P
Phil Hobbs
Guest
John Larkin <jlarkin@highlandSNIPMEtechnology.com> wrote:
Cheers
Phil Hobbs
--
Phil Hobbs
Cheers
Phil Hobbs
--
Phil Hobbs
J
John Larkin
Guest
On Tue, 9 May 2023 19:50:42 -0000 (UTC), Phil Hobbs
<pcdhSpamMeSenseless@electrooptical.net> wrote:
Start with three balls and a spring.
<pcdhSpamMeSenseless@electrooptical.net> wrote:
Start with three balls and a spring.
R
RichD
Guest
On May 9, Martin Brown wrote:
How does it create a torque, when it\'s RADIAL, i.e. perpendicular to the
tangential velocity?
> Kinetic energy on its own is *NOT* conserved. *TOTAL* energy *IS*.
um, yes. But one has to identify ALL the pertinent forms of energy,
and quantify them.
That\'s no analog: there\'s no external energy supplied to the system.
The spin rate slows through conservation of angular momentum.
No energy gain/loss through a fallacious \'potential energy\'.
You\'re unclear on the concept. In mechanics, potential energy is
always associated with a field. As an object moves point to point, it
gains/loses energy from the field.
No field, in this case. Her arms, outstretched, don\'t contain potential
energy, only the kinetic energy of their motion. And that isn\'t a net
energy resource. The \'field\' is the ATP reactions -
--
Rich
How does it create a torque, when it\'s RADIAL, i.e. perpendicular to the
tangential velocity?
> Kinetic energy on its own is *NOT* conserved. *TOTAL* energy *IS*.
um, yes. But one has to identify ALL the pertinent forms of energy,
and quantify them.
That\'s no analog: there\'s no external energy supplied to the system.
The spin rate slows through conservation of angular momentum.
No energy gain/loss through a fallacious \'potential energy\'.
You\'re unclear on the concept. In mechanics, potential energy is
always associated with a field. As an object moves point to point, it
gains/loses energy from the field.
No field, in this case. Her arms, outstretched, don\'t contain potential
energy, only the kinetic energy of their motion. And that isn\'t a net
energy resource. The \'field\' is the ATP reactions -
--
Rich
R
RichD
Guest
On May 9, Jasen Betts wrote:
All right, I see it. The arms don\'t move inertially, like a projectile.
She has to pull continuously, to overcome the centrifugal force. So
they needn\'t stop suddenly, losing kinetic energy (into heat), they
can stop gradually.
I still don\'t see where the tangential torque comes from - it\'s
buried somewhere in the math.
--
Rich
All right, I see it. The arms don\'t move inertially, like a projectile.
She has to pull continuously, to overcome the centrifugal force. So
they needn\'t stop suddenly, losing kinetic energy (into heat), they
can stop gradually.
I still don\'t see where the tangential torque comes from - it\'s
buried somewhere in the math.
--
Rich
C
Clive Arthur
Guest
On 09/05/2023 21:55, RichD wrote:
> On May 9, Martin Brown wrote:
<snip>
To simplify, let\'s say she has very thin arms and is holding weights.
Let\'s also say that she takes half a rotation to pull the weights fully
in. Looking down from a stationary position above, as she spins and
draws the weights inwards, they describe a curve. A mass going round a
curve makes a force, in this case, some of that is tangential.
--
Cheers
Clive
> On May 9, Martin Brown wrote:
<snip>
To simplify, let\'s say she has very thin arms and is holding weights.
Let\'s also say that she takes half a rotation to pull the weights fully
in. Looking down from a stationary position above, as she spins and
draws the weights inwards, they describe a curve. A mass going round a
curve makes a force, in this case, some of that is tangential.
--
Cheers
Clive
J
John Larkin
Guest
On Tue, 9 May 2023 13:55:10 -0700 (PDT), RichD
<r_delaney2001@yahoo.com> wrote:
A compressed spring has potential energy. What field is that?
Her outstretched arms have gravitational potential energy, spinning or
not. That helps her pull her arms in towards her body, so is converted
into rotational energy.
<r_delaney2001@yahoo.com> wrote:
A compressed spring has potential energy. What field is that?
Her outstretched arms have gravitational potential energy, spinning or
not. That helps her pull her arms in towards her body, so is converted
into rotational energy.
P
Phil Hobbs
Guest
On 2023-05-09 16:55, RichD wrote:
It doesn\'t need to create a torque. Torque is the time derivative of
angular momentum, just as force is the time derivative of linear momentum.
What is observed is a change in the skater\'s angular _velocity_, not her
angular momentum. She achieves this with zero applied torque, by
changing her moment of inertia. Standard stuff from freshman physics.
Conservation of energy is a red herring here.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
It doesn\'t need to create a torque. Torque is the time derivative of
angular momentum, just as force is the time derivative of linear momentum.
What is observed is a change in the skater\'s angular _velocity_, not her
angular momentum. She achieves this with zero applied torque, by
changing her moment of inertia. Standard stuff from freshman physics.
Conservation of energy is a red herring here.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
J
John Larkin
Guest
On Tue, 9 May 2023 18:39:59 -0400, Phil Hobbs
<pcdhSpamMeSenseless@electrooptical.net> wrote:
But it\'s a great simplifiction here. It short-cuts a lot of math.
<pcdhSpamMeSenseless@electrooptical.net> wrote:
But it\'s a great simplifiction here. It short-cuts a lot of math.
P
Phil Hobbs
Guest
On 2023-05-09 22:22, John Larkin wrote:
Conservation laws are super useful like that, I agree.
Conservation of energy is one of the most useful ones, but isn\'t easily
applicable to the skater problem.
There\'s no potential energy in the problem as stated, so it isn\'t at all
like a planetary orbit, or a falling mass, or a vibrating string, or
lossless billiard balls on a perfectly-elastic table, where we can see
potential and kinetic energy changing oppositely, so that their sum is
constant.
The skater is using muscle power for this maneuver, so to do an actual
energy analysis would involve examining her metabolism.
She\'s burning glucose, which came from food, using air, which comes from
outside the rotating system (i.e. her body, skates, and clothing). Her
temperature is not in equilibrium with her environment, to the tune of
about 8 MJ for 50-kg of mostly water, 40 K warmer than the ice. That
energy difference equivalent to the kinetic energy of the same mass
moving at nearly 1300 miles per hour.
And then there\'s the actual physiology and the trajectory of her arm
motions.
Messy to analyze, and needs a lot of accurate bookkeeping.
From a kinematic point of view, the problem is far simpler. We have a
rotating body, isolated from external torques, whose angular momentum is
therefore a constant of the motion.
The body\'s moment of inertia changes due to internal rearrangement of
mass, which requires an amount of mechanical work that is easily
calculated, even if we have no very good estimate of the amount of
chemical and thermal energy required to perform the work.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
Conservation laws are super useful like that, I agree.
Conservation of energy is one of the most useful ones, but isn\'t easily
applicable to the skater problem.
There\'s no potential energy in the problem as stated, so it isn\'t at all
like a planetary orbit, or a falling mass, or a vibrating string, or
lossless billiard balls on a perfectly-elastic table, where we can see
potential and kinetic energy changing oppositely, so that their sum is
constant.
The skater is using muscle power for this maneuver, so to do an actual
energy analysis would involve examining her metabolism.
She\'s burning glucose, which came from food, using air, which comes from
outside the rotating system (i.e. her body, skates, and clothing). Her
temperature is not in equilibrium with her environment, to the tune of
about 8 MJ for 50-kg of mostly water, 40 K warmer than the ice. That
energy difference equivalent to the kinetic energy of the same mass
moving at nearly 1300 miles per hour.
And then there\'s the actual physiology and the trajectory of her arm
motions.
Messy to analyze, and needs a lot of accurate bookkeeping.
From a kinematic point of view, the problem is far simpler. We have a
rotating body, isolated from external torques, whose angular momentum is
therefore a constant of the motion.
The body\'s moment of inertia changes due to internal rearrangement of
mass, which requires an amount of mechanical work that is easily
calculated, even if we have no very good estimate of the amount of
chemical and thermal energy required to perform the work.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
J
Jasen Betts
Guest
On 2023-05-09, RichD <r_delaney2001@yahoo.com> wrote:
By reducing the circumference it travels on how can it not orbit faster?
--
Jasen.
ðºð¦ Слава УкÑаÑнÑ
By reducing the circumference it travels on how can it not orbit faster?
--
Jasen.
ðºð¦ Слава УкÑаÑнÑ
M
Martin Brown
Guest
On 10/05/2023 04:20, Phil Hobbs wrote:
Yes there is Phil. Her arms outstretched have stored potential energy by
virtue of their mass and being in a rotating frame.
When she moves them inwards she feels a net torque as she does work
against the centrefugal/petal force due to her rotation and feels a
Coriolis force. It is the effect of that Coriolis force that spins her
up. You should get *exactly* the same result by appealing to *either* of
conservation of energy or conservation of angular momentum.
One way is *MUCH* easier than the other though!
But in some problems like this one it is hard to see the wood for the
trees and some people appear to be unable to see at all
Newtonian dynamics requires corrections by fictitious forces before you
can sensibly apply it in a rotating frame of reference. One of my
physics supervisors worked on rotating radars in WWII and had stories of
people knocking themselves out reaching for spanner too quickly and
forgetting that they were in a rotating frame of reference. Apparently
after a while of being rotated you largely desensitise to it.
(until you stop the rotation then the world spins in reverse)
A classic governer of two weights on a hinged light inextensible arms is
the simplest analogue of the dancer that can be more easily analysed to
understand the split between stored potential energy and kinetic energy
as the device is spun up.
https://en.wikipedia.org/wiki/Centrifugal_governor
--
Martin Brown
Yes there is Phil. Her arms outstretched have stored potential energy by
virtue of their mass and being in a rotating frame.
When she moves them inwards she feels a net torque as she does work
against the centrefugal/petal force due to her rotation and feels a
Coriolis force. It is the effect of that Coriolis force that spins her
up. You should get *exactly* the same result by appealing to *either* of
conservation of energy or conservation of angular momentum.
One way is *MUCH* easier than the other though!
But in some problems like this one it is hard to see the wood for the
trees and some people appear to be unable to see at all
Newtonian dynamics requires corrections by fictitious forces before you
can sensibly apply it in a rotating frame of reference. One of my
physics supervisors worked on rotating radars in WWII and had stories of
people knocking themselves out reaching for spanner too quickly and
forgetting that they were in a rotating frame of reference. Apparently
after a while of being rotated you largely desensitise to it.
(until you stop the rotation then the world spins in reverse)
A classic governer of two weights on a hinged light inextensible arms is
the simplest analogue of the dancer that can be more easily analysed to
understand the split between stored potential energy and kinetic energy
as the device is spun up.
https://en.wikipedia.org/wiki/Centrifugal_governor
--
Martin Brown
M
Martin Brown
Guest
On 09/05/2023 23:39, Phil Hobbs wrote:
There is no externally applied torque but to move her arms inwards in a
rotating frame of reference she does real work against the forces.
Conservation of *TOTAL* energy still holds good but it is *much* more
difficult to compute those correctly in these two configurations.
(and even harder to compute all the forces acting and their torques)
Conservation of angular momentum in the isolated system yields the
correct answer in just a few lines of algebra repeated here now at more
than 6 times by three different authors.
He really needs to throw that brick up in the air and stand under it
only then will it possibly get this concept through his thick skull.
Kinetic energy on its own is not a conserved quantity!
--
Martin Brown
There is no externally applied torque but to move her arms inwards in a
rotating frame of reference she does real work against the forces.
Conservation of *TOTAL* energy still holds good but it is *much* more
difficult to compute those correctly in these two configurations.
(and even harder to compute all the forces acting and their torques)
Conservation of angular momentum in the isolated system yields the
correct answer in just a few lines of algebra repeated here now at more
than 6 times by three different authors.
He really needs to throw that brick up in the air and stand under it
only then will it possibly get this concept through his thick skull.
Kinetic energy on its own is not a conserved quantity!
--
Martin Brown
P
Phil Hobbs
Guest
On 2023-05-10 04:34, Martin Brown wrote:
I invite you to formulate the problem that way, in the fully co-rotating
frame (in which the skater remains stationary throughout the motion).
The constant-Omega Newtonian case that one normally encounters in
ballistics and meteorology isn\'t that difficult.
Kinematically, you just add a term to all the time derivatives of
vectors: for instance, a velocity dX\'/dt in the rotating frame is
dX\'/dt = dX/dt + Omega cross X,
where X = (x, y, z)^T is in the lab frame, and we\'re performing the
usual vector cross product. This leads to the so-called \'fictitious
forces\': centrifugal, which applies to everything, and Coriolis, which
applies only to moving objects.
Angular acceleration of the frame brings in stuff that makes
centrifugal and Coriolis force look trivial--as one small example, you
need a definition of energy that leaves the KE of the arena and the
spectators constant, even though its undergoing rapid angular
acceleration in the fully co-rotating frame.
Anytime you\'re doing something complicated like that, you constantly
check your work by doing the lab-frame calculation in parallel to
double-check, so it\'s a bit of a cheat to say that you\'re analyzing it
in the weird reference frame.
Those old-timey playground carousels were a good lab course too. A
couple of kids running at top speed to spin it up, then pull themselves
on board. They had tangential rails along the rim, and radials from rim
to pivot. Pulling yourself along one of the radials towards the axis
involves a lot of centrifugal force, but the really surprising thing is
the strength of the Coriolis pulling your feet sideways as you move them.
The governor + engine system is easily analyzed as a second-order ODE
with a mild nonlinearity, rather like a driven pendulum. Unlike the
skater problem, the rotating balls of the governor are a tiny
perturbation on a huge rotating assembly with otherwise constant moment
of inertia. That lets you ignore a whole lot of stuff.
And of course this digression is a classic SED trope, i.e. pulling out
bigger and bigger magnifiers as the hairs get split finer and finer.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
I invite you to formulate the problem that way, in the fully co-rotating
frame (in which the skater remains stationary throughout the motion).
The constant-Omega Newtonian case that one normally encounters in
ballistics and meteorology isn\'t that difficult.
Kinematically, you just add a term to all the time derivatives of
vectors: for instance, a velocity dX\'/dt in the rotating frame is
dX\'/dt = dX/dt + Omega cross X,
where X = (x, y, z)^T is in the lab frame, and we\'re performing the
usual vector cross product. This leads to the so-called \'fictitious
forces\': centrifugal, which applies to everything, and Coriolis, which
applies only to moving objects.
Angular acceleration of the frame brings in stuff that makes
centrifugal and Coriolis force look trivial--as one small example, you
need a definition of energy that leaves the KE of the arena and the
spectators constant, even though its undergoing rapid angular
acceleration in the fully co-rotating frame.
Anytime you\'re doing something complicated like that, you constantly
check your work by doing the lab-frame calculation in parallel to
double-check, so it\'s a bit of a cheat to say that you\'re analyzing it
in the weird reference frame.
One way is *MUCH* easier than the other though!
But in some problems like this one it is hard to see the wood for the
trees and some people appear to be unable to see at all
Newtonian dynamics requires corrections by fictitious forces before you
can sensibly apply it in a rotating frame of reference. One of my
physics supervisors worked on rotating radars in WWII and had stories of
people knocking themselves out reaching for spanner too quickly and
forgetting that they were in a rotating frame of reference. Apparently
after a while of being rotated you largely desensitise to it.
(until you stop the rotation then the world spins in reverse)
Those old-timey playground carousels were a good lab course too. A
couple of kids running at top speed to spin it up, then pull themselves
on board. They had tangential rails along the rim, and radials from rim
to pivot. Pulling yourself along one of the radials towards the axis
involves a lot of centrifugal force, but the really surprising thing is
the strength of the Coriolis pulling your feet sideways as you move them.
The governor + engine system is easily analyzed as a second-order ODE
with a mild nonlinearity, rather like a driven pendulum. Unlike the
skater problem, the rotating balls of the governor are a tiny
perturbation on a huge rotating assembly with otherwise constant moment
of inertia. That lets you ignore a whole lot of stuff.
And of course this digression is a classic SED trope, i.e. pulling out
bigger and bigger magnifiers as the hairs get split finer and finer.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
P
Phil Hobbs
Guest
On 2023-05-10 04:35, Martin Brown wrote:
You read the whole thread? I\'m in awe.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
You read the whole thread? I\'m in awe.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
J
John Larkin
Guest
On Wed, 10 May 2023 08:28:19 -0400, Phil Hobbs
<pcdhSpamMeSenseless@electrooptical.net> wrote:
Looking at the skater case from the POE perspective is an enormous
simplification.
At any point in the pull-in process, an incremental
force-times-distance adds energy to the system which can only go into
more spin. At that same point in later extending her arms, the force
is the same but the distance is reversed, so that bit of energy is
exactly negated.
So pulling in her arms adds to the spin energy and later extending
them buys it all back. POE requires that.
<pcdhSpamMeSenseless@electrooptical.net> wrote:
On 2023-05-10 04:34, Martin Brown wrote:
On 10/05/2023 04:20, Phil Hobbs wrote:
On 2023-05-09 22:22, John Larkin wrote:
On Tue, 9 May 2023 18:39:59 -0400, Phil Hobbs
pcdhSpamMeSenseless@electrooptical.net> wrote:
On 2023-05-09 16:55, RichD wrote:
On May 9, Martin Brown wrote:
The skater expends chemical energy, in her muscles, to
pull her arms
Work done is torque x angular displacement. No torque= No
work, and no work means no change in energy. Rotational
kinetic energy does not change.
The skater did work to pull her arms and legs in towards the
spin axis. Where did that energy come from and where did it
go?
As the arms move closer to the axis, the centripetal force
decreases considerably, requiring less work on the skater. But
it makes no difference whatsoever because just bringing the
arms inward in no way shape or form applies a force tangential
to the radius connecting the axis, which is what you need to
torque up the rotational velocity. So why does the skater mass
speed up? The answer is Coriolis.
https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/12%3A_Non-inertial_Reference_Frames/12.08%3A_Coriolis_Force#:~:text=An%20interesting%20application%20of%20the,external%20forces%20which%20is%20correct.
That article is instructive, but his final claim is incorrect.
Retracting the arms injects kinetic energy, from the muscles. The
author\'s claim is that this eventually adds to the total
rotational energy, as the arms merge with the torso.
That is *exactly* what happens. The skater does real work by moving
her arms inwards in a rotating frame of reference and it creates a
torque that spins her up faster.
How does it create a torque, when it\'s RADIAL, i.e. perpendicular to
the tangential velocity?
It doesn\'t need to create a torque. Torque is the time derivative of
angular momentum, just as force is the time derivative of linear
momentum.
What is observed is a change in the skater\'s angular _velocity_, not
her
angular momentum. She achieves this with zero applied torque, by
changing her moment of inertia. Standard stuff from freshman physics.
Kinetic energy on its own is *NOT* conserved. *TOTAL* energy *IS*.
um, yes. But one has to identify ALL the pertinent forms of energy,
and quantify them.
Conservation of energy is a red herring here.
But it\'s a great simplifiction here. It short-cuts a lot of math.
Conservation laws are super useful like that, I agree.
Conservation of energy is one of the most useful ones, but isn\'t
easily applicable to the skater problem.
There\'s no potential energy in the problem as stated, so it isn\'t at
all like a planetary orbit, or a falling mass, or a vibrating string,
or lossless billiard balls on a perfectly-elastic table, where we can
see potential and kinetic energy changing oppositely, so that their
sum is constant.
Yes there is Phil. Her arms outstretched have stored potential energy by
virtue of their mass and being in a rotating frame.
When she moves them inwards she feels a net torque as she does work
against the centrefugal/petal force due to her rotation and feels a
Coriolis force. It is the effect of that Coriolis force that spins her
up. You should get *exactly* the same result by appealing to *either* of
conservation of energy or conservation of angular momentum.
I invite you to formulate the problem that way, in the fully co-rotating
frame (in which the skater remains stationary throughout the motion).
The constant-Omega Newtonian case that one normally encounters in
ballistics and meteorology isn\'t that difficult.
Kinematically, you just add a term to all the time derivatives of
vectors: for instance, a velocity dX\'/dt in the rotating frame is
dX\'/dt = dX/dt + Omega cross X,
where X = (x, y, z)^T is in the lab frame, and we\'re performing the
usual vector cross product. This leads to the so-called \'fictitious
forces\': centrifugal, which applies to everything, and Coriolis, which
applies only to moving objects.
Angular acceleration of the frame brings in stuff that makes
centrifugal and Coriolis force look trivial--as one small example, you
need a definition of energy that leaves the KE of the arena and the
spectators constant, even though its undergoing rapid angular
acceleration in the fully co-rotating frame.
Anytime you\'re doing something complicated like that, you constantly
check your work by doing the lab-frame calculation in parallel to
double-check, so it\'s a bit of a cheat to say that you\'re analyzing it
in the weird reference frame.
One way is *MUCH* easier than the other though!
But in some problems like this one it is hard to see the wood for the
trees and some people appear to be unable to see at all
Newtonian dynamics requires corrections by fictitious forces before you
can sensibly apply it in a rotating frame of reference. One of my
physics supervisors worked on rotating radars in WWII and had stories of
people knocking themselves out reaching for spanner too quickly and
forgetting that they were in a rotating frame of reference. Apparently
after a while of being rotated you largely desensitise to it.
(until you stop the rotation then the world spins in reverse)
Those old-timey playground carousels were a good lab course too. A
couple of kids running at top speed to spin it up, then pull themselves
on board. They had tangential rails along the rim, and radials from rim
to pivot. Pulling yourself along one of the radials towards the axis
involves a lot of centrifugal force, but the really surprising thing is
the strength of the Coriolis pulling your feet sideways as you move them.
A classic governer of two weights on a hinged light inextensible arms is
the simplest analogue of the dancer that can be more easily analysed to
understand the split between stored potential energy and kinetic energy
as the device is spun up.
https://en.wikipedia.org/wiki/Centrifugal_governor
The governor + engine system is easily analyzed as a second-order ODE
with a mild nonlinearity, rather like a driven pendulum. Unlike the
skater problem, the rotating balls of the governor are a tiny
perturbation on a huge rotating assembly with otherwise constant moment
of inertia. That lets you ignore a whole lot of stuff.
And of course this digression is a classic SED trope, i.e. pulling out
bigger and bigger magnifiers as the hairs get split finer and finer.
Cheers
Phil Hobbs
Looking at the skater case from the POE perspective is an enormous
simplification.
At any point in the pull-in process, an incremental
force-times-distance adds energy to the system which can only go into
more spin. At that same point in later extending her arms, the force
is the same but the distance is reversed, so that bit of energy is
exactly negated.
So pulling in her arms adds to the spin energy and later extending
them buys it all back. POE requires that.
P
Phil Hobbs
Guest
On 2023-05-10 09:27, John Larkin wrote:> On Wed, 10 May 2023 08:28:19
-0400, Phil Hobbs
If all you mean by \'conservation of energy\' is that increase in the
rotational energy equals the mechanical work done, sure. I think of
that as being inherent in the definition of mechanical energy--it
wouldn\'t be a useful concept if you couldn\'t do that, whether it was or
was not conserved in the larger sense.
The great 19th C statistical mechanics guys established the equivalence
of work with heat, enabling them to formulate the law of conservation of
energy in its usual form (the first law of thermodynamics).
Before that, what we now call mechanical energy was called \"vis viva\",
and the adding and subtracting was done as above. It had been done that
way for a century or more by that point.
Actually setting up the equations is helpful--you find the simple
approach pretty fast when you do that. Invoking conservation of angular
momentum lets you solve the problem trivially, in two lines of algebra.
If you try the full dynamical calculation, you need a full functional
description of the time-dependent mass distribution of the skater\'s arms
and her instantaneous angular velocity. Because the centrifugal and
Coriolis forces depend on radial position, you have to compute some very
complicated volume integral over her arms--otherwise you can\'t compute
the force she exerts. And of course you don\'t really have separate
objects to consider--where her arms begin and end (i.e. the volume of
integration) is arbitrary at some level.
So that\'s not a super-useful way to frame the calculation.
Using angular momentum, you don\'t have to care about any of that, just
the initial and final values of the skater\'s moment of inertia, however
she chose to make the change.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
-0400, Phil Hobbs
pcdhSpamMeSenseless@electrooptical.net> wrote:
On 2023-05-10 04:34, Martin Brown wrote:
On 10/05/2023 04:20, Phil Hobbs wrote:
On 2023-05-09 22:22, John Larkin wrote:
On Tue, 9 May 2023 18:39:59 -0400, Phil Hobbs
pcdhSpamMeSenseless@electrooptical.net> wrote:
On 2023-05-09 16:55, RichD wrote:
On May 9, Martin Brown wrote:
The skater expends chemical energy, in her muscles, to
pull her arms
Work done is torque x angular displacement. No torque= No
work, and no work means no change in energy. Rotational
kinetic energy does not change.
The skater did work to pull her arms and legs in towards the
spin axis. Where did that energy come from and where did it
go?
As the arms move closer to the axis, the centripetal force
decreases considerably, requiring less work on the skater. But
it makes no difference whatsoever because just bringing the
arms inward in no way shape or form applies a force tangential
to the radius connecting the axis, which is what you need to
torque up the rotational velocity. So why does the skater mass
speed up? The answer is Coriolis.
https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/12%3A_Non-inertial_Reference_Frames/12.08%3A_Coriolis_Force#:~:text=An%20interesting%20application%20of%20the,external%20forces%20which%20is%20correct.
That article is instructive, but his final claim is incorrect.
Retracting the arms injects kinetic energy, from the muscles. The
author\'s claim is that this eventually adds to the total
rotational energy, as the arms merge with the torso.
That is *exactly* what happens. The skater does real work by
moving
her arms inwards in a rotating frame of reference and it creates a
torque that spins her up faster.
How does it create a torque, when it\'s RADIAL, i.e.
perpendicular to
the tangential velocity?
It doesn\'t need to create a torque. Torque is the time
derivative of
angular momentum, just as force is the time derivative of linear
momentum.
What is observed is a change in the skater\'s angular _velocity_, not
her
angular momentum. She achieves this with zero applied torque, by
changing her moment of inertia. Standard stuff from freshman
physics.
Kinetic energy on its own is *NOT* conserved. *TOTAL* energy *IS*.
um, yes. But one has to identify ALL the pertinent forms of
energy,
and quantify them.
Conservation of energy is a red herring here.
But it\'s a great simplifiction here. It short-cuts a lot of math.
Conservation laws are super useful like that, I agree.
Conservation of energy is one of the most useful ones, but isn\'t
easily applicable to the skater problem.
There\'s no potential energy in the problem as stated, so it isn\'t at
all like a planetary orbit, or a falling mass, or a vibrating string,
or lossless billiard balls on a perfectly-elastic table, where we can
see potential and kinetic energy changing oppositely, so that their
sum is constant.
Yes there is Phil. Her arms outstretched have stored potential
energy by
virtue of their mass and being in a rotating frame.
When she moves them inwards she feels a net torque as she does work
against the centrefugal/petal force due to her rotation and feels a
Coriolis force. It is the effect of that Coriolis force that spins her
up. You should get *exactly* the same result by appealing to
*either* of
conservation of energy or conservation of angular momentum.
I invite you to formulate the problem that way, in the fully co-rotating
frame (in which the skater remains stationary throughout the motion).
The constant-Omega Newtonian case that one normally encounters in
ballistics and meteorology isn\'t that difficult.
Kinematically, you just add a term to all the time derivatives of
vectors: for instance, a velocity dX\'/dt in the rotating frame is
dX\'/dt = dX/dt + Omega cross X,
where X = (x, y, z)^T is in the lab frame, and we\'re performing the
usual vector cross product. This leads to the so-called \'fictitious
forces\': centrifugal, which applies to everything, and Coriolis, which
applies only to moving objects.
Angular acceleration of the frame brings in stuff that makes
centrifugal and Coriolis force look trivial--as one small example, you
need a definition of energy that leaves the KE of the arena and the
spectators constant, even though its undergoing rapid angular
acceleration in the fully co-rotating frame.
Anytime you\'re doing something complicated like that, you constantly
check your work by doing the lab-frame calculation in parallel to
double-check, so it\'s a bit of a cheat to say that you\'re analyzing it
in the weird reference frame.
One way is *MUCH* easier than the other though!
But in some problems like this one it is hard to see the wood for the
trees and some people appear to be unable to see at all
Newtonian dynamics requires corrections by fictitious forces before you
can sensibly apply it in a rotating frame of reference. One of my
physics supervisors worked on rotating radars in WWII and had
stories of
people knocking themselves out reaching for spanner too quickly and
forgetting that they were in a rotating frame of reference. Apparently
after a while of being rotated you largely desensitise to it.
(until you stop the rotation then the world spins in reverse)
Those old-timey playground carousels were a good lab course too. A
couple of kids running at top speed to spin it up, then pull themselves
on board. They had tangential rails along the rim, and radials from rim
to pivot. Pulling yourself along one of the radials towards the axis
involves a lot of centrifugal force, but the really surprising thing is
the strength of the Coriolis pulling your feet sideways as you move
them.
A classic governer of two weights on a hinged light inextensible
arms is
the simplest analogue of the dancer that can be more easily analysed to
understand the split between stored potential energy and kinetic energy
as the device is spun up.
https://en.wikipedia.org/wiki/Centrifugal_governor
The governor + engine system is easily analyzed as a second-order ODE
with a mild nonlinearity, rather like a driven pendulum. Unlike the
skater problem, the rotating balls of the governor are a tiny
perturbation on a huge rotating assembly with otherwise constant moment
of inertia. That lets you ignore a whole lot of stuff.
And of course this digression is a classic SED trope, i.e. pulling out
bigger and bigger magnifiers as the hairs get split finer and finer.
v
Looking at the skater case from the POE perspective is an enormous
simplification.
At any point in the pull-in process, an incremental
force-times-distance adds energy to the system which can only go into
more spin. At that same point in later extending her arms, the force
is the same but the distance is reversed, so that bit of energy is
exactly negated.
So pulling in her arms adds to the spin energy and later extending
them buys it all back. POE requires that.
If all you mean by \'conservation of energy\' is that increase in the
rotational energy equals the mechanical work done, sure. I think of
that as being inherent in the definition of mechanical energy--it
wouldn\'t be a useful concept if you couldn\'t do that, whether it was or
was not conserved in the larger sense.
The great 19th C statistical mechanics guys established the equivalence
of work with heat, enabling them to formulate the law of conservation of
energy in its usual form (the first law of thermodynamics).
Before that, what we now call mechanical energy was called \"vis viva\",
and the adding and subtracting was done as above. It had been done that
way for a century or more by that point.
Actually setting up the equations is helpful--you find the simple
approach pretty fast when you do that. Invoking conservation of angular
momentum lets you solve the problem trivially, in two lines of algebra.
If you try the full dynamical calculation, you need a full functional
description of the time-dependent mass distribution of the skater\'s arms
and her instantaneous angular velocity. Because the centrifugal and
Coriolis forces depend on radial position, you have to compute some very
complicated volume integral over her arms--otherwise you can\'t compute
the force she exerts. And of course you don\'t really have separate
objects to consider--where her arms begin and end (i.e. the volume of
integration) is arbitrary at some level.
So that\'s not a super-useful way to frame the calculation.
Using angular momentum, you don\'t have to care about any of that, just
the initial and final values of the skater\'s moment of inertia, however
she chose to make the change.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
F
Fred Bloggs
Guest
On Wednesday, May 10, 2023 at 11:13:38â¯AM UTC-4, Phil Hobbs wrote:
That doesn\'t answer the original question as to why the skater\'s rotation speeds up, and why the kinetic energy of rotation increases.
The answer is Coriolis force, Fc, something that\'s been known for nearly 400 years.
Fc= -2m w* X v\'\'*
m is particle mass, or differential to be part of a sum
w* is angular velocity of rotating frame a ***vector****
v\'\'* is velocity of particle relative to the rotating frame, a ***vector***
r\'\'* radial vector to m in rotating frame ( r, theta, z coordinates)
X is vector cross-product
v\'\'* has a component due to dr\'\'*/dt, velocity radially inward, or outward, as the arms extend etc. Here r\'\'* is radial vector to mass m in the rotating reference frame (cylindrical r, theta, z axes). Use the right hand rule w* aligned with axis of rotation, r\'\'* oriented as usual, and you get exactly a force applied tangentially to the radius of travel of m. The force materializes by way of acceleration of m in the rotating reference frame. If dr\'\'*/dt is negative, arms pulling in, the tangential force is parallel with tangential v\'\'*, making mass speed up. If dr\'\'*/dt is positive, arms extending, the tangential force is in opposite direction to tangential v\'\'*, making the mass slow down.
Energy of rotation is (angular momentum)^2/(2I) , so it obviously increases, greatly in the case of the spinning skater, with no apparent application of external torque.
I don\'t think it has much effect in the renegade tire problem. But then that\'s just my opinion.
On 2023-05-10 09:27, John Larkin wrote:> On Wed, 10 May 2023 08:28:19
-0400, Phil Hobbs
pcdhSpamM...@electrooptical.net> wrote:
On 2023-05-10 04:34, Martin Brown wrote:
On 10/05/2023 04:20, Phil Hobbs wrote:
On 2023-05-09 22:22, John Larkin wrote:
On Tue, 9 May 2023 18:39:59 -0400, Phil Hobbs
pcdhSpamM...@electrooptical.net> wrote:
On 2023-05-09 16:55, RichD wrote:
On May 9, Martin Brown wrote:
The skater expends chemical energy, in her muscles, to
pull her arms
Work done is torque x angular displacement. No torque= No
work, and no work means no change in energy. Rotational
kinetic energy does not change.
The skater did work to pull her arms and legs in towards the
spin axis. Where did that energy come from and where did it
go?
As the arms move closer to the axis, the centripetal force
decreases considerably, requiring less work on the skater. But
it makes no difference whatsoever because just bringing the
arms inward in no way shape or form applies a force tangential
to the radius connecting the axis, which is what you need to
torque up the rotational velocity. So why does the skater mass
speed up? The answer is Coriolis.
https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/12%3A_Non-inertial_Reference_Frames/12.08%3A_Coriolis_Force#:~:text=An%20interesting%20application%20of%20the,external%20forces%20which%20is%20correct.
That article is instructive, but his final claim is incorrect.
Retracting the arms injects kinetic energy, from the muscles. The
author\'s claim is that this eventually adds to the total
rotational energy, as the arms merge with the torso.
That is *exactly* what happens. The skater does real work by
moving
her arms inwards in a rotating frame of reference and it creates a
torque that spins her up faster.
How does it create a torque, when it\'s RADIAL, i.e.
perpendicular to
the tangential velocity?
It doesn\'t need to create a torque. Torque is the time
derivative of
angular momentum, just as force is the time derivative of linear
momentum.
What is observed is a change in the skater\'s angular _velocity_, not
her
angular momentum. She achieves this with zero applied torque, by
changing her moment of inertia. Standard stuff from freshman
physics.
Kinetic energy on its own is *NOT* conserved. *TOTAL* energy *IS*.
um, yes. But one has to identify ALL the pertinent forms of
energy,
and quantify them.
Conservation of energy is a red herring here.
But it\'s a great simplifiction here. It short-cuts a lot of math.
Conservation laws are super useful like that, I agree.
Conservation of energy is one of the most useful ones, but isn\'t
easily applicable to the skater problem.
There\'s no potential energy in the problem as stated, so it isn\'t at
all like a planetary orbit, or a falling mass, or a vibrating string,
or lossless billiard balls on a perfectly-elastic table, where we can
see potential and kinetic energy changing oppositely, so that their
sum is constant.
Yes there is Phil. Her arms outstretched have stored potential
energy by
virtue of their mass and being in a rotating frame.
When she moves them inwards she feels a net torque as she does work
against the centrefugal/petal force due to her rotation and feels a
Coriolis force. It is the effect of that Coriolis force that spins her
up. You should get *exactly* the same result by appealing to
*either* of
conservation of energy or conservation of angular momentum.
I invite you to formulate the problem that way, in the fully co-rotating
frame (in which the skater remains stationary throughout the motion).
The constant-Omega Newtonian case that one normally encounters in
ballistics and meteorology isn\'t that difficult.
Kinematically, you just add a term to all the time derivatives of
vectors: for instance, a velocity dX\'/dt in the rotating frame is
dX\'/dt = dX/dt + Omega cross X,
where X = (x, y, z)^T is in the lab frame, and we\'re performing the
usual vector cross product. This leads to the so-called \'fictitious
forces\': centrifugal, which applies to everything, and Coriolis, which
applies only to moving objects.
Angular acceleration of the frame brings in stuff that makes
centrifugal and Coriolis force look trivial--as one small example, you
need a definition of energy that leaves the KE of the arena and the
spectators constant, even though its undergoing rapid angular
acceleration in the fully co-rotating frame.
Anytime you\'re doing something complicated like that, you constantly
check your work by doing the lab-frame calculation in parallel to
double-check, so it\'s a bit of a cheat to say that you\'re analyzing it
in the weird reference frame.
One way is *MUCH* easier than the other though!
But in some problems like this one it is hard to see the wood for the
trees and some people appear to be unable to see at all
Newtonian dynamics requires corrections by fictitious forces before you
can sensibly apply it in a rotating frame of reference. One of my
physics supervisors worked on rotating radars in WWII and had
stories of
people knocking themselves out reaching for spanner too quickly and
forgetting that they were in a rotating frame of reference. Apparently
after a while of being rotated you largely desensitise to it.
(until you stop the rotation then the world spins in reverse)
Those old-timey playground carousels were a good lab course too. A
couple of kids running at top speed to spin it up, then pull themselves
on board. They had tangential rails along the rim, and radials from rim
to pivot. Pulling yourself along one of the radials towards the axis
involves a lot of centrifugal force, but the really surprising thing is
the strength of the Coriolis pulling your feet sideways as you move
them.
A classic governer of two weights on a hinged light inextensible
arms is
the simplest analogue of the dancer that can be more easily analysed to
understand the split between stored potential energy and kinetic energy
as the device is spun up.
https://en.wikipedia.org/wiki/Centrifugal_governor
The governor + engine system is easily analyzed as a second-order ODE
with a mild nonlinearity, rather like a driven pendulum. Unlike the
skater problem, the rotating balls of the governor are a tiny
perturbation on a huge rotating assembly with otherwise constant moment
of inertia. That lets you ignore a whole lot of stuff.
And of course this digression is a classic SED trope, i.e. pulling out
bigger and bigger magnifiers as the hairs get split finer and finer.
v
Looking at the skater case from the POE perspective is an enormous
simplification.
At any point in the pull-in process, an incremental
force-times-distance adds energy to the system which can only go into
more spin. At that same point in later extending her arms, the force
is the same but the distance is reversed, so that bit of energy is
exactly negated.
So pulling in her arms adds to the spin energy and later extending
them buys it all back. POE requires that.
If all you mean by \'conservation of energy\' is that increase in the
rotational energy equals the mechanical work done, sure. I think of
that as being inherent in the definition of mechanical energy--it
wouldn\'t be a useful concept if you couldn\'t do that, whether it was or
was not conserved in the larger sense.
The great 19th C statistical mechanics guys established the equivalence
of work with heat, enabling them to formulate the law of conservation of
energy in its usual form (the first law of thermodynamics).
Before that, what we now call mechanical energy was called \"vis viva\",
and the adding and subtracting was done as above. It had been done that
way for a century or more by that point.
Actually setting up the equations is helpful--you find the simple
approach pretty fast when you do that. Invoking conservation of angular
momentum lets you solve the problem trivially, in two lines of algebra.
If you try the full dynamical calculation, you need a full functional
description of the time-dependent mass distribution of the skater\'s arms
and her instantaneous angular velocity. Because the centrifugal and
Coriolis forces depend on radial position, you have to compute some very
complicated volume integral over her arms--otherwise you can\'t compute
the force she exerts. And of course you don\'t really have separate
objects to consider--where her arms begin and end (i.e. the volume of
integration) is arbitrary at some level.
So that\'s not a super-useful way to frame the calculation.
Using angular momentum, you don\'t have to care about any of that, just
the initial and final values of the skater\'s moment of inertia, however
she chose to make the change.
That doesn\'t answer the original question as to why the skater\'s rotation speeds up, and why the kinetic energy of rotation increases.
The answer is Coriolis force, Fc, something that\'s been known for nearly 400 years.
Fc= -2m w* X v\'\'*
m is particle mass, or differential to be part of a sum
w* is angular velocity of rotating frame a ***vector****
v\'\'* is velocity of particle relative to the rotating frame, a ***vector***
r\'\'* radial vector to m in rotating frame ( r, theta, z coordinates)
X is vector cross-product
v\'\'* has a component due to dr\'\'*/dt, velocity radially inward, or outward, as the arms extend etc. Here r\'\'* is radial vector to mass m in the rotating reference frame (cylindrical r, theta, z axes). Use the right hand rule w* aligned with axis of rotation, r\'\'* oriented as usual, and you get exactly a force applied tangentially to the radius of travel of m. The force materializes by way of acceleration of m in the rotating reference frame. If dr\'\'*/dt is negative, arms pulling in, the tangential force is parallel with tangential v\'\'*, making mass speed up. If dr\'\'*/dt is positive, arms extending, the tangential force is in opposite direction to tangential v\'\'*, making the mass slow down.
Energy of rotation is (angular momentum)^2/(2I) , so it obviously increases, greatly in the case of the spinning skater, with no apparent application of external torque.
I don\'t think it has much effect in the renegade tire problem. But then that\'s just my opinion.
C
Clive Arthur
Guest
On 10/05/2023 21:25, Fred Bloggs wrote:
<snipped>
That\'s all true I suspect, but IMO doesn\'t really demonstrate where the
Coriolis force comes from.
If she pulls a mass in slowly compared to the rotational speed, it will
of course follow a spiral path from the PoV of a stationary observer,
but that\'s not too helpful either for visualisation of Coriolis force.
However, if she pulls a mass in quickly, in say half a turn, it will
describe something like a semicircle. It\'s then easy to see that the
\'centrifugal\' force due to this semicircular mass motion provides the
tangential Coriolis force which speeds up the rotation.
And then it\'s easy to see that a spiral path does indeed do the same,
only producing less force for a longer time.
--
Cheers
Clive
<snipped>
That\'s all true I suspect, but IMO doesn\'t really demonstrate where the
Coriolis force comes from.
If she pulls a mass in slowly compared to the rotational speed, it will
of course follow a spiral path from the PoV of a stationary observer,
but that\'s not too helpful either for visualisation of Coriolis force.
However, if she pulls a mass in quickly, in say half a turn, it will
describe something like a semicircle. It\'s then easy to see that the
\'centrifugal\' force due to this semicircular mass motion provides the
tangential Coriolis force which speeds up the rotation.
And then it\'s easy to see that a spiral path does indeed do the same,
only producing less force for a longer time.
--
Cheers
Clive
F
Fred Bloggs
Guest
On Wednesday, May 10, 2023 at 5:50:26â¯PM UTC-4, Clive Arthur wrote:
Right, it doesn\'t say anything about how the force arises. It was first observed by 17th century Italian experimenters who were working precision measurements of objects dropped from towers. The object always hit east of a straight drop landing. This was eventually connected to the Coriolis due rotation of the Earth.
In the end it makes no difference how the final angular moment of inertia was attained, fast or slow. The net gain in kinetic energy is the same.
Right, it doesn\'t say anything about how the force arises. It was first observed by 17th century Italian experimenters who were working precision measurements of objects dropped from towers. The object always hit east of a straight drop landing. This was eventually connected to the Coriolis due rotation of the Earth.
In the end it makes no difference how the final angular moment of inertia was attained, fast or slow. The net gain in kinetic energy is the same.
F
Fred Bloggs
Guest
On Wednesday, May 10, 2023 at 5:50:26â¯PM UTC-4, Clive Arthur wrote:
I should add the Coriolis force is not a real force. It\'s called a fictitious force because the Coriolis effect is that which would arise from the fictitious force acting on the system. Mass in rotational motion is constantly accelerating due to the constant change in direction. So I suspect the Coriolis effect can be derived from first principles of that dynamic without resort to a force, but with great complexity. The model using the force vastly simplifies things.
I should add the Coriolis force is not a real force. It\'s called a fictitious force because the Coriolis effect is that which would arise from the fictitious force acting on the system. Mass in rotational motion is constantly accelerating due to the constant change in direction. So I suspect the Coriolis effect can be derived from first principles of that dynamic without resort to a force, but with great complexity. The model using the force vastly simplifies things.