
Which weight will lift first as the rope is pulled? - brlewis
http://friendfeed.com/rahsheen/b7e00bb3/which-weight-will-lift-first-as-rope-is-pulled
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mkn
I loved these pulley problems in statics.

Assume the weights are resting on a surface, the pulleys and rope are
massless, the pulleys are frictionless, and the system is maintained in quasi-
equilibrium as the rope is pulled (steady state & small accelerations). In
this case, the tension T in the rope is constant everywhere. Now, take a
horizontal section through the ropes. ("Cut" them and replace the missing
portions of the rope with the tension.) Each weight is experiencing an upward
force of 2T.

When 2T >= 20, or T = 10, weight A begins to rise. Once a hits a stop, T must
be increased to just above 20 to get Weight B to rise. Similarly, T just above
30 causes C to rise after B stops.

The "trick" with these pulley problems is to section the problem through the
cables and show the tension, T. Then you've just got free body problems, in
this case subject to the floor constraint.

Oh, also, while the first weight is being lifted, the floor beneath weight B
experiences 40 - 20 = 20 units of force, and the floor under C experiences 60
- 20 = 40 units of force. Once B is lifted, the floor under C experiences 60 -
40 = 20 units of force. (Presuming that the labels are weights, and not
masses.)

~~~
lazyant
"When ... T = 10, weight A begins to rise".

No; the Tension the guy is supporting at rest is bigger than 10 (there are 2
other weights) so at T = 10 he would be moving backwards. He needs to apply a
bit more than the tension at rest.

(edit: this is for when all 3 weights are in the air, I see now some people
see them in a floor that is not drawn)

~~~
brlewis
The bottom of each weight is at about the same level as the bottom of the
man's feet. A floor is the natural thing to assume. Otherwise it would be
difficult to get them into this configuration.

~~~
leif
A good way to think about this is to replace the man with a fourth weight,
fixed to the rope (as opposed to a pulley). Now it should be clear that the
weights would not rest in that configuration unless they were supported by a
floor.

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RiderOfGiraffes
If the system is frictionless, the rope is weightless, and the weights are not
supported, then the lighter weight will rise and the heavier weight(s) will
fall. Therefore if the rope is pulled very slowly, the lighter weight will
rise first.

If the rope really is weightless and the pulleys really are fictionless (and
inertialess) then it doesn't matter how hard or fast you pull, the lighter
weight will rise first. This is at odds with your intuition simply because you
have no (or insufficient) experience with weightless and frictionless
environments. This is one reason why space is so bloody dangerous, in addition
to the dangers posed by, say, diving, where similarly to space, your equipment
has to work perfectly or you die.

In the real world, pulling fast enough will make the closer weight rise first.

The truth lies somewhere in between.

~~~
mcn
>If the rope really is weightless and the pulleys really are fictionless (and
inertialess) then it doesn't matter how hard or fast you pull, the lighter
weight will rise first.

I'm considering a thought experiments that make me believe that this is not
the whole story.

Imagine a two weight system set up similar to the original diagram in which
the weights are the same weight, and the gravity is very little. Yanking on
the rope I imagine them rising at the same speed. Now, we take a very small
flake off of one weight and repeat the experiment. It seems clear that both
weights will still rise from the start, just the lighter weight will rise at a
faster speed.

I think this should extend to three weights of any positive mass -- if you
yank the rope fast enough (and it might be very fast) all three should rise
from the start.

~~~
brlewis
In your thought experiment you make the weight difference so small that
otherwise insignificant factors (e.g. friction) take over. Go the opposite
direction and make the weight difference enormous.

~~~
mcn
Assuming no friction and given a fast tug of the rope I still don't think the
near equal weight would hold still or sink while the other near equal weight
flew up quickly.

With the large weight difference, I think the required speed of pulling the
rope to make them both rise just becomes impractically fast.

Imagine the system in space - all the weights would move up, so they all have
an upward force applied to them from the rope tugging. It's just a matter of
pulling so fast that that upward force overcomes gravity.

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cynicalkane
Well, the diagram is impossible unless the weights are all resting on the
floor, so let's assume they are. This system will find a steady state only at
a local minimum of potential energy, so the lightest weight will be lifted
before the heavier two get off the ground.

~~~
psycocoffey
Assumptions:

    
    
      pulleys: idealized, massless, frictionless
    
      rope: idealized, massless, doesn't stretch
    
      weights: resting on ground.
    
    

As tension is applied to the rope, Weight A will be lifted first, until it is
lifted to the ceiling. Then Weight B, and finally Weight A.

It helps to visualize Weight A as being massless. In that case, there would
just be extra slack in the rope, and B&C would not move until the slack was
taken in.

~~~
dkokelley
* Then Weight B, and finally Weight A.*

I believe you meant C, correct?

~~~
psycocoffey
Yes, I meant C. Too late to go and fix it now. Good catch

------
hop
This tricks people because it doesn't start out in static equilibrium. For it
to look like this, the weights would have to be resting on a table to relieve
the tension in the rope.

If the man does nothing, the heaviest weight will fall and the lightest will
rise. If it's frictionless and he starts pulling, the same thing will happen,
only the lengths will lessen.

~~~
EventHorizon
Could it not be in a meta stable equilibrium like this if the weights were of
equal mass? The numbers of the weights are unit-less. We assume they represent
the mass, but for all I know the number is the serial number of the weight.

~~~
hop
For all we know, it could be in space, they could be heavily magnetized,
100m/s wind, .5 CoF in the pulleys, the rope could stretch, and the system
could be traveling near the speed of light.

------
tankenmate
I'm with Rider, if you view it as a "classic" physics class problem, ie
weightless ropes and pulleys (and hence no angular inertia for the pulleys,
and no inertia due to the ropes), and no friction, weight A would rise even
without pulling the rope, weight B would remain static, and weight C would
drop at the same rate that weight A would go up. If you pulled on the rope you
could never make weight C go up faster than weight A until it hit the top and
stopped moving. So the pulling bit is a red herring, as weight A would always
hit the top before the other two, either pulling or holding it still. Assuming
the weights are in kilograms you would have to let the rope go at a rate of
3.13 m/s (11.27 kph, 7 mph) to prevent weight A from rising.

~~~
tankenmate
If you go with the weights on the floor model, then barrkel's comment a few
below is correct.

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JoeAltmaier
Its not a static system; the question is unfair.

~~~
cynicalkane
Why does that make it unfair?

~~~
lolipop1
Is the person currently pulling, are the weights on the ground, are the
weigths currently moving to get into a stable position? Friction, weight of
cord, acceleration, etc. etc. Any combination of the previous?

Depending on your level with maths/physics you'll probably give different
answers and make different assumptions.

~~~
cynicalkane
In physics problems things are assumed idealized unless they aren't. Also,
velocity doesn't matter here, and no math is required to solve the problem.

~~~
lolipop1
I didn't know physics problems are assumed to be idealized... When you build a
bridge for example, will the physics try to be simplified to the max? (Honest
question here).

Velocity would matter -- it seems to me -- if there is friction and probably a
few other factors included like elasticity. No?

Personally, I always considered physicists to be applied mathematicians (not
the other way around although I've seen physics problems thrown in university
level math classes). That's why I put that there, so assuming a high level of
math skills, you'd probably change your way of thinking quite a bit.

~~~
cynicalkane
When you build a bridge, that's engineering, not physics. It's a physics
convention that unspecified factors are assumed to be unimportant. If they
were important, they would be specified.

Anyway, it's implausible that the elasticity of the rope or the friction of
the pulleys are going to matter. Unless you have really rusty pulleys, or
something.

------
amalcon
This reminds me of the airplane on the conveyor belt, in that the only
confusion comes from the question being insufficiently specified.

    
    
      - Friction of the pulleys
      - Mass of the pulleys
      - Moment of inertia of the pulleys
      - Mass of the rope
      - Unit of mass of the weights
      - Is there a surface that the weights are resting on?
      - What's the local gravity like?
      - Others
    

If we assume the things we're likely supposed to (rope mass, pulley friction,
pulley mass and moment of inertia all insignificant, gravity tending down,
resting on a surface), it's clear that the lightest weight will rise first.
If, on the other hand, we make ridiculous assumptions (weights mass in AMU, in
a no-gravity environment, high moment of inertia pulleys), then the "heavy"
weight will lift first (because it's easier to lift the weight than to spin
the pulleys).

~~~
brlewis
The airplane on the conveyor belt is more interesting in this respect, in that
different people have different ideas about how the question should be
interpreted.

For this question, I think everyone will agree on the likely expected
assumptions. Incidentally, the mass of the rope doesn't affect the answer so
long as it's uniform, and the mass of the pulleys doesn't matter so long as
the pulleys directly attached to the weights all have the same mass. And once
you assume the pulleys are frictionless, their moment of inertia doesn't
affect the answer either.

~~~
amalcon
True. The airplane problem had several things that were ambiguous; this
problem only has things that are unspecified.

------
farmerbuzz
I would love to see a video demonstration of the answer (my intuition was
apparently wrong which would make a video that much more entertaining).

------
ck2
None. Instead he'll pull the ceiling down as they failed to screw it into the
studs. ;-)

~~~
someone_here
There's an elephant in the way?

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lazyant
The length of the rope is constant. For every weigh with a pulley the weigh
moves 1/2 length unit for every length unit it's being pulled. Weights don't
matter; C will go up first, then B then A (at a relative 1/2 length ratio).

~~~
lazyant
funny the right answer being downvoted without reason

~~~
loup-vaillant
You go against the consensus without a solid explanation. For instance, why
don't the weights matter? Would they matter if they were respectively 1mg,
1kg, and 1megaton?

~~~
lazyant
Fair enough. I gave the summarized version. I'll try to explain better.

I'll only take basic classic mechanics assumptions: that the rope is of
constant length (ie is like a cable that doesn't compress or expand). The
framework is quasi-static classical mechanics; the results of the guy pulling
slowly a little bit can solve the problem or be generalized (I won't consider
the situation of the guy jerking quickly the rope etc).

The guy's hand is under a calculable tension Tw that will be 60 units or
whatever, it doesn't matter. The problem asks what happens when the guy pulls,
so we suppose that he's not the one being pulled but he moves forward to the
right. The tension Tg that he applies doesn't matter; as long as it's bigger
than the one from the weights (Tg > Tw) he'll move the rope (we discard
friction since he moves slowly or if you take into account friction he just
needs more force, it doesn't matter). So the distribution of weights or their
actual measure don't matter so far. (of course if you have a million tons and
you blow the guy away weighs matter).

Now since the cable/rope has constant length (there are no slacks etc since
it's moving) when the guy pulls 1mm then than length needs to be taken from
somewhere in the pulley systems.

The effect of a pulley is to divide the length of rope you take in two (one
has to go to the left vertical part of rope and the other one to the right
one); this is why the tension in each side of a pulley is 1/2 of the total
tension and you can pull with 1/2T a weight of T with a pulley. So with this
we can straightforwardly calculate the tensions everywhere but we don't need
that.

So the that 1mm is taken from the system and the more pulleys the rope has to
go through the less is taken (because of this 1/2 I explained above), so the
weight closer to the guy (with less pulleys) will raise first, then the next
one in the middle, then the next one etc; weighs don't matter.

clarification update: poor conclusion wording: if weights are in the air all 3
weights go up at the same time but C will move more than B and B more than A.

~~~
cynicalkane
When a weight is at rest, the tension in the rope must be less then or equal
to half the weight. The problem asks which weight will be first to not be at
rest. The answer should now be obvious.

~~~
lazyant
When a weight is at rest the tension in the rope is _exactly_ half its weight.

Problem asks which weight would be first to _raise_ (not "not be at rest" that
could be going down) so that's why I'm supposing the guy can pull the whole
thing. I'm also saying the first one is C (closer to guy).

~~~
cynicalkane
The tension in the rope is greater than 30lbs, yet only one weight moves? I
give up.

~~~
lazyant
sorry; I made an edit with a clarification.

Basically if the weights are at rest on the floor then mkn's answer is the
correct one. If they are at rest on the air, then my answer is the correct
one.

~~~
loup-vaillant
No: <http://news.ycombinator.com/item?id=1659676>

(this comment assumes that the weights are unsupported).

~~~
lazyant
I supposed the system was at equilibrium without making the numbers. If it's
not (as it seems) then sure, my reasoning is invalid and he's right, no
problem.

