[Phys-L] Re: Spring question

[jsd at AV8N.COM (John Denker)]

Savinainen Antti wrote:
> A vertical spring (spring constant k) is hanging freely.
> A block of mass m is attached to the spring and the spring
> streches. At the equilibrium position of the block, where it
> hangs motionless, the spring has streched by x. Determine the
.       ^^^^^^^^^^
> mass of the block.
>
> Solution 1:
>
> At the equilibrium the net force on the block is zero
> and hence kx = mg which gives m = kx/g.
>
> Solution 2:
>
> The change in gravitational potential energy of the block
> is mgx. This energy must be in the mass-spring system and
> equals 1/2kx*x (assuming no friction in the spring).
> But this calculation gives m = 1/2* kx/g!!
>
> Which solution is correct? What's wrong with the other
> solution?

The force-balance solution is what you want.  The alleged
energy solution neglects the energy that leaves the system
when the block is brought to rest after falling the distance
x, accelerating the whole time.

The initial conditions with constant energy would result
in an oscillating mass, not a mass that "hangs motionless".

In physics, momentum is conserved (which implies force
balance) *and* energy is conserved.  Momentum can't be
hidden, but sometimes keeping track of the energy requires
a bit of thought.

There are a jillion problems with hidden energy, including
the infamous railroad cars that bang into each other and
become coupled.