# Is this oscillator driven?

A mass $m$ is attached to a vertical massless spring or a spring constant $k$. Originally, the spring was relaxed because the mass was held by a clip. Suddenly the clip was released. THe mass dropped down and the maximal elongation of the spring was recorded as $l$. What is the resonant frequency of the system? The gravitational constant is $g = 9.8 \frac{m}{s}$.

I assume (based on instructions from the professor) the intention of the question is that there is some damping going on here (damping constant $b$). What I'm confused about is if the gravitational force here makes this a driven damped oscillator or not. The "driving" force is a constant, so it isn't changing the oscillations at all (I think?). Meaning, is my equation (for $\omega = \sqrt{\frac{k}{m}}$ and $\beta = \frac{b}{2m}$):

$$\ddot{x} + 2\beta\dot{x} +\omega^2x = 0$$

or

$$\ddot{x} + 2\beta\dot{x} +\omega^2x = g$$

If it is driven, is my resonant frequency $\omega_r = \sqrt{\omega_0^2 - 2\beta^2}$? What else would resonant frequency mean in this case?

• I don't think any damping is implied. This is a regular SHO and you just need to calculate the force constant $k$ so you can calculate the frequency. You get the force constant from how much the spring extends before the mass is brought to a stop. Apr 2, 2014 at 6:21
• @JohnRennie That's what it would seem, but the professor who assigned the problem assured me that he intended for there to be implied damping. I can clarify that in the question. Apr 2, 2014 at 6:30
• It's rather odd that your prof would "intend" for "implied" damping without specifying the source of the damping force. Does he typically assign such "open-ended" questions, expecting you to posit something like heat loss in spring deformation or (miniscule) air resistance? Apr 2, 2014 at 11:44

Your equation is Newton's second law. That's always true for any system in Newtonian mechanics. So it's very straightforward to figure out which equation of motion applies to this system: write out $\sum F = ma$, plug in the forces, and simplify.
• Mathematically, any simple harmonic oscillator (driven or not, damped or not) has an equilibrium position, and it's conventional to choose a coordinate $q$ such that the equilibrium position is at $q = 0$. With that in mind, consider the coordinate transformation $q = x - \frac{g}{\omega^2}$. I'll let you work out the implications of that. :-) (Fun fact: this is mathematically equivalent to the Higgs mechanism.)
• If it's not driven, then what is the resonant frequency? How can you have resonance on an underdamped spring? Does that mean the frequency for resonance if the system were to be driven? ($\sqrt{\omega^2 - 2\beta^2}$?) Apr 2, 2014 at 19:32