A motorcycle rides on a flat treadmill moving at 40 mph, so that it stays stationary relative to an observer beside it.

First, the setup:
  the treadmill is a flat belt which is driven by a motor to go a fixed
velocity -- its important to realize that is NOT a friction-less, mass-less
belt that is driven by the motorcycle.  If you tried to do this, it would
be impossible stay on the treadmill.
  the motorcycle runs on the belt and maintains its position by using the
motor to stay on the treadmill -- the idea is to show that the motor must
do additional work to stay on the belt as the inclination is increased.

Now the physics:
In order for the bike to remain stationary, the sum of all the forces on
it must be zero.
So let's look at the forces:
1) gravity works straight down,
2) we have the wheel against the belt -- this force will be
   perpendicular to the surface of the belt.

If the treadmill is level, that's it.  Even if its moving at 40 mph, the
only effort the motor in the motorcycle needs to do is overcome friction
internal to the bike -- i.e. axle and engine friction.

Now, what happens when the treadmill in inclined?  Force (1) stays the
same but force(2) changes direction.  It leans backward by the same angle
that the treadmill is inclined.  So now we have a horizontal component
which will tend to make the bike roll backward.  The vertical component is
still exactly cancelled out by gravity.  We must now introduce a third
force:
3) friction between tire and belt which is parallel to the surface of the
belt.  The tire pushes back on the belt and the belt pushes forward on the
tire.

To maintain the motorcycle in a stationary position force(3) must exactly
counteract the backward component of force(2).  As several people observed,
if the treadmill is not moving this is quite trivial -- just apply the brakes.
But with the treadmill moving the only way to accomplish this is with
the motor exerting extra energy to push back on the belt as well as maintain
the rotation of the wheels.

Technically speaking:
Energy = Work / Time = Force * Distance / Time = Force * Velocity
   where Force = weight_of_bike * sin(angle_of_inclination)
     and Velocity = 40 mph

-- Roy Harrington <roy@harrington.com>

> But with the treadmill moving the only way to accomplish this is with
> the motor exerting extra energy to push back on the belt as well as maintain
> the rotation of the wheels.

That part I don't follow. The last sentence.

Is the idea that when the treadmill is inclined, the friction must be
greater, so there must be heat expended by this system that isn't expended
in the system if the treadmill is not inclined? I can *almost* see this,
but not quite. The wheels of the bike do not rub against the fabric of the
treadmill at all. On the other hand, I can certainly imagine doing this
trick: I stop the treadmill and incline it; I want to keep the motorcycle
stationary without applying the brake; so I run the motor just a little
bit.

But now imagine this. Instead of a motorcycle, let's use a bicycle. It has
no brakes. I want to keep it in one place on an inclined surface, so I
just lean down on a pedal. I am doing no work at all, I expend no energy.
This is why it's so hard to believe that the motorcycle has to use more
gasoline when the treadmill is inclined.
But then again, when the treadmill is moving, if I *am* pushing down
harder, then naturally I am expending more energy.
So now I'm confused. (I mean, I was confused before, too, but now I am
aware of being confused.)

> Technically speaking:
> Energy = Work / Time = Force * Distance / Time = Force * Velocity

For one thing, energy does not equal work/time. Power = work/time.

But that's not really what I'm having a problem with.

-- Jamie Dreier <pl436000@brownvm.brown.edu>

I think there's two different things getting mixed up here.  First,
there is the contact between the tire and the belt.  There's no slipping
here at all, there's friction between the two to keep them moving
together at the same speed whether its 0 mph or 40 mph.  Since there is
no slipping, there's no heat, no energy expended etc -- its just a
transfer of forces back and forth to keep from slipping.  The second
thing is the internals of the treadmill.  There's a motor in it whose
job is to keep the belt moving at a constant velocity -- if something
tries to slow it down the motor pushes harder and if something tries to
speed it up the motor resists and slows it down.  This is sort of the
weird part because in the inclined scenario, the motorcycle is trying to
speed it up so the treadmill's motor is resisting that speed up.

> But now imagine this. Instead of a motorcycle, let's use a bicycle. It has
> no brakes. I want to keep it in one place on an inclined surface, so I
> just lean down on a pedal. I am doing no work at all, I expend no energy.

This is good, I like the bicycle idea because its a little clearer.
Anyway, you're exactly right, with the treadmill stationary, you just have
to apply a force on the pedal to keep the bike stationary.  There's no
work involved because there's no motion.

Now if we start moving, we still have to apply the force on the pedals but
they start to move too.  So now we have that same force applied over a
distance -- its rotational but Work = Force * Distance still.

> This is why it's so hard to believe that the motorcycle has to use more
> gasoline when the treadmill is inclined.
> But then again, when the treadmill is moving, if I *am* pushing down
> harder, then naturally I am expending more energy.

For the same incline you're not pushing HARDER you're just moving as well
as pushing -- thus now expending energy.

Just to tie this all back together.  The force you're applying to the pedals
is transferred to the wheels and tires.  The tires push backward on the belt
trying to speed the belt up but of course the motor in the treadmill
resists that and uses up the exact same energy you expended pushing on the
pedals.  The forces between the tire and belt are exactly what are needed
to keep the bike from rolling off the treadmill and incidentally are
identical to the forces that would be needed to climb a real hill at the
same velocity.

> For one thing, energy does not equal work/time. Power = work/time.

Yes, I should of left the "Energy = " off.  I mainly wanted to show what
force was needed.

-- Roy Harrington <roy@harrington.com>

Got it!

Thanks.

(Now I'm reminded of that joke, the one whose punchline is: "Yes, I was
right, it *is* obvious.")

-- Jamie Dreier <pl436000@brownvm.brown.edu>