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Spring Paradox (apparently). Energy problem with one vertical spring. 
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#1
Dec812, 08:45 PM

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Apologies if this is too basic, but I've been studying for a while and I'm stuck.
1. The problem statement, all variables and given/known data Spring paradox. What is wrong with the following argument? Consider a mass m held at rest at y = 0, the end of an unstretched spring hanging vertically. The mass is now attached to the spring, which will be stretched because of the gravitational force mg on the mass. When the mass has lost gravitational potential energy mgy and the spring has gained the same amount of potential energy so that mgy= 1/2 cy^{2} the mass will come to equilibrium. Therefore the position of equilibrium is given by y= (2mg)/C 2. Relevant equations Conservation of total mechanical energy [itex]K_1 + U_1 = K_2 + U_2[/itex] Potential Energy (gravitational) U=mgy Potential Energy (elastic) 1/2 cy^{2} Kinetic Energy 1/2 mv^{2} 3. The attempt at a solution At first glance, I can't seem to figure out what is wrong with the argument. So I began recreating the whole thing. I started drawing it this way: A is the intial situation where the spring is at rest, not supporting the mass. It's just there. B is the situation where the mass has been attached to the spring which supports the mass' weight. The blue line depicts y=0. Since no nonconservative forces seem to be involved here, I applied the conservation of total mechanical energy, this makes: E_{A}=E_{B} K_{A}+U_{A}=K_{B}+U_{B} Since the A situation is at the assigned zero, both elastic potential and gravitational potential will be 0. It's at rest so kinetic is also 0. In short, E_{A}=0 0=1/2 cy^{2}  mgy + 1/2 mv^{2} The spring would go up and down and eventually reach equilibrium, where the kinetic energy is zero. 0=1/2 cy^{2}  mgy So far nothing wrong has been found about the problem given. Because this leads to: mgy=1/2 cy^{2} And then, solving for y, it becomes y= (2mg)/C Again, this matches the results given. So I can't find what's wrong. Is it a trick question and nothing is wrong? Am I missing something? Thanks! 


#2
Dec812, 08:57 PM

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#3
Dec812, 09:08 PM

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*slaps forehead* I should have seen that. O_O
The paradox is clear now. What's not clear is how did this happen. There are no other forces acting on it. That 2 shouldn't be there. I'm beyond exhausted, going to have to sleep on it. Thank you very much! (It seems weird to say 'thank you very much, Dick', but yeah. Thank you.) 


#4
Dec812, 09:10 PM

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Spring Paradox (apparently). Energy problem with one vertical spring.



#5
Dec812, 09:23 PM

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Oh wait. If... there are no other forces, wouldn't it oscillate instead of just sitting there? When you release the mass, realistically it would go beyond the equilibrium point then back up, and keep oscillating. I assumed that 'eventually', it would stop moving. But that's only because in reality, oscillators are damped. By friction, usually. So the paradox happens because this system is not really free from nonconservative forces?



#6
Dec812, 09:30 PM

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