Dumb wheel physics question

On Fri, 30 Mar 2007 15:45:35 -0800, "Phil Holman"

piholmanc@yourservice said:

<[email hidden]> wrote in message
news:[email hidden]...

Quoted message said:

A friend was looking at the current thread about standing-start
downhill coasting tests.

He asked me whether the mass of the tires would make a difference.

I cleverly replied, "Huh?"

He expanded his question to something like this:

Given the same CRR and wind drag, will a wheel with more mass on its
rim (heavier tire) coast downhill from a standing start faster,
slower, or the same as the less massive wheel?

Same.

M = 6 kg and 12 kg
All the weight is at the rim circumference.
Starting height is 10m
Calculate V at bottom of hill

PE = mgh = 588.6 J and 1177.2 J
KE = ½ mV^2 + ½ mV^2 = 588.6 and 1177.2 (rotational and translational)
V^2 = 588.6/6 and 1177.2/12
V = Sqrt (588.6/6) V = Sqrt (1177.2/12)

V = 9.904m/s V = 9.904m/s

Phil H

Dear Phil,

So far, my impression is that there are two answers, depending on how
the question is framed.

As long as _all_ the mass is at the rim (idealized pair of hoops of
same dimensions, one of steel, one of aluminum), the "wheels" roll
downhill at the same speed. That's the version that you're solving
with your explicit "All the weight is at the rim circumference"
statement, which nails the point down nicely. Thanks for that and for
the detailed example that shows how the numbers then reduce to the
same velocity.

But all the mass isn't out at the tire in real wheels because they
have hubs, spokes, and rims closer to the axle than the tires.

So the other answer seems to be that increasing the tire weight
changes the proportion of total mass out at the rim, making the wheel
resemble a hollow cylinder more than a solid disk, and the hollow
cylinder rolls a little more slowly down the hill.

It's worth pointing out that I had no idea that there were two
different models--I just assumed that my friend's identical dimension
hoop example was the same, which should teach me to make such
assumptions.

Eventually I noticed that everyone was giving two different answers,
heavier-is-slower if they were using real wheels with some mass closer
to the axle than the tire, heavier-and-lighter-roll-the-same if they
were using idealized "hoops" that avoid any mass other than tires.

So far, this distinction between the two scenarios seems to be holding
up.

Cheers,

Carl Fogel

Carl Fogel said:

Eventually I noticed that everyone was giving two different answers,
heavier-is-slower if they were using real wheels with some mass
closer to the axle than the tire, heavier-and-lighter-roll-the-same
if they were using idealized "hoops" that avoid any mass other than
tires.

Quoted message said:

So far, this distinction between the two scenarios seems to be
holding up.

Not exactly. If the wheel had a massless rim with all its mass in a
small diameter hub, it would accelerate faster, having no rotational
inertia and essentially be frictionlessly traveling down the ramp. If
all its mass were at the periphery it would have significant
rotational inertia making it accelerate more slowly on its descent.

To visualize the effect, one would need to catch the wheel by its axle
at the bottom of the run and notice ho much energy it stored. The one
with all its mass in a small diameter hub would be effortlessly
brought to a stop by rubbing ones thumb while holding the axle. The
Rim mass model would take far longer to bring to a stop. I'm sure
most folks have done tis with a real wheel. Just a spinning hub is
stopped instantly.

Jobst Brandt

Ron Ruff said:

Quoted message said:

You're saying (I think) that they'd accelerate from a standing start
at the same speed, just like a BB and a cannonball dropped in a
vacuum, right?

Yes, *any* object that can be idealized to have all of it's mass at
the outer edge, will roll downhill at the same speed (same
acceleration). The "hoops" can have any diameter or mass. In this case
the inertial mass is always double the static mass... ie M/Mi =.5

Quoted message said:

(If it does, I get to tell my friend that his stinking improved
example with the steel and aluminum hoops was entirely different and
not like real bicycle wheels at all, nyah-nyah-nyah!)

Looks that way to me... On a bike, if you increase the hoop weight
then both the inertial and static masses go up by the same amount, but
the ratio for the whole system, M/Mi goes down... so the rate of
acceleration also drops.

2 words:

rolling resistance

A vacuum does not change the fact that there is still rolling
resistance (aka frictional losses) of the wheels rolling down the
inclined plane.

One could *assume* that these are equal ... but an object with a
larger mass would encounter a larger (in magnitude) normal force,
hence a larger frictional force (unless mu(sub)s is exceeded, in which
the wheel is no longer purely rolling and is now sliding - mu(sub)k.

at some non-arbitrary angle, this will occur (tan theta = mu(sub)k
IIRC).

hth.

-bdbafh

Ben C said:

Quoted message said:

On Thu, 29 Mar 2007 08:23:27 GMT, Ron Hardin <[email hidden]>
wrote:

Quoted message said:

Quoted message said:

Some of the kinetic energy is in the spinning wheel, at the bottom
of the hill, so the bike itself has to be going forward slower by
that amount of energy. More mass in the rims, more energy in the
spinning wheel, and less in forward movement.

Quoted message said:

Dear Ron,

Quoted message said:

Another email from my friend just boiled the question down by
eliminating the bike and asking me to place my bet:

Quoted message said:

Two hoops of identical dimensions, one made of aluminum, the other of
steel, roll down an incline in a vacuum from a standing start.

Quoted message said:

If they were balls falling in a vacuum, it would be a tie because the
mass accelerates at the same rate, no matter how much there is of it.

Quoted message said:

Your explanation for the denser-heavier tire being slower sounds good
to me at first . . .

Quoted message said:

Until I wonder if the extra energy is supplied by the extra
gravitational pull on the extra mass, so it all cancels out, and hoops
of the same dimensions (and CRR) will accelerate uniformly down a
slope in a vacuum.

I think they will.

Two wheels with the same mass, diameter and centre of mass but different
distribution of mass is instructive. At the bottom energy is the same, so
rotational speed can't be. The wheel with the mass in the centre must have a
higher rotational speed to have the same total kinetic energy. This is what I
think Ron Hardin said.

Those two wheels have the same mass, but I don't think it matters what
that actual mass is. If I use two wheels just the same (same mass,
diameter and centre of mass, but one with the mass more in the centre)
but made of lead instead of steel, I expect the same linear and
rotational speeds at the bottom (we're in a vacuum, no tyre losses etc.
etc.).

So when it comes to the aluminium and steel rims of identical
dimensions, since they have the same mass distributions, I think they
will have the same rotational and linear speeds as each other at the
bottom of the slope.

It doesn't matter what either wheel is made of. It's the mass
distribution that makes the difference.

Another way of looking at it is to say that the force applied to the
centre of mass of each wheel scales with its mass since it's a
gravitational force (gravity applies a bigger force to more massive
things). But it's the reaction to that force at the ground that applies
a torque that turns the wheel. The force (and therefore torque) doesn't
scale with the moment of inertia of the wheel, but with its mass. So if
you increase the moment of inertia without changing the mass, the same
force will result in a slower rate of angular acceleration.

Quoted message said:

But Ron Ruff says (I think) that a heavier wheel is a little slower
down the slope in some model that he worked up, so that's two votes
from two Rons for heavier being slower down the slope.

Ah but Ron's heaver wheel might have also have had more of its mass
around the rim, so different mass distribution (and therefore moment of
inertia-- resistance to being spun).

Quoted message said:

So then I wonder if that means that a heavy enough wheel would refuse
to move at all down the slope, while a light enough wheel would reach
the speed of light. (I'm pretty sure that this logic must have a
gaping hole in it somewhere, but I'm floundering.)

When you approach the speed of light relativistic effects start to
become significant, and the mass increases, or something. This is why
it's so hard to develop a warp drive.

If the light wheel had the mass of a photon (very low, almost nothing) I
expect it might approach the speed of light. But in that case it
couldn't be made of aluminium or steel since a single atom of either
requires more material than a photon.

The heavy wheel will always move in the idealized situation. A flea can
move a space station, you will just get a very slow acceleration.
There's no static friction threshold to be overcome, and neither is
there in our wheel rolling experiment (although there does have to be
friction or the wheels wouldn't roll).

Ben,

The term that you are searching for is "moment of inertia".

Quoted message said:

It doesn't matter what either wheel is made of. It's the mass
distribution that makes the difference.

agreed, provided that both wheels do not slip.

-bdbafh

Carl Fogel said:

Eventually I noticed that everyone was giving two different answers,
heavier-is-slower if they were using real wheels with some mass
closer to the axle than the tire, heavier-and-lighter-roll-the-same
if they were using idealized "hoops" that avoid any mass other than
tires.

Quoted message said:

So far, this distinction between the two scenarios seems to be
holding up.

Not exactly. If the wheel had a massless rim with all its mass in a
small diameter hub, it would accelerate faster, having no rotational
inertia as it traveled frictionlessly down the ramp. If all its mass
were at the periphery it would have significant rotational inertia
making it accelerate more slowly on its descent.

To visualize the effect, one would need to catch the wheels by the
axle at the bottom of the run and notice how much energy was stored.
The one with all its mass in the hub would be quickly and effortlessly
brought to a stop by rubbing ones thumb on the hub while holding the
axle while the rim mass model would take far longer to bring to a
stop. I'm sure most folks have done this with a real wheel. In
contrast a spinning hub is stopped almost instantly.

Jobst Brandt

Ben C said:

Quoted message said:

On Thu, 29 Mar 2007 08:23:27 GMT, Ron Hardin <[email hidden]>
wrote:

Quoted message said:

Quoted message said:

Some of the kinetic energy is in the spinning wheel, at the bottom
of the hill, so the bike itself has to be going forward slower by
that amount of energy. More mass in the rims, more energy in the
spinning wheel, and less in forward movement.

Quoted message said:

Dear Ron,

Quoted message said:

Another email from my friend just boiled the question down by
eliminating the bike and asking me to place my bet:

Quoted message said:

Two hoops of identical dimensions, one made of aluminum, the other of
steel, roll down an incline in a vacuum from a standing start.

Quoted message said:

If they were balls falling in a vacuum, it would be a tie because the
mass accelerates at the same rate, no matter how much there is of it.

Quoted message said:

Your explanation for the denser-heavier tire being slower sounds good
to me at first . . .

Quoted message said:

Until I wonder if the extra energy is supplied by the extra
gravitational pull on the extra mass, so it all cancels out, and hoops
of the same dimensions (and CRR) will accelerate uniformly down a
slope in a vacuum.

I think they will.

Two wheels with the same mass, diameter and centre of mass but different
distribution of mass is instructive. At the bottom energy is the same, so
rotational speed can't be. The wheel with the mass in the centre must have a
higher rotational speed to have the same total kinetic energy. This is what I
think Ron Hardin said.

Those two wheels have the same mass, but I don't think it matters what
that actual mass is. If I use two wheels just the same (same mass,
diameter and centre of mass, but one with the mass more in the centre)
but made of lead instead of steel, I expect the same linear and
rotational speeds at the bottom (we're in a vacuum, no tyre losses etc.
etc.).

So when it comes to the aluminium and steel rims of identical
dimensions, since they have the same mass distributions, I think they
will have the same rotational and linear speeds as each other at the
bottom of the slope.

It doesn't matter what either wheel is made of. It's the mass
distribution that makes the difference.

Another way of looking at it is to say that the force applied to the
centre of mass of each wheel scales with its mass since it's a
gravitational force (gravity applies a bigger force to more massive
things). But it's the reaction to that force at the ground that applies
a torque that turns the wheel. The force (and therefore torque) doesn't
scale with the moment of inertia of the wheel, but with its mass. So if
you increase the moment of inertia without changing the mass, the same
force will result in a slower rate of angular acceleration.

Quoted message said:

But Ron Ruff says (I think) that a heavier wheel is a little slower
down the slope in some model that he worked up, so that's two votes
from two Rons for heavier being slower down the slope.

Ah but Ron's heaver wheel might have also have had more of its mass
around the rim, so different mass distribution (and therefore moment of
inertia-- resistance to being spun).

Quoted message said:

So then I wonder if that means that a heavy enough wheel would refuse
to move at all down the slope, while a light enough wheel would reach
the speed of light. (I'm pretty sure that this logic must have a
gaping hole in it somewhere, but I'm floundering.)

When you approach the speed of light relativistic effects start to
become significant, and the mass increases, or something. This is why
it's so hard to develop a warp drive.

If the light wheel had the mass of a photon (very low, almost nothing) I
expect it might approach the speed of light. But in that case it
couldn't be made of aluminium or steel since a single atom of either
requires more material than a photon.

The heavy wheel will always move in the idealized situation. A flea can
move a space station, you will just get a very slow acceleration.
There's no static friction threshold to be overcome, and neither is
there in our wheel rolling experiment (although there does have to be
friction or the wheels wouldn't roll).

<snip>
It doesn't matter what either wheel is made of. It's the mass
distribution that makes the difference.
</snip>

unless one is red, as it will go faster.

-bdbafh

more brit aerospace?
home of the massless rim.
rimless mass?

bdbafh said:

Ron Ruff said:

Quoted message said:

You're saying (I think) that they'd accelerate from a standing start
at the same speed, just like a BB and a cannonball dropped in a
vacuum, right?

Yes, *any* object that can be idealized to have all of it's mass at
the outer edge, will roll downhill at the same speed (same
acceleration). The "hoops" can have any diameter or mass. In this case
the inertial mass is always double the static mass... ie M/Mi =.5

Quoted message said:

(If it does, I get to tell my friend that his stinking improved
example with the steel and aluminum hoops was entirely different and
not like real bicycle wheels at all, nyah-nyah-nyah!)

Looks that way to me... On a bike, if you increase the hoop weight
then both the inertial and static masses go up by the same amount, but
the ratio for the whole system, M/Mi goes down... so the rate of
acceleration also drops.

2 words:

rolling resistance

A vacuum does not change the fact that there is still rolling
resistance (aka frictional losses) of the wheels rolling down the
inclined plane.

One could *assume* that these are equal ... but an object with a
larger mass would encounter a larger (in magnitude) normal force,
hence a larger frictional force (unless mu(sub)s is exceeded, in which
the wheel is no longer purely rolling and is now sliding - mu(sub)k.

The frictional force in this experiment is not dissipative. If we assume
the tyre doesn't skid then it is a static friction contact at all times.

You will lose some energy by flexing the tyre, and that would probably
be a bit more for the heavier wheel assuming they both had the same
tyres at the same pressures. But I think it's OK for the sake of
argument to ignore rolling resistance losses.

Quoted message said:

Eventually I noticed that everyone was giving two different answers,
heavier-is-slower if they were using real wheels with some mass closer
to the axle than the tire, heavier-and-lighter-roll-the-same if they
were using idealized "hoops" that avoid any mass other than tires.

That isn't my answer. As long as the two wheels have the same
*distribution* of mass, ie same ratio of inertial to static mass, they
will roll down the hill at the same speed. This is for wheels by
themselves. If you connect the wheel to a bike with rider though, the
wheel with greater rotational inertia will be slower.

Ron Ruff said:

Quoted message said:

Eventually I noticed that everyone was giving two different answers,
heavier-is-slower if they were using real wheels with some mass closer
to the axle than the tire, heavier-and-lighter-roll-the-same if they
were using idealized "hoops" that avoid any mass other than tires.

That isn't my answer. As long as the two wheels have the same
*distribution* of mass, ie same ratio of inertial to static mass, they
will roll down the hill at the same speed. This is for wheels by
themselves.

I agree with that part.

Quoted message said:

If you connect the wheel to a bike with rider though, the wheel with
greater rotational inertia will be slower.

Why?

Just to clarify: identical bikes, identical riders, wheels exactly the
same shape and therefore mass distribution, but made of aluminium on one
bike and lead on the other. Tyres assumed to lose no energy or the same
amount, roll down hill in vacuum, I think they'll be going the same
speed at the bottom.

Ben C? said:

Quoted message said:

Quoted message said:

Eventually I noticed that everyone was giving two different
answers, heavier-is-slower if they were using real wheels with
some mass closer to the axle than the tire,
heavier-and-lighter-roll-the-same if they were using idealized
"hoops" that avoid any mass other than tires.

Quoted message said:

Quoted message said:

That isn't my answer. As long as the two wheels have the same
*distribution* of mass, ie same ratio of inertial to static mass,
they will roll down the hill at the same speed. This is for wheels
by themselves.

Quoted message said:

I agree with that part.

Quoted message said:

Quoted message said:

If you connect the wheel to a bike with rider though, the wheel
with greater rotational inertia will be slower.

Quoted message said:

Why?

Quoted message said:

Just to clarify: identical bikes, identical riders, wheels exactly
the same shape and therefore mass distribution, but made of
aluminium on one bike and lead on the other. Tyres assumed to lose
no energy or the same amount, roll down hill in vacuum, I think
they'll be going the same speed at the bottom.

The one with the heavier (lead) wheels will have converted more energy
to rotating momentum in its wheels than lighter one so it will have a
slower forward speed. A magnetic levitated sled would be faster than
either rolling vehicle having no energy used in generating rotational
momentum (neglecting any rolling resistance for the wheeled vehicles).

Jobst Brandt

Quoted message said:

Ben C? said:

Quoted message said:

> Eventually I noticed that everyone was giving two different
> answers, heavier-is-slower if they were using real wheels with
> some mass closer to the axle than the tire,
> heavier-and-lighter-roll-the-same if they were using idealized
> "hoops" that avoid any mass other than tires.

Quoted message said:

Quoted message said:

That isn't my answer. As long as the two wheels have the same
*distribution* of mass, ie same ratio of inertial to static mass,
they will roll down the hill at the same speed. This is for wheels
by themselves.

Quoted message said:

I agree with that part.

Quoted message said:

Quoted message said:

If you connect the wheel to a bike with rider though, the wheel
with greater rotational inertia will be slower.

Quoted message said:

Why?

Quoted message said:

Just to clarify: identical bikes, identical riders, wheels exactly
the same shape and therefore mass distribution, but made of
aluminium on one bike and lead on the other. Tyres assumed to lose
no energy or the same amount, roll down hill in vacuum, I think
they'll be going the same speed at the bottom.

The one with the heavier (lead) wheels will have converted more energy
to rotating momentum in its wheels than lighter one

Yes, but it also has more (gravitational potential) energy to start
with-- it's heavier.

Quoted message said:

so it will have a slower forward speed.

It occurs to me now that the bike + rider assembly with heavier wheels
has a higher percentage of its total weight in the wheels. This does
mean that its total energy at the bottom will be distributed with more
in rotational energy and therefore less in forward energy, so I think I
can see why Ron Ruff is right. This may have been your point as well.

Quoted message said:

A magnetic levitated sled would be faster than either rolling vehicle
having no energy used in generating rotational momentum (neglecting
any rolling resistance for the wheeled vehicles).

Yes, although if you get to cruise up another hill the other side, it
should all work out, with those heavy flywheels giving you back the
energy they've stored.

<[email hidden]> wrote in message
news:[email hidden]...

Quoted message said:

Ben C? said:

Quoted message said:

> Eventually I noticed that everyone was giving two different
> answers, heavier-is-slower if they were using real wheels with
> some mass closer to the axle than the tire,
> heavier-and-lighter-roll-the-same if they were using idealized
> "hoops" that avoid any mass other than tires.

Quoted message said:

Quoted message said:

That isn't my answer. As long as the two wheels have the same
*distribution* of mass, ie same ratio of inertial to static mass,
they will roll down the hill at the same speed. This is for wheels
by themselves.

Quoted message said:

I agree with that part.

Quoted message said:

Quoted message said:

If you connect the wheel to a bike with rider though, the wheel
with greater rotational inertia will be slower.

Quoted message said:

Why?

Quoted message said:

Just to clarify: identical bikes, identical riders, wheels exactly
the same shape and therefore mass distribution, but made of
aluminium on one bike and lead on the other. Tyres assumed to lose
no energy or the same amount, roll down hill in vacuum, I think
they'll be going the same speed at the bottom.

The one with the heavier (lead) wheels will have converted more energy
to rotating momentum in its wheels than lighter one so it will have a
slower forward speed. A magnetic levitated sled would be faster than
either rolling vehicle having no energy used in generating rotational
momentum (neglecting any rolling resistance for the wheeled vehicles).

Jobst Brandt

A heavier wheel with more rotational inertia also has more potential
energy to start with and these things balance so the wheels will roll at
the same speed. The slower wheel will be the one whose radius of
gyration is larger.

Phil H

Alternative explanation, not using moments of inertia.

The potential energy of the bike at the top of the hill is
converted to kinetic energy at the bottom.

The kinetic energy is the sum of every bit of mass times its
speed squared (over two, but forget that).

The tops of the wheels are going at twice the speed of the
bicycle. So those bits of mass have four times the energy
of bits of mass at (say) the hub, which is going at the speed
of the rest of the bicycle.

The mass at the top can be averaged with the mass diametrically
opposite, giving an average kinetic energy 2 times that of a bit
of mass on the bicycle itself, for the vertical diameter, and less
for lesser diameters, but always more than for mass on the bicycle
frame.

Since the mass on the wheel soaks up extra kinetic energy, the kinetic
energy on the frame must be less than it would be if the wheels
weren't rotating (but rather sliding frictionlessly, or rolling
masslessly). Hence the bike goes slower with mass on rotating wheel
rims, for there's only a fixed quantity of potential energy
to be converted into kinetic energy.

It goes fastest when every bit of mass on the bike goes at the
same speed.

This is attained when it slides frictionlessly without rotating
wheels, or if there's no mass on the wheels away from the hub,
to name two cases.

A statistical formulation : the bike goes fastest when the standard
deviation of the speeds of bits of mass on it is zero. The rms
speed of bits of mass equals the speed of the center of mass.
--
Ron Hardin
[email hidden]

On the internet, nobody knows you're a jerk.

Ron Hardin said:

The mass at the top can be averaged with the mass diametrically
opposite, giving an average kinetic energy 2 times that of a bit
of mass on the bicycle itself, for the vertical diameter, and less
for lesser diameters, but always more than for mass on the bicycle
frame.

Actually it may be constant, at twice the energy of the bits of
mass on the frame. Somebody can work it out.

For a horizontal diameter, the bits of mass are going forward
at the speed of the bicycle, and upwards at the speed of the bicycle,
so have speed squared twice that of the bicycle speed squared, so
again it seems you get 2, as you did for the vertical diameter.

Probably sin^2 and cos^2 work out at every other angle too. I don't
feel like hunting up a pencil to diagram it.

--
Ron Hardin
[email hidden]

On the internet, nobody knows you're a jerk.

Ben C? said:

Quoted message said:

Quoted message said:

>> Eventually I noticed that everyone was giving two different
>> answers, heavier-is-slower if they were using real wheels with
>> some mass closer to the axle than the tire,
>> heavier-and-lighter-roll-the-same if they were using idealized
>> "hoops" that avoid any mass other than tires.

Quoted message said:

Quoted message said:

Quoted message said:

> That isn't my answer. As long as the two wheels have the same
> *distribution* of mass, ie same ratio of inertial to static mass,
> they will roll down the hill at the same speed. This is for
> wheels by themselves.

Quoted message said:

Quoted message said:

Quoted message said:

I agree with that part.

Quoted message said:

Quoted message said:

Quoted message said:

> If you connect the wheel to a bike with rider though, the wheel
> with greater rotational inertia will be slower.

Quoted message said:

Quoted message said:

Quoted message said:

Why?

Quoted message said:

Quoted message said:

Quoted message said:

Just to clarify: identical bikes, identical riders, wheels exactly
the same shape and therefore mass distribution, but made of
aluminium on one bike and lead on the other. Tyres assumed to lose
no energy or the same amount, roll down hill in vacuum, I think
they'll be going the same speed at the bottom.

Quoted message said:

Quoted message said:

The one with the heavier (lead) wheels will have converted more
energy to rotating momentum in its wheels than lighter one

Quoted message said:

Yes, but it also has more (gravitational potential) energy to start
with-- it's heavier.

I thin k the example with the wheels with identical weight but with
mass concentrated at the rim and mass concentrated at the hub should
make that clear. A heavier bicycle wheel has most of its added weight
at the periphery. Kinetic energy going as the square of velocity
makes it a non linear comparison. The added weight on the periphery
does not affect the forward acceleration proportional to its increase
because the rotational energy goes as the square of linear velocity.

Quoted message said:

Quoted message said:

so it will have a slower forward speed.

Quoted message said:

It occurs to me now that the bike + rider assembly with heavier
wheels has a higher percentage of its total weight in the
wheels. This does mean that its total energy at the bottom will be
distributed with more in rotational energy and therefore less in
forward energy, so I think I can see why Ron Ruff is right. This may
have been your point as well.

Quoted message said:

Quoted message said:

A magnetic levitated sled would be faster than either rolling
vehicle having no energy used in generating rotational momentum
(neglecting any rolling resistance for the wheeled vehicles).

Quoted message said:

Yes, although if you get to cruise up another hill the other side,
it should all work out, with those heavy flywheels giving you back
the energy they've stored.

Yes, that is in keeping with conservation of energy. Just the same,
let's not lose perspective on the relatively small effect this has in
the realm of tire weight differences.

Jobst Brandt

On 31 Mar 2007 10:10:32 -0700, "Ron Ruff" <[email hidden]>

Quoted message said:

Quoted message said:

Eventually I noticed that everyone was giving two different answers,
heavier-is-slower if they were using real wheels with some mass closer
to the axle than the tire, heavier-and-lighter-roll-the-same if they
were using idealized "hoops" that avoid any mass other than tires.

That isn't my answer. As long as the two wheels have the same
*distribution* of mass, ie same ratio of inertial to static mass, they
will roll down the hill at the same speed. This is for wheels by
themselves. If you connect the wheel to a bike with rider though, the
wheel with greater rotational inertia will be slower.

Dear Ron,

Maybe I've confused things, so I'll try to work through them again.

One case is a plain hoop with no hub, spokes, or rim to complicate
things (not a real wheel). Most posters seem to agree that plain hoops
of identical dimensions and same mass distribution will roll down the
hill in a vacuum at the same speed, whether they're light or heavy.

Phil Holman points out that the special case of solid hoops of the
same dimensions (disks) still produces the same result. The
identically shaped disks will roll downhill in a vacuum at the same
speed, whether they're made of steel, aluminum, or lead.

But a real wheel with hub, spokes, and rim (but no bicycle attached)
behaves differently with lighter or heavier tires. A heavier tire
moves the moment of inertia out toward the rim, making the whole wheel
closer to a slower-rolling hollow cylinder than to a faster-rolling
solid disk.

1) Same-dimension hoops roll the same, regardless of mass, same moment
of inertia.

2) Adding a heavier tire slows a real wheel down, moment of inertia
moves out toward rim.

But you seem to be saying that attaching the real rotating wheels to
the non-rotating bicycle mass somehow changes things.

Does connecting the wheel to a bicycle and rider change things? Or am
I just misreading your reply?

Adding a heavier tire slows down a real wheel, so it seems as if it
should also slow down anything attached.

If a bicycle's normal wheels and tires are replaced with solid disks
of identical dimensions made of lead and steel, it seems as if these
solid-hoop disks (and attached bicycle) should roll downhill at the
same speed.

Sorry if this is all just another misunderstanding on my part.

Cheers,

Carl Fogel

Ron Hardin said:

Ron Hardin said:

The mass at the top can be averaged with the mass diametrically
opposite, giving an average kinetic energy 2 times that of a bit
of mass on the bicycle itself, for the vertical diameter, and less
for lesser diameters, but always more than for mass on the bicycle
frame.

Actually it may be constant, at twice the energy of the bits of
mass on the frame.

I think that's right. If the bike is travelling at 20mph, the bottom of
the wheel is stationary relative to the road, and therefore moving
relative to the bike at 20mph backwards, while the top of the wheel is
moving relative to the bike at 20mph forwards (and therefore at 40mph
relative to the road). At all times all parts of the rim are moving
relative to the bike at 20mph magnitude while the bike moves relative to
the road at 20mph.

Suppose the final speed is 20mph.

The linear kinetic energy of the whole system is 1/2 mv^2 where m
includes the mass of the wheels. Never mind that the wheels are
spinning, the centre of mass of the whole system is moving at 20mph. So
far so good.

Then we can work out the rotational kinetic energy of the wheels in the
reference frame of the bike, and add that to the total so far. If the
mass of each wheel is _all_ at the rim, then every particle of rotating
mass has a speed of 20mph magnitude relative to the bike. So effectively
this rotating mass gets counted twice-- it's doing 20mph relative to the
road, and another 20mph relative to the bike.

If the rotating mass isn't all at the rim, but distributed throughout
the wheel as it in reality the calculations are more complex.

Quoted message said:

For a horizontal diameter, the bits of mass are going forward
at the speed of the bicycle, and upwards at the speed of the bicycle,
so have speed squared twice that of the bicycle speed squared, so
again it seems you get 2, as you did for the vertical diameter.

Yes, and this is the approach of working out the rotational kinetic
energy in the reference frame of the road rather than in that of the
bike.

It's justified to use the bike's reference frame for the rotational
energy and the road's frame for the linear energy because at the start
of the experiment, the wheels were stationary relative to the bike and
the bike was stationary relative to the road. Since that implies that
the wheels were also stationary relative to the road, choosing the
road's frame for both energy components is also correct, but may be more
confusing.

Quoted message said:

But you seem to be saying that attaching the real rotating wheels to
the non-rotating bicycle mass somehow changes things.

Does connecting the wheel to a bicycle and rider change things? Or am
I just misreading your reply?

You *are* reading me correctly.

With no confusing air resistance or rolling resistance, the equation
of motion for coasting down the hill can be represented by:

a=-g*G*M/Mi

a= acceleration
g= gravity
G= grade
M= static mass
Mi= inertial mass, ie an equivalent mass for acceleration

For any rolling object that has all if it's mass at the periphery,
Mi=2*M. Lets assume that for the sake of simplicity. *Any* rolling
object that has this characteristic will roll down the hill at the
same rate of acceleration (and therefore speed). Size and weight are
completely irrelevant. Such an object will roll at 1/2 the
acceleration rate of any object which has all of its mass concentrated
at the center (Mi=M), which is also true of all non-rotating mass. But
if you attach the one of these to a bike, then it is no longer able to
roll freely, but must go along with the bike... which *wants* to go
faster than the wheel would on it's own. That's is why the rotating
wheel slows things down, and the more inertial mass the wheel has the
more it slows things down. Basically the wheel wants to go downhill at
half the speed that the rest of the bike does.