My 3rd grader and I have completed most of her science fair project. 
The project involved building two identical vehicles out of styrofoam.
 The vehicles are exactly alike EXCEPT for the size of the wheels. We
have been able to complete the entire project except for background
knowledge/research.  We have determined through a series of "races"
that the vehicle with the smaller wheels goes faster down the inclined
plane.  After spending several hours on the internet, I have been
unable to find out WHY the smaller wheels go faster.  I found one
reference to it on a message board, but the terminology was way beyond
her comprehension --- and mine!  Please help!!  WHY DO SMALLER WHEELS
GO FASTER THAN LARGER WHEELS??

---

Request for Question Clarification by sublime1-ga on 02 Apr 2006 14:33 PDT

4448don...

In the simplest terms, based on the math at the site pinkfreud-ga
pointed to, the relative slowness of the larger wheels is due to
the increased inertia (tendency for an object at rest to stay at
rest) arising from the increased mass of the larger wheels. Even
more simply, it takes them longer to respond to the force of
gravity, given their greater inertia.

Let me know if this interpretation satisfies your interests...

sublime1-ga

Hi Don,

Nice experiment!  In this explanation, I will try to give you both the
physics and pointers to  a way to help your daughter understand it.

You have several factors present in your experiment:

1) total mass of the two vehicles (You should weigh both, to determine their mass)
    
    I expect the one with bigger wheels will be a bit heavier - 
    so you have masses m.A and m.B
    and slightly different inital energies 
                      (E= m * g * h)  with (m = m.A or m=m.B)  

   
2) Friction - The bigger wheels result in smaller rolling friction 
    (This is why bicycles are made with bigger wheels.)

    These two factors would favor the conclusion that vehicle B (=Bigger wheels)
    will be faster, the opposite of what you found.

 That leaves  the third factor to explain your data:

3) Inertia: 
 a) linear inertia of the vehicle and  b) rotational inertia  of the wheels 
   

     Inertia opposes acceleration and does in two ways 
              a) vehicle B (with total mass m.B) has harder time 
picking up speed (this cancels advantage 1)
              b) the vehicle with wheels which have larger inertia has
harder time getting them rolling.

http://commons.bcit.ca/physics/demos1103/DLD/DLD19.htm

 It is more difficult -- takes more energy --
 to get larger wheels rolling; they have a harder time picking up speed.

 If your daughter has a bicycle, you could call her attention to the
 force she has to exert 
 to get it started (overcoming inertia and friction) vs
 to keep it going (overcoming frictio only) vs
  going up the hill (overcoming friction and gravity)

____________________________

 I disagree with the comment by qed100-ga on 02 Apr 2006 13:21 PDT
that to do experiment is sufficient.
 It's great that you want to understand the result. I suggest you do
additional observations to verify  understanding of your results:

  a) You may estimate the kinetic energy of the vehicle at the bottom of the ramp. 
 To do that, terminate the ramp with a gentle curve (to avoid impact
with the table) and observe how far and how quickly
 vehicles  A and B  run on the table after they leave the ramp.

this is a picture of 'gentle termination' of a ramp
http://www.andamooka.org/newtphys/figs/bk1/ch03/ramp.JPG
taken from here:
http://www.andamooka.org/reader.pl?pgid=newtphysbk1ch03

 Kinetic energy of  vehicles will be stored in forward  movement of
the whole vehicle (forward speed v) and in the rotation of the wheels
(angular velocity or RPM)
 If factor #3  is significant, vehicle B will run further before it
stops. You still have two factors here less friction, and possible
larger inertia. So - how can you separate them?

 b)  Attach weights  I) to the body    II) to the wheels   (In two
ways, close to and farther from the axis)

 that will affect factors #1) and #3) differently and will not change
factor #2) the friction too much.
 Make an estimate/guess of the effect of weights and see if the
additional experiments confirm your understanding and data.

 Have fun, and do not hesitate to post an RFC  (and/or results)
__________________________________________
Few links for the background and formulas: 

The effect 'rotational inertia sucking away energy' is actually a physics FAQ
best illustrated by:
"For a final activity take a ring and a solid disk [same mass] and
roll them down a ramp..."
http://www.iit.edu/~smile/ph9117.html
http://csats.psu.edu/files/GREATT/Flywheels/Flywheels.htm


Simple explanation is here:  (moment of inertia = rotational inertia ,
which resists change in angular velocity)

 The more mass is distributed farther from the axis, the greater the
moment of inertia.  And the greater the moment of inertia, the more
resistance to changes in rotational motion.  So as objects roll down
the ramp, potential energy is transformed into translational and
rotational kinetic energies, but the object with the greatest moment
of inertia rotates more slowly.
http://acad.udallas.edu/physics/GP1/rotational_motion_ii_faqs.htm

SEARCH TERMS: rotational inertia ramp
 brings more info on Moment of Inertia (of full and hollow cylinder)
http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.inertia.html
http://physics.ucsd.edu/students/courses/summer2002/ss1/physics2bl/EXPER2_overview.htm
 also
SEARCH TERMS: rolling friction ramp
such as:
http://teacher.scholastic.com/dirt/roll.htm

Do not miss:
http://images.google.com/images?client=opera&rls=en&q=rolling%20friction%20ramp

Picture at  www.andamooka.org (used above) shows a 'gentle ramp termination' 

and leads to some  'Balls and Ramps Study' of other researchers
http://www.belmont.k12.ma.us/winnbrook/index.html?class_pages/first.htm


If you find a way to change the shape of your ramp, you can ask
which shape will bring the ball down fastest - and (re) discover the
Brachistochrone curve
http://en.wikipedia.org/wiki/Brachistochrone
http://www.du.edu/~etuttle/math/brach.htm
http://images.google.com/images?q=Brachistochrone

 
 This sets of slides
 Rolling Cylinders Demonstration (This may be too advanced, but the
pictures may be of interest)

         "..involves different cylinders rolling down a slope
Depending on their moment of inertia, they accelerate at different rates. 
Download a PowerPoint presentation explaining the experiment..."
  http://www.physics.gla.ac.uk/misc/teachers/index.html#rollingCylinders

This article may provide a bit of insight. It's not written at the
third grade level, however:

http://hea-www.harvard.edu/~fine/opinions/wheelsize.html

Not sure if this is the forum you found, but it may help:

http://www.physicsforums.com/archive/index.php/t-3071.html

Rainbow~

Here's something which may give you a lift. Consider this: You have
performed an experiment. You've controlled all the relevant variables.
You kept all the variables constant except one, the wheel radius, and
you've faithfully collected data on speed vs wheel radius. You notice
that, all other things kept equal (including the energy source), the
speed of the car is related -statistically- to the wheel radius.

   But- you don't know at this point exactly *what* the deep
connection is between speed & radius. Does this mean that you and your
daughter have failed the science project? Not at all! :)

   The backbone of scientific exploration, before theory, is
experiment. And in experiment, there is no promise, no pretense, of
arriving at a deep theory to connect & explain all the data. As long
as your experiment has been done rigorously and honestly, then you can
also honestly be said to have done a good work of science.

   ...Now of course you still *want* to know a good theory of the
contribution of wheel size to speed. You are prodded to ask for such a
theory by the conspicuous correlation in your data. It's something to
aspire toward. But, the question always comes before the answer, and
there's no guarantee of having the answer. Many questions in science
persist in going unanswered, even after decades, or even centuries, of
research.

   So to reiterate, if you've done the experiment, then you've done as
much science as can be expected of anyone. You can look for some
source of insight into this question (and if you're really curious,
you will). But even without that answer your child can write up a
perfectly legitimate summary of the methods & results of her research,
and could be deserving of a high grade for her work. Be proud.

Here is the key point--there are 2 different things that move down the
inclined plane:

1) the wheels

2) the body of the car.

Both start from a standstill, and gravity pulls on them, gradually
speeding them up down the plane.  BUT there is a difference between
the two.  The body just has to move down the plane, while the wheels
have to move down the plane AND start to spin.  So it's harder for the
wheels to speed up than it is for the body.  Assuming you have no
significant friction in your bearings, if you just take a wheel of one
of the cars and roll it down the ramp by itself, you will find that it
goes slower than either one of the cars.  It has a speed it 'wants' to
go (this speed is the same for a big wheel and a little wheel).  The
body 'wants' to go faster, and if you could somehow get rid of
friction, the cars would slide down at this faster speed.  So,  wheels
tend to slow down the bodies, and the bigger wheels, being a bigger
fraction of the total mass of the vehicle, are able to 'get their way'
more, and so the car with the big wheels goes slower.

Bottom line: the fact that wheels have to spin as well as move down
the plane slows them down.

It's a concept of "Work", not exactly 3rd grader material but not too
bad.  Gravity is doing all the work on your box rolling down an
incline. for all purposes, we ignore everything else.  the work
gravity has done from top to bottom is the same whether you have small
wheels or big wheels, neglecting mass difference.  that equation is
mass * gravitational acceleration constant * vertical height, or mgh.
this will be the same for value for both cars. that term is equal to
the kinetic engergy at the end, which, without wheels, or with very
very small wheels, would be 1/2 * m * velocity^2.  but with wheels,
the mgh term is equal to 1/2 *m * velocity^2 plus a new term
accounting for the work done rotating the wheels themselves. that term
is 1/2 * I * angular velocity^2. the higher this 1/2Iw^2 term is , the
lower the 1/2mv^2 has to be, so the lower your velocity down the
track.  the I is "second moment of inertia", which is 1/2 * mass of
wheel * radius^2.   I have no idea what kind of math skills a 3rd
grader has so this is likely worthless info.

If you wanna explain it to a 3rd grader, I'd say... 

Take a full soup can. Spin it around in your hand. Notice it takes
effort to spin it.  A larger can will be harder to spin. That's like
your wheels. Gravity will give your carts the same amount of energy
regardless of the wheel size. But the more energy that's used to spin
those wheels, the less that's available to move the actual cart down
the hill.


Initial potential energy = Final kinetic energy

Greetings,

This problem has to do with a measure of the wheel called the "moment
of inertia."  Before I explain what this term represents, it is a good
idea to review basic kinematic concepts.  Typically, in basic physics
problems like this, we start off by dealing with what is called
"linear motion," that is, things which move in a straight line.  These
things can obviously have a position, speed, acceleration, and so on. 
One other thing they have is a mass.  In linear motion problems (this
problem of a car, specifically a wheel, is NOT a linear problem - more
on that in a moment) there is also a mass which acts at the "center of
mass" of the object in question.

Let me just clarify that for a moment.  If you think of a basketball,
it has some weight, maybe a quarter pound or half a pound.  This
basketball is also quite large, but in physics the size is not a
problem.  The problems treats the basketball as if all of its mass
were concentrated at the center of the ball.  This is called the
center of mass. http://en.wikipedia.org/wiki/Center_of_mass

In rotational motion (a wheel rotates, like I said before it is not
linear) there is something sort of like mass.  However, this quantity
has to do with how the mass of your object is arranged around its
center of mass.  This is the moment of inertia that I was talking
about a moment ago. http://en.wikipedia.org/wiki/Moment_of_inertia
http://hyperphysics.phy-astr.gsu.edu/HBASE/mi.html


In a basketball, all of the mass is on the outside, since inside there
is nothing but air (very low mass).  Thus, the moment of inertia will
be different than a big red dodgeball which weighs the same, but who's
mass is on the inside as well as the outside.

There is another important concept to introduce before the solution to
your probelm will be apparent: kinetic energy.  Specifically, there
are two types of kinetic energy we want to consider: that due to the
linear motion, and that due to the rotating motion of the wheels.  The
linear kinetic energy depends on two things: mass and speed of an
object.  Similarly, the rotational kinetic energy depends on two
things: moment of inertia and ROTATIONAL speed of an object (think RPM
- revolutions per minute).  More on kinetic energy and rotational
kinetic energy here:
http://theory.uwinnipeg.ca/mod_tech/node50.html

Now, there is one last concept called work to deal with.  Fortunately,
work is just a fancy name for "change in kinetic energy."  The
"Work-Energy Theorem" says precisely that - Work is equal to the
change in kinetic energy.  In this problem, the only work done on your
wheel is the work done by gravity.  The car starts at the top of the
hill at rest.  At the bottom it is moving.  The motion is due to
gravity, hence the work is done by gravity.  Now, if both of your cars
weight the same amount, then this work will be the same amount (see
this link for the equations:
http://hyperphysics.phy-astr.gsu.edu/HBASE/gpot.html ).

Since the motion at the top (when you start the car on the ramp) is
nothing, there is no kinetic energy.  Since the work is the same, the
cars must have the same kinetic energy at the bottom of the hill. 
Remember, kinetic energy is broken into two parts:

kinetic energy = rotational + linear

Now, remember that rotational kinetic energy depends on the moment of
inertia, and the moment of inertia in turn depends on two things:

moment of inertia = mass x distance (from the center of mass of the object)

So, what happens if you make the distance smaller (i.e. smaller wheel)?

Smaller wheel  -->  smaller moment of inertia  -->  smaller rotational
kinetic energy.  But if the rotational kinetic energy is smaller for
the small wheel than the big wheel, but the TOTAL kinetic energy is
the same, we have the following situation:

TOTAL energy (small wheel) = TOTAL energy (big wheel)
rotational (small wheel) + linear (small wheel) = rotational (big
wheel) + linear (big wheel)

But rotational (small wheel) is less than rotational (big wheel). 
This means that linear (small wheel) is BIGGER than linear (big
wheel).  If the LINEAR kinetic energy is bigger, we must go back to
the formula for linear kinetic energy:  kinetic energy depends on mass
and speed.  If the mass stays the same, and overall the amount must
increase, only one thing is left to do - increase the speed.

I hope this wasn't too technical an argument, I will summarize it below:

-----------------------------

(Legend: K = Kinetic energy, I = moment of Inertia, M = Mass, D =
Distance, S = Speed, R = Rotations per minute (RPM), ~ = "depends on")

K ~ rotational K, linear K
rotational K ~ I, R
linear K ~ M, S

Now,
I ~ M, D

For a small wheel, obviously D is smaller, thus

I(big) is larger than I(small).

This implies that

rotational K(big) is larger than rotational K(small).

Since final K(big) is exactly the same as final K(small), we know that

linear K(big) is SMALLER than linear K(small).

But now linear K(small) ~ M, S.

M does not change, so the only thing left to change is S, the speed of
the smaller wheel, and it must become larger than the speed of the
bigger wheel.

-------------------------------------

If you want to do the calculations, all you would need is a standard
calculator and the equations, which can be found on any introductory
physics website, such as:

Example problems
http://www.dac.neu.edu/physics/a.cromer/Physics1192/Dynamics.html

Common moments of inertia
http://kosmoi.com/Science/Physics/Mechanics/tpecp21.gif

Equations / lecture notes
http://www.colorado.edu/physics/phys2010/phys2010_sm99/NOTES/Lecture13.html

If you can provide all of the measurements necessary (radius of the
wheels, type of wheel i.e. solid or just a hoop, mass of wheel, mass
of car, angle of ramp, height of ramp) I would be happy to walk you
through the calculation, it is not too complicated once you have the
numbers.