I have encountered a problem with a mass $m$ hanging from a spring. I want to find the maximum distance it will be stretched from the spring's equilibrium. At the maximum distance the spring is stretched, the mass will be at rest so the weight will cancel out the spring force, so we should have,
$$ky = mg$$
or
$$y = \frac{mg}{k}$$
Where $k$ is the spring constant, $y$ is the maximum distance, $m$ is the mass and $g$ is gravitational acceleration. But, if you look at this problem through the conservation of energy, you would conclude energy is conserved, since we have only a spring force and gravity acting. If we define our origin to be the spring's equilibrium, then the initial energy of the system is zero. The energy at all other times must be zero, so at maximum distance, we have,
$$\frac{ky^2}{2} + mg(-y) = 0$$
Which yields a different answer of
$$y = \frac{2mg}{k}$$
What's going on? Where am I going wrong in approaching this problem?
Answer
What's going on? Where am I going wrong in approaching this problem?
Carefully consider your 3rd statement:
At the maximum distance the spring is stretched, the mass will be at rest so the weight will cancel out the spring force
At maximum distance, the mass will certainly be at rest but is it necessarily true that the weight and spring force cancel?
Put another way, does that fact that an object is (instantaneously) at rest imply that the acceleration of the object is zero?
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