(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I work a simple problem in two different ways and get two different answers. I'd like to know which way is wrong, and why.

This is a simple mass-spring problem. The spring hangs from a ceiling, and a mass is attached to its end.

We know the spring constant, k, and the mass, m, The problem is to find x, how much the spring has stretched from its natural length by the mass.

3. The attempt at a solution

First way - Forces in equilibrium

The mass is acted upon by two forces, its weight downward, and the spring tension upward. Since the mass is not moving, the forces must cancel.

kx = mg. or x = mg/k (answer one)

Second way - energy bookkeeping

After the mass is attached to the end of the spring, it falls the distance x. It loses potential energy mgx. This energy must be stored in the spring. Since F = -kx = -dV/dx where V is the springs potential energy, then V = kx**2/2 . This must be equal to the potential energy lost by the mass.

kx**2/2 = mgx or x = 2mg/k (answer two)

Where are my analyses going astray?

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# Simple mass-spring problem

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