| Topic: |
Science > Physics |
| User: |
"mathlover" |
| Date: |
20 Aug 2006 10:04:12 PM |
| Object: |
a strange discrepancy in a mechanic problem |
Dear all,
The problem is:
Consider a rope (with length density u) putted on a horizontal smooth
table, then a man takes one end and travels with constant velocity v
(assuming the remaining rape is still stationary on the table), what is
strange is, if we calculated the power done by the man and the
increasing rate of kinetic energy of the rope, they are different !!!
(I give my answers for the two in the bottom)
I'm wandering, this problem really confused me so much. I inspected
the calculation of the above two answers for many times, I can't find
any step unreasonable, and if these answers were correct, then where is
the missing energy? In the interface of stationary and moving ropes?
Does that potential energy in the interface cause the strong force to
accelerate the rope element from speed 0 to speed v???
welcom any ideas to point out my errors, I'm really so urged to solve
this discrepansy, thanks a lot !!!
Sincerely
--------------------------
my answer is uv^3 for the power; and 1/2 u v^3 for the rate of kinetic
energy.
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| User: "mathlover" |
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| Title: Re: a strange discrepancy in a mechanic problem |
21 Aug 2006 03:41:37 AM |
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mathlover =E5=AF=AB=E9=81=93=EF=BC=9A
Dear all,
The problem is:
Consider a rope (with length density u) putted on a horizontal smooth
table, then a man takes one end and travels with constant velocity v
(assuming the remaining rape is still stationary on the table), what is
strange is, if we calculated the power done by the man and the
increasing rate of kinetic energy of the rope, they are different !!!
(I give my answers for the two in the bottom)
I'm wandering, this problem really confused me so much. I inspected
the calculation of the above two answers for many times, I can't find
any step unreasonable, and if these answers were correct, then where is
the missing energy? In the interface of stationary and moving ropes?
Does that potential energy in the interface cause the strong force to
accelerate the rope element from speed 0 to speed v???
welcom any ideas to point out my errors, I'm really so urged to solve
this discrepansy, thanks a lot !!!
Sincerely
--------------------------
my answer is uv^3 for the power; and 1/2 u v^3 for the rate of kinetic
energy.
---------------------------------------------------------------------------=
------------------
I found another problem. If we let the rope drop from a table, and let
x denotes the distance from the table to the end of the rope, find
x(t).
This question is from the standard text: Classical dynamics of
particles and systems by Marion Thornton. 5th edition. (problem 9-15)
The solution given by Marion is also surprised me, since it's wrong
obviously.
his equation of motion is: ugx =3D d/dt (uxx') this is obviously wrong,
since his constant of motion is : ux^2 x'^2 - (2/3)ugx^3 <=3D=3D=3D its
dimension is not energy!
i'm really confused by these kind of problem, I come up with a feeling
that to assume the stationary of part of rope is not feasible. It is a
impossible system under the assumtion.
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| User: "GrosBouddha" |
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| Title: Re: a strange discrepancy in a mechanic problem |
24 Aug 2006 04:28:00 AM |
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mathlover a =E9crit :
I found another problem. If we let the rope drop from a table, and let
x denotes the distance from the table to the end of the rope, find
x(t).
I suppose that x denotes the length of the vertical length of the rope.
Obvioulsy all the points of the rope move with the same velocity
v=3Ddx/dt (The x-axis is vertical and downward)
So the KE for a rope of lenght L and mass per unith length =B5
1/2*=B5Lv^2.
The potential energy PE is (placing the zero reference at the table)
-1/2*u*x*x*g
because the center of mass of the vertical rope of lenght x is x/2
under the table and the mass of this part of the rope is =B5x.
If there is no energy dissipation (by friction on the table and in the
rope) then the mechanical energy is ME=3DPE+KE=3Dconstant.
By derivatin versus the time then
d^2x/dt^2-g/Lx=3D0 is the equation of the movement.
I hope to be understandable (I'm not a native english speaker as you
surely guessed !)
See you
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| User: "mathlover" |
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| Title: Re: a strange discrepancy in a mechanic problem |
24 Aug 2006 09:42:04 AM |
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GrosBouddha =E5=AF=AB=E9=81=93=EF=BC=9A
mathlover a =C3=A9crit :
I found another problem. If we let the rope drop from a table, and let
x denotes the distance from the table to the end of the rope, find
x(t).
Sorry for that I forgot to clarify there is also an assumtion : the
rope is coiled (a rope round or something like that), so some part is
falling; some part is stationary. So there is more and more mass
joining the falling part of the rope.
I suppose that x denotes the length of the vertical length of the rope.
Obvioulsy all the points of the rope move with the same velocity
v=3Ddx/dt (The x-axis is vertical and downward)
So the KE for a rope of lenght L and mass per unith length =C2=B5
1/2*=C2=B5Lv^2.
The potential energy PE is (placing the zero reference at the table)
-1/2*u*x*x*g
because the center of mass of the vertical rope of lenght x is x/2
under the table and the mass of this part of the rope is =C2=B5x.
If there is no energy dissipation (by friction on the table and in the
rope) then the mechanical energy is ME=3DPE+KE=3Dconstant.
By derivatin versus the time then
d^2x/dt^2-g/Lx=3D0 is the equation of the movement.
I hope to be understandable (I'm not a native english speaker as you
surely guessed !)
The above derivation is quite clear and welldone. I think it is correct
for a falling rope which is straight in advance.
But if the situation is what I described above, I found something
strange.
Consider the whole rope as the system,
Let x denotes the length from the table to the end of the rope
Then the momentum of the whole rope is :
p =3D (u*x) x' (where x' =3D dx/dt)
Thus the external force exert on the rope is :
f =3D dp/dt =3D u x'^2 + u x x'' ---- (a)
If we demand the conservation of mechanical energy :
1/2 (ux) x'^2 =3D (ux) g (x/2) (with some reference
point chosen)
then equation (a) becomes :
f =3D (ux) g + u x x''
The above equation is unreasonable, since the external force should be
"uxg", but now a strange term is added!
It's really strange...
=20
See you
Thanks for your discussion, sincerely.
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| User: "GrosBouddha" |
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| Title: Re: a strange discrepancy in a mechanic problem |
23 Aug 2006 02:33:24 AM |
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mathlover a =E9crit :
Dear all,
The problem is:
Consider a rope (with length density u) putted on a horizontal smooth
table, then a man takes one end and travels with constant velocity v
(assuming the remaining rape is still stationary on the table),
Could you describe precisely how you achieve the rest of the rope to
stay with no movement, while a end move with constant velocity (and
direction ?). I could then try to calculate the power done by the man
and the rate of increase of kinetic energy.
See you
.
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| User: "mathlover" |
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| Title: Re: a strange discrepancy in a mechanic problem |
23 Aug 2006 10:19:53 AM |
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GrosBouddha =E5=AF=AB=E9=81=93=EF=BC=9A
mathlover a =C3=A9crit :
Dear all,
The problem is:
Consider a rope (with length density u) putted on a horizontal smooth
table, then a man takes one end and travels with constant velocity v
(assuming the remaining rape is still stationary on the table),
Could you describe precisely how you achieve the rest of the rope to
stay with no movement, while a end move with constant velocity (and
direction ?). I could then try to calculate the power done by the man
and the rate of increase of kinetic energy.
Good question. This is just an rough assumption. I guess this is a very
possible point where the strange result come up with. Since this
assumption demands the discontinuity in the tension of the rope; but
I'm not quite sure. However, it's impossible to achive this in reality.
By the way, one of the problem in the standard text book of Mechanics
(by Marion & Thornton) also makes this assumption. (PS. It's solution
given by the instructor's manual is naive and obviously wrong) And from
this I learned, the famous book may be wrong somewhere too. :P
Thanks for your ideas, Sincerely.
=20
See you
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| User: "Doc" |
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| Title: Re: a strange discrepancy in a mechanic problem |
23 Aug 2006 10:31:45 AM |
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What problem # is it? I'm not sure I understand your description of
the problem fully.
mathlover wrote:
GrosBouddha =E5=AF=AB=E9=81=93=EF=BC=9A
mathlover a =C3=A9crit :
Dear all,
The problem is:
Consider a rope (with length density u) putted on a horizontal smoo=
th
table, then a man takes one end and travels with constant velocity v
(assuming the remaining rape is still stationary on the table),
Could you describe precisely how you achieve the rest of the rope to
stay with no movement, while a end move with constant velocity (and
direction ?). I could then try to calculate the power done by the man
and the rate of increase of kinetic energy.
Good question. This is just an rough assumption. I guess this is a very
possible point where the strange result come up with. Since this
assumption demands the discontinuity in the tension of the rope; but
I'm not quite sure. However, it's impossible to achive this in reality.
By the way, one of the problem in the standard text book of Mechanics
(by Marion & Thornton) also makes this assumption. (PS. It's solution
given by the instructor's manual is naive and obviously wrong) And from
this I learned, the famous book may be wrong somewhere too. :P
=20
Thanks for your ideas, Sincerely.
=20
See you
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| User: "Randy Poe" |
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| Title: Re: a strange discrepancy in a mechanic problem |
23 Aug 2006 11:04:29 AM |
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mathlover wrote:
Dear all,
The problem is:
Consider a rope (with length density u) putted on a horizontal smooth
table, then a man takes one end and travels with constant velocity v
(assuming the remaining rape is still stationary on the table), what is
strange is, if we calculated the power done by the man and the
increasing rate of kinetic energy of the rope, they are different !!!
(I give my answers for the two in the bottom)
I think there is some information missing. I think that
the "stationary" end of the rope is coiled and lying on
the table. So what is happening over time is that more
and more rope is being added to the moving section.
If the rope is uncoiling at velocity v, then each unit time
a mass uv is accelerating to velocity v, being given
kinetic energy 0.5*uv^3.
my answer is uv^3 for the power; and 1/2 u v^3 for the rate of kinetic
energy.
OK, I agree with the second. Where did you get the first?
- Randy
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| User: "mathlover" |
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| Title: Re: a strange discrepancy in a mechanic problem |
23 Aug 2006 10:30:13 PM |
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Randy Poe =E5=AF=AB=E9=81=93=EF=BC=9A
mathlover wrote:
Dear all,
The problem is:
Consider a rope (with length density u) putted on a horizontal smooth
table, then a man takes one end and travels with constant velocity v
(assuming the remaining rape is still stationary on the table), what is
strange is, if we calculated the power done by the man and the
increasing rate of kinetic energy of the rope, they are different !!!
(I give my answers for the two in the bottom)
I think there is some information missing. I think that
the "stationary" end of the rope is coiled and lying on
the table. So what is happening over time is that more
and more rope is being added to the moving section.
yes. The stationary part of the rope is assumed to be coiled to be like
a roll or something. So there is more and more mass to participate in
the moving part.
If the rope is uncoiling at velocity v, then each unit time
a mass uv is accelerating to velocity v, being given
kinetic energy 0.5*uv^3.
my answer is uv^3 for the power; and 1/2 u v^3 for the rate of kinetic
energy.
OK, I agree with the second. Where did you get the first?
Assuming there is length x to be pulled out( in constant v),
considering the momentum of the whole rope, it's (ux)v
so the rate of momentum change of the system(whole rope),
i=2Ee. the external force, is just uv^2
so the power done by the external force is uv^2 * v =3D uv^3
- Randy
Thanks for your discussion, Sincerely.
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| User: "Mike" |
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| Title: Re: a strange discrepancy in a mechanic problem |
23 Aug 2006 07:45:07 AM |
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mathlover wrote:
Dear all,
The problem is:
Consider a rope (with length density u) putted on a horizontal smooth
table, then a man takes one end and travels with constant velocity v
(assuming the remaining rape is still stationary on the table), what is
strange is, if we calculated the power done by the man and the
increasing rate of kinetic energy of the rope, they are different !!!
(I give my answers for the two in the bottom)
The man must do Work to move the rope and thus his KE cannot be
constant. Your little thought experiment is ill-concieved.
I'm wandering, this problem really confused me so much. I inspected
the calculation of the above two answers for many times, I can't find
any step unreasonable, and if these answers were correct, then where is
the missing energy? In the interface of stationary and moving ropes?
Does that potential energy in the interface cause the strong force to
accelerate the rope element from speed 0 to speed v???
The only thing that is missing is a complete understanding of mechanics
by your part.
welcom any ideas to point out my errors, I'm really so urged to solve
this discrepansy, thanks a lot !!!
There is no discrepancy to be found in the whole subject of mechanics,
a science that has been confirmed experimentally to an accuracy of more
than a part in a trillion over and over again.
my answer is uv^3 for the power; and 1/2 u v^3 for the rate of kinetic
energy.
Not even close, considering the units are wrong anyway.
Mike
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| User: "mathlover" |
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| Title: Re: a strange discrepancy in a mechanic problem |
23 Aug 2006 10:11:27 AM |
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Mike =E5=AF=AB=E9=81=93=EF=BC=9A
mathlover wrote:
Dear all,
The problem is:
Consider a rope (with length density u) putted on a horizontal smooth
table, then a man takes one end and travels with constant velocity v
(assuming the remaining rape is still stationary on the table), what is
strange is, if we calculated the power done by the man and the
increasing rate of kinetic energy of the rope, they are different !!!
(I give my answers for the two in the bottom)
The man must do Work to move the rope and thus his KE cannot be
constant. Your little thought experiment is ill-concieved.
So the work done by the man must equal the energy obtained by the
rope, but now it seems not so. That's why I'm wondering whether I
missed some energy or I calculated wrong.
I'm wandering, this problem really confused me so much. I inspected
the calculation of the above two answers for many times, I can't find
any step unreasonable, and if these answers were correct, then where is
the missing energy? In the interface of stationary and moving ropes?
Does that potential energy in the interface cause the strong force to
accelerate the rope element from speed 0 to speed v???
The only thing that is missing is a complete understanding of mechanics
by your part.
I have a somewhat different reflection---this problem is subtle.
welcom any ideas to point out my errors, I'm really so urged to solve
this discrepansy, thanks a lot !!!
There is no discrepancy to be found in the whole subject of mechanics,
a science that has been confirmed experimentally to an accuracy of more
than a part in a trillion over and over again.
Of course, the classical mechanics is self-consistent to some extent.
So where this problem valuable is that it tells us how to use the
classical mechanics carefully.
my answer is uv^3 for the power; and 1/2 u v^3 for the rate of kinetic
energy.
Not even close, considering the units are wrong anyway.
I guess I may not give the definitions of these symbols clearly, since
they obviously have the correct units.
The dimension analysis is correct as follows :
u has the dimension of Mass/Length
v has the dimension of Length/Time
Thus, uv^3 has the dimension of "power"(rate of work done by the man)
So, it's ok.
Thanks for your attention.
=20
Mike
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