forward, the pedal doesn't move quite as far forward as the wheel
does. __center__ of
the wheel. Notice that although the wheel turns and the pedal turns with it,
the center of the wheel definitely moves forward. And if the center of the
wheel moves forward, then so does all the rest of the trike - seat, frame,
handlebars, etc.

You
can also solve equations 1-3 for f, which gives:

f
= - F(I + mRr)/(I + mR²) (5)

Notice
that f is definitely negative. That is, the frictional force always is in
the backward direction. Thus the frictional force, if it acted alone, would
indeed push the trike backward. But in our case, as long as R > r, **f**
is not large enough to offset the force that __we__ exert, **F**.

What
if r were greater than or equal to R? That is what if the length of the pedal
arm were as large as the radius of the wheel or even greater? Eq. (4) says that if r = R, then a =0, and
the wheel won't move. And that equation also says that if r > R, then a <
0, that is, that the trike will indeed movebackward.

Normally, you can't have r that large, but you can achieve that condition by putting an extension on the pedal and then resting the whole thing at the edge of a table, so that the “pedal” is down below the edge of the table:

Now
if you push forward just at the edge of the table ( r = R ), you'll find that
the trike won't budge - until you push it so hard that it begins to skid,
without rolling. (We assumed rolling in our analysis of the problem, so our
theory can't be expected to make any sense if
it skids.) And if you apply your forward force down below the edge
of the table (r>R), as in the sketch just above, you'll find that the trike
does begin to move backward.

**Alternative
Theory:**

(This is a careful version of the initial discussion given at the beginning,
what we called the "semi-intuitive" version.) In the preceding analysis,
we calculated torques about the center of mass. It is also legitimate to take
torques about any __fixed__ point, if we're careful, and sometimes it's
simpler to do so. Taking torques about the point in the ground where the wheel
touches the ground, we have:

F(R
- r) = I’a (6)

where
I’ is the moment of inertia *with respect to that new point*. By the
principal axis theorem (from introductory physics),

I’
= I + mR² ,

and,
with this alternative theory, a is still
equal to a/R.

Putting
those results into Eq. (6) leads to the same expression as before for a, Eq.
(4). This approach is algebraically much simpler. We don't have to introduce
f, since it exerts zero __torque__ about that point in the ground. Although
this approach is simpler and more elegant, it may seem a bit too simple to
be reliable, and it's reassuring to know that we can also get the same result
by our first approach, calculating torques about the center of mass. And part
of the fun of doing physics is that often there are several different and
equally correct ways of tackling the same problem.