Apologies if this is too basic, but I've been studying for a while and I'm stuck.(adsbygoogle = window.adsbygoogle || []).push({});

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 non-conservative 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!

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

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