What is the angular frequency of oscillations in a SHM system with weight?

[applestrudle]

Homework Statement

A mass hanging on a spring is oscillating with SHM. The initial displacement due to gravity is 0.1m By considering the forces acting on the mass at equilibrium, calculate the angular frequency of the oscillations.

Homework Equations

$$\frac{d^2x}{dt^2} + {w}^{2}x = g$$

The Attempt at a Solution

I used the integrating factor method to get

$$x = \frac{m}{kg} (1-{e}^{-kt/m} -0.1{e}^{-kt/m}$$

I'm not really sure where to go from there

I always get stuck on questions when they have weight!

 

[applestrudle]

okay dw i got it now, 9.9 s-1 you just equate the forces at equilibrium with x = 0.1

 

[Simon Bridge]

$$\frac{d^2x}{dt^2} + {w}^{2}x = g$$

The Attempt at a Solution

I used the integrating factor method to get

$$ x = \frac{m}{kg} (1-{e}^{-kt/m} -0.1{e}^{-kt/m}) $$

I'm not really sure where to go from there

... I think you are getting ahead of yourself.
The weight adds an extra force into your free-body diagram.

If you have positive displacements downwards, measured from the unstretched length position,
then $-ky+mg=m\ddot y$ (you don't need to solve it!)
... divide through by m gets the same relation you have - see what $\omega^2$ is equal to?

You don't know k or m.
But you do know what the equilibrium displacement is
- what is special about the forces at equilibrium?