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Having a vertical spring, if we hang a mass to it, what is the maximum stretch length of spring? and how big should be mass to break the spring?

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    $\begingroup$ Well, that depends on the spring (its material, thickness, etc.) and will probably not be uniquely answerable as the spring will cease to behave like a spring under Hooke's law before breaking. $\endgroup$
    – ACuriousMind
    Sep 16, 2014 at 17:30

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The force applied to the spring is:

$$F = -mg$$

where m is the mass, g is the acceleration due to gravity near the earth's surface (9.8 m/s^2)

The equation relating distance and force for a spring is:

$$F = -kx$$

where k is the spring constant and x is the distance the spring is stretched from equilibrium.

When the mass is attached to the spring and settled down, these forces are equal, so you can set them equal to each other:

$$ -mg = -kx$$

and solve for x to see how far the spring is stretched:

$$ \frac{mg}{k} = x$$

(note this assumes the spring itself has no mass)

Regarding breaking the spring, we need to get into some more advanced material properties. What is the spring made of? In the simplest case, what will happen first is you could imagine the spring being pulled until it is "straight" and then figuring out what is the force to break a straight piece of the material that the spring is made of.

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